# N-body problem

*is a*

**N-body problem****draft**programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

The -body problem is an inherent feature of a physical description of mechanical system of interacting objects. The system is described by Newton's Law

with continious combined force from other bodies

Exact formulation of first mechanical problem is, given initial coordinates and velocities , to find coordinates and velocities at with accuracy more then given relative value

As well known from physical background, only the choice of can be analytically solved.

- Task

Write a simulation of three masses interacting under gravitation and show their evolution over a number of time-steps (at least 20).

## C[edit]

### A little history[edit]

A long, long time ago ( well, more than 10 years, which is a very long time when it comes to computing ), I implemented a BGI program in C for the Mitchell-Green Gravity Set. A chaotic dynamical system, the Gravity set, MGGS in short, is a plot of the paths of massive bodies as they move under the influence of gravity. Those were the days when I was crazy about Graphics and the MGGS orbits brought to life all of the high school Physics I had learnt. Orbits, oscillations, parabolic and hyperbolic trajectories suddenly came to life on my computer screen. I didn't stop there and even implemented a 3D version in OpenGL. This was on a Pentium-4 system with 512 MB RAM and I remember the machine groaning when the OpenGL simulation ran for even 10 bodies.

And then I got a job, and in case you are wondering, yes, I can't find either program anywhere. Sad, because I am sure I implemented far more Physics in them than in these ones. The only bit extra I have built into the two implementations below is collision detection. To keep things simple, all bodies are assumed to be point particles and collisions to be perfectly elastic.

### And Now[edit]

One time step is assumed to be unit time, t = 1, hence there is no time variable in the implementations. The TCL example uses a time step of 10^-12, which frankly is too small for the C Standard Library and needs specialized techniques or libraries to be handled. Input data is read from a file and hence you can plot as many bodies as you want, keep in mind that this is computationally intensive, so intensive that on an i5, 8 GB RAM system, I didn't have to use any delays in the Graphics implementation for animation. Ideally such tasks are well suited for parallel processing. That's why Supercomputers are required to study weather, galaxies, black holes and molecules. All systems, big or small, can be reduced to N-body problems or definitely play a big role in their study.

### File Format[edit]

For both implementations, the file format is as follows :

<Gravitational Constant> <Number of bodies(N)> <Time step> <Mass of M1> <Position of M1 in x,y,z co-ordinates> <Initial velocity of M1 in x,,y,z components> ... <And so on for N bodies>

### The Implementations[edit]

#### Textual or Console[edit]

/*Abhishek Ghosh, 6th October 2017*/

#include<stdlib.h>

#include<stdio.h>

#include<math.h>

typedef struct{

double x,y,z;

}vector;

int bodies,timeSteps;

double *masses,GravConstant;

vector *positions,*velocities,*accelerations;

vector addVectors(vector a,vector b){

vector c = {a.x+b.x,a.y+b.y,a.z+b.z};

return c;

}

vector scaleVector(double b,vector a){

vector c = {b*a.x,b*a.y,b*a.z};

return c;

}

vector subtractVectors(vector a,vector b){

vector c = {a.x-b.x,a.y-b.y,a.z-b.z};

return c;

}

double mod(vector a){

return sqrt(a.x*a.x + a.y*a.y + a.z*a.z);

}

void initiateSystem(char* fileName){

int i;

FILE* fp = fopen(fileName,"r");

fscanf(fp,"%lf%d%d",&GravConstant,&bodies,&timeSteps);

masses = (double*)malloc(bodies*sizeof(double));

positions = (vector*)malloc(bodies*sizeof(vector));

velocities = (vector*)malloc(bodies*sizeof(vector));

accelerations = (vector*)malloc(bodies*sizeof(vector));

for(i=0;i<bodies;i++){

fscanf(fp,"%lf",&masses[i]);

fscanf(fp,"%lf%lf%lf",&positions[i].x,&positions[i].y,&positions[i].z);

fscanf(fp,"%lf%lf%lf",&velocities[i].x,&velocities[i].y,&velocities[i].z);

}

fclose(fp);

}

void resolveCollisions(){

int i,j;

for(i=0;i<bodies-1;i++)

for(j=i+1;j<bodies;j++){

if(positions[i].x==positions[j].x && positions[i].y==positions[j].y && positions[i].z==positions[j].z){

vector temp = velocities[i];

velocities[i] = velocities[j];

velocities[j] = temp;

}

}

}

void computeAccelerations(){

int i,j;

for(i=0;i<bodies;i++){

accelerations[i].x = 0;

accelerations[i].y = 0;

accelerations[i].z = 0;

for(j=0;j<bodies;j++){

if(i!=j){

accelerations[i] = addVectors(accelerations[i],scaleVector(GravConstant*masses[j]/pow(mod(subtractVectors(positions[i],positions[j])),3),subtractVectors(positions[j],positions[i])));

}

}

}

}

void computeVelocities(){

int i;

for(i=0;i<bodies;i++)

velocities[i] = addVectors(velocities[i],accelerations[i]);

}

void computePositions(){

int i;

for(i=0;i<bodies;i++)

positions[i] = addVectors(positions[i],addVectors(velocities[i],scaleVector(0.5,accelerations[i])));

}

void simulate(){

computeAccelerations();

computePositions();

computeVelocities();

resolveCollisions();

}

int main(int argC,char* argV[])

{

int i,j;

if(argC!=2)

printf("Usage : %s <file name containing system configuration data>",argV[0]);

else{

initiateSystem(argV[1]);

printf("Body : x y z | vx vy vz ");

for(i=0;i<timeSteps;i++){

printf("\nCycle %d\n",i+1);

simulate();

for(j=0;j<bodies;j++)

printf("Body %d : %lf\t%f\t%lf\t|\t%lf\t%lf\t%lf\n",j+1,positions[j].x,positions[j].y,positions[j].z,velocities[j].x,velocities[j].y,velocities[j].z);

}

}

return 0;

}

Input file: same data as the TCL example.

0.01 3 20 1 0 0 0 0.01 0 0 0.1 1 1 0 0 0 0.02 0.001 0 1 1 0.01 -0.01 -0.01

Invocation and output (x,y,z denote position components in 3-space, vx,vy,vz denote velocity components) :

C:\rosettaCode>mggs.exe mggsData.txt Body : x y z | vx vy vz Cycle 1 Body 1 : 0.010177 0.000179 0.000002 | 0.010354 0.000357 0.000004 Body 2 : 0.998230 0.998232 0.020002 | -0.003539 -0.003536 0.020004 Body 3 : 0.010177 0.988232 0.988055 | 0.010354 -0.013536 -0.013889 Cycle 2 Body 1 : 0.020709 0.000718 0.000011 | 0.010710 0.000721 0.000014 Body 2 : 0.992907 0.992896 0.039971 | -0.007109 -0.007138 0.019935 Body 3 : 0.020717 0.972888 0.972173 | 0.010727 -0.017153 -0.017876 Cycle 3 Body 1 : 0.031600 0.001625 0.000034 | 0.011072 0.001094 0.000033 Body 2 : 0.983985 0.983910 0.059834 | -0.010735 -0.010835 0.019790 Body 3 : 0.031643 0.953868 0.952235 | 0.011125 -0.020886 -0.021999 Cycle 4 Body 1 : 0.042858 0.002913 0.000081 | 0.011443 0.001481 0.000060 Body 2 : 0.971393 0.971163 0.079509 | -0.014448 -0.014659 0.019561 Body 3 : 0.042981 0.931039 0.928087 | 0.011552 -0.024772 -0.026299 Cycle 5 Body 1 : 0.054492 0.004595 0.000160 | 0.011826 0.001884 0.000097 Body 2 : 0.955030 0.954509 0.098909 | -0.018278 -0.018649 0.019238 Body 3 : 0.054766 0.904225 0.899522 | 0.012018 -0.028857 -0.030829 Cycle 6 Body 1 : 0.066517 0.006691 0.000281 | 0.012224 0.002308 0.000145 Body 2 : 0.934759 0.933760 0.117931 | -0.022265 -0.022849 0.018806 Body 3 : 0.067040 0.873197 0.866280 | 0.012530 -0.033199 -0.035655 Cycle 7 Body 1 : 0.078950 0.009225 0.000456 | 0.012642 0.002759 0.000206 Body 2 : 0.910400 0.908677 0.136456 | -0.026454 -0.027316 0.018244 Body 3 : 0.079856 0.837662 0.828023 | 0.013101 -0.037871 -0.040861 Cycle 8 Body 1 : 0.091815 0.012227 0.000702 | 0.013086 0.003245 0.000284 Body 2 : 0.881722 0.878958 0.154340 | -0.030902 -0.032122 0.017523 Body 3 : 0.093281 0.797239 0.784313 | 0.013749 -0.042975 -0.046559 Cycle 9 Body 1 : 0.105140 0.015737 0.001035 | 0.013564 0.003775 0.000383 Body 2 : 0.848429 0.844216 0.171401 | -0.035684 -0.037362 0.016600 Body 3 : 0.107405 0.751427 0.734579 | 0.014498 -0.048649 -0.052908 Cycle 10 Body 1 : 0.118964 0.019805 0.001481 | 0.014085 0.004362 0.000509 Body 2 : 0.810137 0.803953 0.187408 | -0.040900 -0.043166 0.015414 Body 3 : 0.122346 0.699554 0.678056 | 0.015384 -0.055097 -0.060138 Cycle 11 Body 1 : 0.133337 0.024498 0.002071 | 0.014662 0.005025 0.000672 Body 2 : 0.766343 0.757509 0.202050 | -0.046687 -0.049720 0.013868 Body 3 : 0.138268 0.640690 0.613685 | 0.016460 -0.062633 -0.068603 Cycle 12 Body 1 : 0.148327 0.029907 0.002851 | 0.015317 0.005792 0.000888 Body 2 : 0.716377 0.703998 0.214889 | -0.053246 -0.057302 0.011810 Body 3 : 0.155406 0.573482 0.539941 | 0.017816 -0.071782 -0.078886 Cycle 13 Body 1 : 0.164025 0.036157 0.003887 | 0.016079 0.006709 0.001184 Body 2 : 0.659310 0.642172 0.225282 | -0.060887 -0.066351 0.008976 Body 3 : 0.174112 0.495836 0.454475 | 0.019596 -0.083511 -0.092045 Cycle 14 Body 1 : 0.180564 0.043437 0.005286 | 0.017000 0.007852 0.001613 Body 2 : 0.593807 0.570186 0.232208 | -0.070119 -0.077621 0.004875 Body 3 : 0.194929 0.404136 0.353320 | 0.022038 -0.099890 -0.110265 Cycle 15 Body 1 : 0.198150 0.052049 0.007234 | 0.018171 0.009372 0.002283 Body 2 : 0.517817 0.485100 0.233878 | -0.081861 -0.092550 -0.001535 Body 3 : 0.218605 0.290860 0.228583 | 0.025314 -0.126661 -0.139210 Cycle 16 Body 1 : 0.217126 0.062542 0.010117 | 0.019781 0.011614 0.003484 Body 2 : 0.427899 0.381659 0.226654 | -0.097974 -0.114332 -0.012913 Body 3 : 0.244268 0.131956 0.057562 | 0.026013 -0.191148 -0.202831 Cycle 17 Body 1 : 0.238346 0.076539 0.015221 | 0.022658 0.016380 0.006723 Body 2 : 0.317489 0.248502 0.200967 | -0.122846 -0.151982 -0.038461 Body 3 : 0.075592 -0.559591 -0.487315 | -0.363366 -1.191945 -0.886924 Cycle 18 Body 1 : 0.263123 0.097523 0.026918 | 0.026898 0.025587 0.016672 Body 2 : 0.173428 0.050424 0.112716 | -0.165275 -0.244174 -0.138041 Body 3 : -0.286241 -1.745597 -1.369528 | -0.360299 -1.180066 -0.877501 Cycle 19 Body 1 : 0.270854 0.113045 0.061923 | -0.011436 0.005457 0.053339 Body 2 : 0.199821 -0.093105 -0.208666 | 0.218061 -0.042882 -0.504723 Body 3 : -0.646318 -2.924909 -2.246453 | -0.359856 -1.178559 -0.876350 Cycle 20 Body 1 : 0.258572 0.116046 0.112038 | -0.013129 0.000544 0.046890 Body 2 : 0.426346 -0.111425 -0.681150 | 0.234987 0.006241 -0.440245 Body 3 : -1.006089 -4.103186 -3.122591 | -0.359686 -1.177995 -0.875924

#### Graphics[edit]

One of the golden rules of Physics : Draw a diagram. It doesn't have to be artistic, but a properly drawn diagram whose proportions are realistic simplifies the solution of almost every Physics problem. That was the case with this task as well, when I first ran the console version, I noticed that Body 1's position and velocity is varying wildly. I put it down to the chaotic nature of the system and got busy writing the Graphics part and it's when I found that nothing was being drawn on the screen, I took a good look at the implementation and discovered many glaring mistakes. Gravitation is an attractive force, it never repels ( we will talk about Dark Energy later). When I saw the particles moving away from each other, I knew the acceleration vector had the wrong sign.

So yes, always draw a diagram, always do a Graphics implementation of any simulation. Input file for this differs from the console one since the BGI screen is the positive or 1st quadrant of the XY-plane. Hence z components are 0, you can use non-zero values but you won't see any change ( or really weird ones ) on the screen.

0.01 3 20 1.0 100.0 100.0 0.0 0.0 0.0 0.0 2.0 600.0 100.0 0.0 0.0 0.0 0.0 3 500.0 500.0 0.0 0.0 0.0 0.0

The time step although present in the file, and read, is not used here, the loop goes on indefinitely. Requires the WinBGIm library.

/*Abhishek Ghosh, 6th October 2017*/

#include<graphics.h>

#include<stdlib.h>

#include<stdio.h>

#include<math.h>

typedef struct{

double x,y,z;

}vector;

int bodies,timeSteps;

double *masses,GravConstant;

vector *positions,*velocities,*accelerations;

vector addVectors(vector a,vector b){

vector c = {a.x+b.x,a.y+b.y,a.z+b.z};

return c;

}

vector scaleVector(double b,vector a){

vector c = {b*a.x,b*a.y,b*a.z};

return c;

}

vector subtractVectors(vector a,vector b){

vector c = {a.x-b.x,a.y-b.y,a.z-b.z};

return c;

}

double mod(vector a){

return sqrt(a.x*a.x + a.y*a.y + a.z*a.z);

}

void initiateSystem(char* fileName){

int i;

FILE* fp = fopen(fileName,"r");

fscanf(fp,"%lf%d%d",&GravConstant,&bodies,&timeSteps);

masses = (double*)malloc(bodies*sizeof(double));

positions = (vector*)malloc(bodies*sizeof(vector));

velocities = (vector*)malloc(bodies*sizeof(vector));

accelerations = (vector*)malloc(bodies*sizeof(vector));

for(i=0;i<bodies;i++){

fscanf(fp,"%lf",&masses[i]);

fscanf(fp,"%lf%lf%lf",&positions[i].x,&positions[i].y,&positions[i].z);

fscanf(fp,"%lf%lf%lf",&velocities[i].x,&velocities[i].y,&velocities[i].z);

}

fclose(fp);

}

void resolveCollisions(){

int i,j;

for(i=0;i<bodies-1;i++)

for(j=i+1;j<bodies;j++){

if(positions[i].x==positions[j].x && positions[i].y==positions[j].y && positions[i].z==positions[j].z){

vector temp = velocities[i];

velocities[i] = velocities[j];

velocities[j] = temp;

}

}

}

void computeAccelerations(){

int i,j;

for(i=0;i<bodies;i++){

accelerations[i].x = 0;

accelerations[i].y = 0;

accelerations[i].z = 0;

for(j=0;j<bodies;j++){

if(i!=j){

accelerations[i] = addVectors(accelerations[i],scaleVector(GravConstant*masses[j]/pow(mod(subtractVectors(positions[i],positions[j])),3),subtractVectors(positions[j],positions[i])));

}

}

}

}

void computeVelocities(){

int i;

for(i=0;i<bodies;i++)

velocities[i] = addVectors(velocities[i],accelerations[i]);

}

void computePositions(){

int i;

for(i=0;i<bodies;i++)

positions[i] = addVectors(positions[i],addVectors(velocities[i],scaleVector(0.5,accelerations[i])));

}

void simulate(){

computeAccelerations();

computePositions();

computeVelocities();

resolveCollisions();

}

void plotOrbits(){

int i;

for(i=0;i<bodies;i++)

putpixel(positions[i].x,positions[i].y,i%15 + 1);

}

int main(int argC,char* argV[])

{

int i,j;

if(argC!=4)

printf("Usage : %s <file name containing system configuration data and width and height of window>",argV[0]);

else{

initiateSystem(argV[1]);

initwindow(atoi(argV[2]),atoi(argV[3]),"N Body Simulation");

while(1){

simulate();

plotOrbits();

}

}

return 0;

}

### Further work[edit]

And as in every experiment report/journal paper/whatever, time for some philosophizing, a more rigorous implementation will require the bodies to be realistic, this means they will not be point particles and hence the dynamics will be much more complex. Tidal effects, collisions, orbits will require much more computation. When I say realistic, I don't mean rigid bodies because even rigid bodies are an idealization of reality. And all of this, even with classical mechanics, employing the Newtonian version of Gravitation will be a worthy task for any competent Physicist, let alone a Programmer. Introduce the General Theory of Relativity and everything changes. I can go on and on, but this is a coding, not a rambling site. I will end by saying that the Inverse Square Law, a name often used for Newtonian Gravitation, is just one class of forces studied in Dynamics. The force relation can take any form, the N-body problem can therefore be solved in (countably/uncountably ?) infinite ways.

## Fortran[edit]

### Interpretation[edit]

As stated, the task involves a general rendition of , in the sense that a mass is being accelerated (that is, its position is changing, and the second differential of position against time is the acceleration) and therefore there is a force, but the cause and nature of the acceleration is unstated. Then there is mention of masses mutually interacting according to Newton's law of gravitation, so the formula becomes for each *i* and *j* not equal

The bodies are presumed to be able to move freely (not sliding on a wire or across a surface, etc.) so at any position the forces can be calculated and thereby the accelerations, but, only via a function of position. What is desired however is to determine the positions as a function of time. Analysis of the formulae combine constraints that lead to the conclusion that the paths are conic sections: a circle, ellipse, parabola or hyperbola. Thus, for a small mass *m* in a circular orbit around a large mass *M*, the force required to bend the path into a circle is directed towards the circle's centre and is given by

where *r* is the radius of the orbit and *v* its orbital velocity. This assumes that the gravitational mass and inertial masses are the same - experimentally tested by Newton, and to great accuracy by Eötvös *et seq*. For a constant-velocity circular orbit, the time for one orbit to complete is

The value of *m* can be cancelled from both sides to give a limiting case where *m* is zero, a "test mass" (so this is almost a one-body problem) and

A fuller analysis allowing for elliptical orbits finds that

In the MKS system, G = 6·6742E-11 so for a one-second orbit around a one kilogram mass, r = 0·000119 metres, a tenth of a millimetre! A sphere of that radius has volume 7E-12 cubic metres so the density would have to be at least 1.4E+11 KG/cubic metre; osmium manages 2.25E+4, neutronium 4E+17. Alternatively, with two one-kilogram masses orbiting their centre of mass with r = 1 metre, T = 6·3 days... The force between the two spheres is just G, 6·6742E-11 Newtons. Preventing such a languid orbit from being disrupted by electrostatic forces (due to cosmic rays, radioactive disintegration, etc.) will be difficult. Determining G via an experiment in which two *known* masses orbit each other does not look promising.

For more than two bodies the possible behaviours become numerous while the constraints are too few. Notably, bodies can pass within infinitesimal distances of each other and thereby suffer disruptive accelerations. For three bodies certain arrangements remain easily solved: one body having zero mass, or a symmetric arrangement of all three may provide sufficient constraints such as equal masses at the points of an equilateral triangle with equal velocities. For more numerous collections, analysis is abandoned in favour of computation.

### Numerical approach[edit]

The basic step is for each body, sum the forces on it exerted by every other body, then apply F = ma and divide by its mass to determine its acceleration. The only shortcut lies in noting that the force of body A on body B is exactly that of body B on body A (in the reverse direction along the vector joining them) so that the task can be halved. *In Newtonian physics, there is no time lag across distance.* Then, wrongly supposing those accelerations constant for some small time interval, calculate how they would affect the velocities and the positions over that time interval, using a step size that is small enough for this to be approximately true... Too small a step size not only means a lengthy calculation but introduces error by accumulation as in the formation of the sum of a very numerous collection of values.

The Fortran programme was written in the early 1990s and so employs F77. A F90 compiler was available on the IBM mainframe, but it must have been an early version because it was riddled with errors. A colleague tried its QUADRUPLE PRECISION facility, and found that it crashed the operating system! In-line comments were unavailable, and the twenty-four line IBM3278 display screen discouraged expending screen space on commentary, especially because they were non-scrolling. It was also an exploratory programme, so documenting details that were a variation and might soon be removed was a further discouragement. In other words, I now find it difficult to follow the code. (Stop laughing!) The initial effort was prompted by an article in Scientific American that listed the masses, positions and velocities of the twenty nearest stars and invited calculations. It soon became apparent that the stars were not gravitationally bound to each other, as they simply drifted apart along nearly straight lines.

The other area of interest was the performance of various methods for numerical approximation. The calculation effort is assessed on the basis of the number of evaluations of the differential equation required for each step: outside of textbook examples this is usually a large effort. Here for example it is proportional to the square of the number of bodies, and a globular cluster may contain not just hundreds but millions of stars... Thus, the first-order Euler method requires one calculation of accelerations per step, the second-order method two, and the Runge-Kutta method four. Only if the step size could be increased more than proportionally would higher-order methods be worthwhile - provided that accuracy was maintained! Error bounds can be calculated, both for an individual step and for the result after a sequence of steps, via the usual analysis for a polynomial solution function (and if it were a polynomial then numerical methods would not be needed) and as usual produces results that rely on there being an upper bound to a suitably high differential of the solution function over the interval. But in this problem, there is no such bound. Bodies, as points, can approach arbitrarily closely and the resulting accelerations are arbitrarily high, being 1/d^{2} as d approaches zero. Thus, no assurances are available, and a common method is to run a calculation once with a given step size, then again with half the step size, at a painful cost. The irresponsible make a guess at a suitable step size and worry no further.

All the straightforward methods attempt to assess the conditions over the step, perhaps briefly as with the Euler method using the conditions only at the start, whereas the classic fourth-order Runge-Kutta method first probes half a step forwards, uses those results to probe for a full step, then combines all results to make the full step. Such probing requires many evaluations. Rather than forgetting previous positions so painfully computed, the predictor-corrector methods instead rely on the recent past history as being a good indication of the near future - on the supposition of no step-discontinuities. This problem is slightly more difficult because it is second order (as for acceleration to velocity to position) rather than the first order (say, velocity to position) problems discussed in texts. Specifically, the predictor part is to fit a polynomial to the past few accelerations (e.g. at times t-2h, t-h and t) and integrate to obtain a provisional position for the *end* of the current time step at t+h. Evaluate the acceleration there and use it as the new value to perform the actual advance to the end of the time step, the corrector part. Since this is now the new position (and will differ from the provisional position based on polynomial extrapolation), recalculate the acceleration there to maintain coherence between position and acceleration. There are thus just two evaluations per step, and this applies even if more values are retained for higher-order extrapolation and integration methods.

This could be reduced to one evaluation per step: as before use a polynomial fitted to the past accelerations to enable integration so as to determine the position at t+h. Calculate the acceleration for that position (at time t+h) to complete the step, ready to start the next. A sense of unease should arise at this, made clear by asking "Why bother with the corrector at all?" - but that would mean not involving the actual differential equation! The predictor step involves extrapolating a polynomial fit over t-2h, t-h and t (all being accepted solutions for the differential equation), to t+h but the predicted acceleration at time t+h is likely to match the differential equation's value at the new position determined from it only if the solution is a simple polynomial. Put another way, the second-order method employs the differential equation twice (at both ends of the step), and does much better than the first-order method with one evaluation.

Two temptations are to be avoided: using ever-higher order polynomials for extrapolation and integration amounts to saying that the differential equation is a polynomial, and its is not - no polynomial has a vertical or horizontal asymptote and gravitation has both, so another possibility would be to try fitting the values with other functions (perhaps with negative powers: Laurent series?), not just polynomials. Secondly, avoid the "iteration heresy" (F.S. Acton, *Numerical Methods that Work (Usually)*) whereby there is a temptation to iterate the predictor-corrector cycle for each step until it converges. Alas, it will converge to the solution of a difference equation, and this will only be the solution of the *differential* equation after the step size is reduced to zero. Further, such iteration will only re-evaluate the accelerations at almost the same positions each time, leaving the rest of the step interval uninspected. Traversing that step via multiple smaller steps would be a much better use for all those extra evaluations, spreading them evenly.

The simple methods (Euler, second-order, Runge-Kutta) can easily accommodate changes to the step size, but guidance for these changes is not easily obtained. The Milne predictor-corrector method is much messier but contains an inbuilt assessment of the possibilities for changing the step size: at the new position at the end of the step, compare the predicted value for the acceleration with the final calculated value. With the "extrapolation to zero step size" methods of Gragg, Bulirsch, and Stoer, arithmetical analysis of the effect of different step sizes is central to its workings. With calculus the purpose is the analysis of the limiting behaviour as a step is reduced to zero with a view to obtaining results spanning some region of interest that may be ±infinity. Only with the GBS approach is there a corresponding analysis, an arithmetical analysis. It does not concern itself with the nature of the functions but only with the limited-precision inspection of values obtained via limited-precision evaluation.

On escalating to the Milne Predictor-Corrector method, there arose a large increase in administration, especially when the time step was to be doubled or halved. Previous methods had involved just a little juggling but now the juggling was so variable that introducing a multi-item stash with an allocation and freeing scheme reduced the strain. Under consideration were questions such as how to compare two vectors for being adequately close (as in deciding whether to change the step size) when their numerical value might vary over a wide range (astronomical units, or MKS?) yet one or both may be zero too, and whether to stick with halving/doubling or consider halving/tripling so that if (say) 2 was too small and 4 too big, intermediate values such as 3 could be reached, and whether this would be worth the floundering about.

### Organisation of arrays[edit]

Aside from the details of the calculations and their method, organising the data structure is important. Accessing an array element is time-consuming, so a particular ploy was to de-reference the arrays: from three dimensions to two (via NEWT) and from two to one (via NEWTON) so that the indexing of the time-consuming computation would be simplified. In other words, the storage for X (position), V (velocity) and A (acceleration) are all 100 (to allow for N bodies) by 3 (for x, y, z components) by 14, this last being sufficient multiple sets of data for MILNE to juggle. It uses Runge-Kutta to initialise its march then proceeds, possibly doubling its step size (so discarding alternate saved historic position sets, thus needing to have eight so that it would be able to retain four) or halving its step size. Notably, for a circular orbit, MILNE developed coefficients equivalent to the first few terms of the series expansion of sine. From F90 there appear arrangements for manipulating arrays, and especially the FOR ALL statement, but their exact behaviour is poorly described. For instance, if the FOR ALL statement were to produce results in a work area then copy that to the destination area, performance may be impeded compared to the update-in-place of the straightforward statements due to the extra data transfer involved.

The arrays are dimensioned as 100,3,14 rather than say 14,3,100 because Fortran orders array elements so that the first index varies most rapidly for adjacent elements in storage - some languages are the other way around and I have never seen an explanation for why this choice was made for Fortran, as it is unhelpful for matrix arithmetic. Thus, element (i,j,k) is followed by element (i + 1,j,k). Accordingly, as NEWTON works along the N bodies, the values for consecutive bodies are adjacent. It is presented with separate X, Y, Z arrays even though these are actually parts of the same big array, so that it may use one-dimensional array indexing (and with the same index for all) rather than the much slower two-dimensional indexing. However, the storage locations of these arrays have fixed relationships with each other. Specifically, each are one hundred elements apart so that Y(i) is one hundred elements along from X(i), and Z(i) another hundred along from Y(i). Thus, there could instead be an array XYZ of 300 elements with the code referencing XYZ(i) to obtain X(i), XYZ(i + 100) for Y(i), and XYZ(i + 200) for Z(i). This would have the obvious advantage of maintaining one-dimensional indexing, and gain the possibility that the compiler might generate code that uses one index register for all the accesses rather than three. This can be achieved for "free" - suppose the determination of the effective address (in memory of a datum) is of the form (base address of XYZ) + (index register: the offset). Rather than calculating offsets from (i) and (i + 100), and (i + 200) the equivalent would be to use an offset of (i) every time, but with (XYZ) and (XYZ + 100) and (XYZ + 200), these all being constants.

This notion could also be extended to the 14 sets of 100 triples. In other words, any multi-dimensional array could be considered as equivalent to a single-dimension array of the same size, and indeed could be made so via an EQUIVALENCE statement. Thus, the notational and organisational conveniences of multi-dimensional arrays could be retained, with the use of single indexing reserved for places of desperate need for speed. However, in the absence of a PARAMETER statement to identify the various constants, any adjustment to the size and shape of the array will be difficult and simple mistakes will often be missed. Serious computational programmes often involve large and complex formulae, so the task is not small. Depending on one's ingenuity, there are many possible schemes to choose amongst and a great deal of time can vanish, possibly greater than any recovered by faster running...

All of this could, and indeed should, be done via the compiler's analysis rather than have a human disappear down a rabbit hole. First Fortran (1958) engaged in intensive analysis, guided by a Monte Carlo simulation possibly assisted by the programmer supplying hints via the FREQUENCY statement and produced surprisingly cunning code - in part because the (debugged) compiler would engage in complex arithmetic that a human programmer would avoid due to the risk of making a mistake. Subsequent compilers are often praised, but inspection of the code is less impressive, and the task is not well done. For instance, simple changes to the source file result in faster-running programmes, and these changes should not have made any difference. Thus, NEWTON puts values from arrays into simple variables rather than reference the same array element for each usage: the compiler should be able to notice this. There is certainly talk about "invariant expressions" and the like, but, the run times *are* different. It is also possible that the code produced by the "optimising" compiler runs slower than that without the "optimising" active, despite the longer compile time. This is not just me: the British Meteorological Office has admitted to similar experiences, during a job interview.

All calculations are in double precision. Test runs with near-approach paths demonstrated the need, even though the constant of gravitation is known to well less than single-precision accuracy. Startling plots resulted, because as a body approaches a mass its velocity builds up to a high value and thus, covering a lot of distance near to the mass in the same time step means that curvature is missed. Further, the build up from a small velocity to high velocities during the approach must be matched by their reduction during the departure, but limited precision means that the details of the low velocity have been lost to truncation/rounding while the number was large and so the departure path will not be the mirror of the approach path. For example, imagine that a single-precision variable holds the value of pi, then add 1, 10, 100, 1000, 10000, 100000, followed by subtracting 100000, 10000, ... , 1. The value of pi will have been badly damaged. Only in double precision would that single precision accuracy have been maintained.

### Source[edit]

I have left the source file pretty much as it was, so F77 rules. This was written in part to exercise the graph-plotting facilities, which served a number of output devices (screen to paper) but in a page-only mode: no moving pictures. Rather than attempt to develop a library of plot assistance routines, for exploration it was easier to incorporate (and possibly fiddle) the most helpful source from other programmes - the Fortran compiler ran at about 50,000 lines/minute, and also I'd had odd experiences with the linkage loader. Those plotting routines are long gone, and there are no similar routines to hand now so they have here been made into dummy routines. Variable DIVE enables runs with plotting deactivated so that text output could be concentrated upon, and also suppressed tedious requests to name the plotting device.

There are also slight changes - the IBM MVS/XA with TSO system did not support the open-file-by-name protocol, so the source now contains an `OPEN (IN,FILE="TCL.dat")`

to read the parameters. During a TSO session, one would assign a named file to a unit number (like a DD statement in JCL for batch jobs) and then run the programme as often as one wished. Similarly, as scrolling screens are now available, there are some additional output statements such as a confirmatory display of what was read from the input file. Previously, one would begrudge every output as it used up one more of the available twenty-three lines, each of seventy-nine characters. The IBM3270 (*et al*) display screens divided the screen space into display fields, each starting with a control field (stating input allowed or not, digits only, may be skipped, bright/dim, etc.) that took up a character space. If when viewing the next screen's output one wished to refer back to previous output, it was too late, there was no scrolling back... One could write helpful output to a text file, but, there was no provision for multiple "windows" on the screen to view the graph and the text together, and not much space anyway even if there were. Typically, one would hit the enter key until the "READY" was close to the bottom of the screen then start the run. This way the resulting output would spill over to a fresh screen and so enable as much as possible to be seen together. To assist in this, there was a facility to specify IST and LST, the first and last body for which output was to be shown, here set to 1 and 3 for the specific three-body example being run.

More seriously, later Fortran has additional routines, and confusion can arise: thus subroutine FREE was renamed to SFREE, and array INDEX renamed to INDEXS, though calling it SINDEX was the first idea - but that would require additional declarations so as to keep it integer. The habit of not bothering with minor declarations lingers on with exploratory programmes...

SUBROUTINE PLOTS(DX,DY,DEVICE)

INTEGER DEVICE(10)

COMMON LINPR

WRITE (LINPR,*) "PlotS:",DX,DY,DEVICE

END

SUBROUTINE PLOT(X,Y,IHIC)

COMMON LINPR

WRITE (LINPR,*) "Plot:",X,Y,IHIC

END

SUBROUTINE FACTOR(F)

COMMON LINPR

WRITE (LINPR,*) "PlotFactor:",F

END

SUBROUTINE SYMBOL(X,Y,H,T,A,N)

INTEGER T(1)

COMMON LINPR

WRITE (LINPR,*) "PlotSymbol:",X,Y,H,T,A,N

END

SUBROUTINE NUMBER(X,Y,H,V,A,N)

COMMON LINPR

WRITE (LINPR,*) "PlotNumber:",X,Y,H,V,A,N

END

C==================The above suppresses attempts to access a long-lost plotting system

c FUNCTION CPUTIM(I)

c CALL INTVAL(-3,IWASTE)

c CPUTIM = FLOAT(IWASTE)/1000

c IF (I .EQ. 0) CALL STIME

c RETURN

c END

LOGICAL FUNCTION PRANGE(X1,X2,Y1,Y2)

LOGICAL ROTATE,DIVE

INTEGER DEVICE(10),CARDS,BLANK

COMMON /PLOTIT/ ROTATE,DIVE,DEVICE,BLOAT

COMMON LINPR,CARDS

DATA BLANK/' '/

PRANGE = DIVE

IF (DIVE) RETURN

IF (DEVICE(1).NE.BLANK) GO TO 10

WRITE (LINPR,1)

1 FORMAT (' Name your output device (TEK618,GDDM78,..)')

READ (CARDS,2) DEVICE(1),DEVICE(2)

2 FORMAT (2A4)

10 DEVICE(3) = 0

DEVICE(4) = 4*10

DX = X2 - X1

DY = Y2 - Y1

IF (.NOT.ROTATE) CALL PLOTS(DX,DY,DEVICE)

IF ( ROTATE) CALL PLOTS(DY,DX,DEVICE)

PRANGE = DEVICE(3).EQ.0

IF (PRANGE) GO TO 100

WRITE (LINPR,11) (DEVICE(I),I = 1,4)

11 FORMAT (' Muckup.',2A4,2I9)

RETURN

100 CALL PLOT(0.0,0.0,3)

D = AMAX1(DX,DY)

IF ( ROTATE) CALL FACTOR(BLOAT)

IF (.NOT.ROTATE) CALL PLOT(-X1,-Y1,-3)

IF ( ROTATE) CALL PLOT( Y2,-X1,-3)

RETURN

END

SUBROUTINE PLOTXY(X,Y,I)

LOGICAL ROTATE,DIVE

COMMON /PLOTIT/ ROTATE,DIVE

C (x,y)*(0,1) = (-y,x), a 90-degree rotation.

IF (DIVE) RETURN

IF (.NOT.ROTATE) CALL PLOT( X,Y,I)

IF ( ROTATE) CALL PLOT(-Y,X,I)

RETURN

END

SUBROUTINE SYMBXY(X,Y,H,T,A,N)

LOGICAL ROTATE,DIVE

INTEGER T(1)

COMMON /PLOTIT/ ROTATE,DIVE

IF (DIVE) RETURN

IF (.NOT.ROTATE) CALL SYMBOL(X,Y,H,T,A,N)

IF ( ROTATE) CALL SYMBOL(-Y,X,H,T,A + 90,N)

RETURN

END

SUBROUTINE NUMBXY(X,Y,H,V,A,N)

LOGICAL ROTATE,DIVE

COMMON /PLOTIT/ ROTATE,DIVE

IF (DIVE) RETURN

IF (.NOT.ROTATE) CALL NUMBER(X,Y,H,V,A,N)

IF ( ROTATE) CALL NUMBER(-Y,X,H,V,A + 90,N)

RETURN

END

SUBROUTINE DOBOX(X1,Y1,X2,Y2)

CALL PLOTXY(X1,Y1,3)

CALL PLOTXY(X2,Y1,2)

CALL PLOTXY(X2,Y2,2)

CALL PLOTXY(X1,Y2,2)

CALL PLOTXY(X1,Y1,2)

RETURN

END

C============Above are routines for producing plots.

SUBROUTINE SWAPI(I,J)

IT = I

I = J

J = IT

RETURN

END

SUBROUTINE NEWTON(X,Y,Z,AX,AY,AZ,M,N)

IMPLICIT REAL*8 (A-H,O-Z)

REAL*8 X(N),Y(N),Z(N),AX(N),AY(N),AZ(N),M(N)

REAL*8 MI,MJ

COMMON /CONST/ G,CLOSE

DO 1 I = 1,N

AX(I) = 0.0

AY(I) = 0.0

1 AZ(I) = 0.0

NL1 = N - 1

DO 3 I = 1,NL1

AXI = AX(I)

AYI = AY(I)

AZI = AZ(I)

MI = M(I)

XI = X(I)

YI = Y(I)

ZI = Z(I)

J = I + 1

DO 2 J = J,N

MJ = M(J)

DX = X(J) - XI

DY = Y(J) - YI

DZ = Z(J) - ZI

D2 = DZ*DZ + DY*DY + DX*DX

IF (D2 .LT. CLOSE) CLOSE = D2

F = G/(D2*DSQRT(D2))

AIJ = F*DX

AXI = AXI + MJ*AIJ

AX(J) = AX(J) - MI*AIJ

AIJ = F*DY

AYI = AYI + MJ*AIJ

AY(J) = AY(J) - MI*AIJ

AIJ = F*DZ

AZI = AZI + MJ*AIJ

2 AZ(J) = AZ(J) - MI*AIJ

AX(I) = AXI

AY(I) = AYI

3 AZ(I) = AZI

RETURN

END

SUBROUTINE NEWT(X,A,N)

IMPLICIT REAL*8 (A-H,O-Z)

REAL*8 X(100,3),A(100,3),M(100)

COMMON /CONST/ G,CLOSE,M

CALL NEWTON(X(1,1),X(1,2),X(1,3),A(1,1),A(1,2),A(1,3),M,N)

RETURN

END

SUBROUTINE PROBE(X,X2,V,A,N,DT)

C Looks ahead one time step.

IMPLICIT REAL*8 (A-H,O-Z)

REAL*8 X(100,3),V(100,3),A(100,3)

REAL*8 X2(100,3)

DO 1 I = 1,N

DO 1 J = 1,3

1 X2(I,J) = X(I,J) + (V(I,J) + 0.5*A(I,J)*DT)*DT

RETURN

END

SUBROUTINE EULER(K,L,N,T,DT)

Computes the first order advance. Uses A at T only.

IMPLICIT REAL*8 (A-H,O-Z)

COMMON /PLACE/ X(100,3,14),V(100,3,14),A(100,3,14)

CALL NEWT(X(1,1,K),A(1,1,K),N)

DO 2 I = 1,N

DO 2 J = 1,3

VIJ = V(I,J,K)

AIJDT = A(I,J,K)*DT

V(I,J,L) = VIJ + AIJDT

2 X(I,J,L) = X(I,J,K) + (VIJ + 0.5*AIJDT)*DT

T = T + DT

RETURN

END

SUBROUTINE LUNGE(K,L,N,T,DT)

Computes a second order advance (Huen's, or 2'nd order Euler).

C Uses A(t) to probe ahead to X(t + dt) to find A(t + dt)

C and then averages the two to advance one step.

IMPLICIT REAL*8 (A-H,O-Z)

COMMON /PLACE/ X(100,3,14),V(100,3,14),A(100,3,14)

CALL NEWT (X(1,1,K),A(1,1,K),N)

CALL PROBE(X(1,1,K),X(1,1,L),V(1,1,K),A(1,1,K),N,DT)

CALL NEWT (X(1,1,L),A(1,1,L),N)

DO 2 I = 1,N

DO 2 J = 1,3

VIJ = V(I,J,K)

AIJDT = (A(I,J,K) + A(I,J,L))*0.5*DT

V(I,J,L) = VIJ + AIJDT

2 X(I,J,L) = X(I,J,K) + (VIJ + 0.5*AIJDT)*DT

T = T + DT

RETURN

END

SUBROUTINE RUNGE(K,L,N,T,DT)

Classic Runge-Kutta fourth order advance.

C 1) Use A(t) to reach x2(t + dt/2) and compute a2(t + dt/2)

C 2) Use a2(t + dt/2) to reach x3(t + dt/2) and compute a3(t + dt/2)

C 3) Use a3(t + dt/2) to reach x4(t + dt) and compute a4(t + dt/2)

C 4) Use a weighted average of a,a2,a3,a4 to make the actual advance.

IMPLICIT REAL*8 (A-H,O-Z)

REAL*8 A3(100,3),A4(100,3)

COMMON /PLACE/ X(100,3,14),V(100,3,14),A(100,3,14)

CALL NEWT (X(1,1,K),A(1,1,K),N)

CALL PROBE(X(1,1,K),X(1,1,L),V(1,1,K),A(1,1,K),N,DT/2)

CALL NEWT (X(1,1,L),A(1,1,L),N)

CALL PROBE(X(1,1,K),X(1,1,L),V(1,1,K),A(1,1,L),N,DT/2)

CALL NEWT (X(1,1,L),A3,N)

CALL PROBE(X(1,1,K),X(1,1,L),V(1,1,K),A3,N,DT)

CALL NEWT (X(1,1,L),A4,N)

DO 2 I = 1,N

DO 2 J = 1,3

VIJ = V(I,J,K)

AIJ = (A(I,J,K) + 2.*(A(I,J,L) + A3(I,J)) + A4(I,J))/6.

A(I,J,L) = AIJ

AIJDT = AIJ*DT

V(I,J,L) = VIJ + AIJDT

2 X(I,J,L) = X(I,J,K) + (VIJ + 0.5*AIJDT)*DT

T = T + DT

RETURN

END

SUBROUTINE SCLEAR

COMMON /STORE/ N,INDEXS(14)

N = 0

DO 1 I = 1,14

1 INDEXS(I) = 14 - I + 1

RETURN

END

SUBROUTINE SGRAB(IT)

COMMON /STORE/ N,INDEXS(14)

IT = INDEXS(N)

N = N - 1

RETURN

END

SUBROUTINE SFREE(IT)

COMMON /STORE/ N,INDEXS(14)

IF (IT .LE. 0) RETURN

N = N + 1

INDEXS(N) = IT

IT = 0

RETURN

END

SUBROUTINE MILNE(N,T,T1,DT,EPS,NSTEP)

Chase along according to the Milne predictor-corrector scheme.

C 1) Predict: Fit a parabola to the last three a's. (k-2,k-1,k)

C Integrate from (k-3) to (k+1) giving v(k+1)

C Fit a parabola to the latest v's (k-1,k,k+1)

C Integrate from (k-1) to (k+1) giving x(k+1)

C Compute a(kp1) at x(k+1), i.e. a(t + dt).

C 2) Correct: Repeat the prediction step, using k-1,k,k+1.

C There are details. By integrating from (k-3) to (k+1) we have a

C symmetrical region of the parabola fitted to (k-1),(k),(k+1),

C which means that P4 is exact for cubics.

C Secondly, the correction step doesn't need to extrapolate

C because we have an estimate for (k+1). (The whole point!)

C 3) Check: The difference between the predicted and corrected a's.

C Instead of iterating the corrector until (pred - corr) is small,

C which involves repeated evaluations of the a's at the various

C corrected x's, all of which are more or less at the same point,

C the scheme here is to refine the sampling of the whole interval

C x(k) to x(k+1) by halving the step size, thereby sampling the

C behaviour at a more even spread of positions.

IMPLICIT REAL*8 (A-H,O-Z)

COMMON /STORE/ NAVAIL,INDEXS(14)

COMMON /PLACE/ X(100,3,14),V(100,3,14),A(100,3,14)

REAL*8 EXTRAP(100,3)

Compute assorted integrals.

P4(Y1,Y2,Y3) = 2.*(Y1 + Y3) - Y2

P3(Y1,Y2,Y3) = Y1 + Y3 + 4.*Y2

P24(Y1,Y2,Y3) = 2.*Y3 - Y1 + 11.*Y2

KMAX = 14

BB = 0

NSTEP = 0

C

Concoct a past history from which to extrapolate.

DO 1 K = 1,3

CALL RUNGE(K,K + 1,N,T,DT)

CALL JOIN(K,K + 1,N)

1 CONTINUE

KP1 = 5

K = 4

KL1 = 3

KL2 = 2

KL3 = 1

KL4 = 0

KL5 = 0

KL6 = 0

NAVAIL = 0

DO 2 I = 6,KMAX

NAVAIL = NAVAIL + 1

2 INDEXS(NAVAIL) = KMAX - I + 6

C

Cook up an estimate of the A's at KP1, one time step ahead.

10 H3 = DT/3

H4 = DT*4/3

C WRITE (6,666) NSTEP,T,DT, NAVAIL,KL6,KL5,KL4,KL3,KL2,KL1,K,KP1

666 FORMAT (I4,F7.2,F9.4,14X ,I3,':',8I3)

DO 11 I = 1,N

DO 11 J = 1,3

VT = V(I,J,KL3) + H4*P4(A(I,J,KL2),A(I,J,KL1),A(I,J,K))

V(I,J,KP1) = VT

X(I,J,KP1) = X(I,J,KL1) + H3*P3(V(I,J,KL1),V(I,J,K),VT)

11 CONTINUE

C

Compute the A's at the extrapolated position, thus involving the DE.

CALL NEWT(X(1,1,KP1),EXTRAP,N)

C

Correct the X's and V's now that the story at KP1 is known (sortof).

20 DO 21 I = 1,N

DO 21 J = 1,3

VT = V(I,J,KL1) + H3*P3(A(I,J,KL1),A(I,J,K),EXTRAP(I,J))

V(I,J,KP1) = VT

X(I,J,KP1) = X(I,J,KL1) + H3*P3(V(I,J,KL1),V(I,J,K),VT)

21 CONTINUE

C

Calculate new A's to ensure a coherent solution of the DE.

CALL NEWT(X(1,1,KP1),A(1,1,KP1),N)

C

Compare the provisional and the accepted A's.

30 B = 0

DO 32 I = 1,N

D = 0.

DP = 0.

DC = 0.

DO 31 J = 1,3

AP = EXTRAP(I,J)

AC = A(I,J,KP1)

D = D + (AP - AC)**2

DP = DP + AP*AP

DC = DC + AC*AC

31 CONTINUE

DPC = DP + DC

IF (DPC .LE. 0.0) DPC = 1

D = D/DPC

32 IF (D .GT. B) B = D

IF (B .GT. BB) BB = B

C WRITE (6,667) B,NAVAIL,KL6,KL5,KL4,KL3,KL2,KL1,K,KP1

667 FORMAT (20X, F14.11,I3,':',8I3)

IF (B .LT. EPS) GO TO 50

C

Chop the step size in half. Interpolate IL1, IL3

40 CALL SFREE(KL6)

CALL SFREE(KL5)

CALL SFREE(KL4)

CALL SFREE(KL3)

CALL SGRAB(IL3)

CALL SGRAB(IL1)

H24 = DT/24

DO 41 I = 1,N

DO 41 J = 1,3

AK = A(I,J,K)

AKL1 = A(I,J,KL1)

AKL2 = A(I,J,KL2)

VKL1 = V(I,J,KL1)

VIL1 = VKL1 + H24*P24(AKL2,AKL1,AK)

VIL3 = VKL1 - H24*P24(AK,AKL1,AKL2)

V(I,J,IL3) = VIL3

V(I,J,IL1) = VIL1

VK = V(I,J,K)

VKL2 = V(I,J,KL2)

XKL1 = X(I,J,KL1)

XIL1 = XKL1 + H24*P24(VKL2,VKL1,VK)

XIL3 = XKL1 - H24*P24(VK,VKL1,VKL2)

X(I,J,IL3) = XIL3

41 X(I,J,IL1) = XIL1

CALL NEWT(X(1,1,IL1),A(1,1,IL1),N)

CALL NEWT(X(1,1,IL3),A(1,1,IL3),N)

KL4 = KL2

KL3 = IL3

KL2 = KL1

KL1 = IL1

DT = DT/2

GO TO 10

C

Complete the step by advancing all fingers.

50 NSTEP = NSTEP + 1

T = T + DT

CALL SFREE(KL6)

KL6 = KL5

KL5 = KL4

KL4 = KL3

KL3 = KL2

KL2 = KL1

KL1 = K

K = KP1

CALL SGRAB(KP1)

CALL JOIN(KL1,K,N)

IF (T .GE. T1) GO TO 100

C

Consider doubling the step size.

60 IF (B .GE. EPS/36) GO TO 10

IF (KL4*KL5*KL6 .LE. 0) GO TO 10

C WRITE (6,668) NAVAIL,KL6,KL5,KL4,KL3,KL2,KL1,K,KP1

668 FORMAT (34X, I3,':',8I3)

CALL SFREE(KL1)

CALL SFREE(KL3)

CALL SFREE(KL5)

KL1 = KL2

KL2 = KL4

KL3 = KL6

KL4 = 0

KL5 = 0

KL6 = 0

DT = DT*2

GO TO 10

Completed.

100 CALL PLOTXY(0.0,0.0,999)

WRITE (6,*) "Bmax=",BB

CALL PRINT(X(1,1,K),V(1,1,K),1,N)

RETURN

END

SUBROUTINE JOIN(K,L,N)

IMPLICIT REAL*8 (A-H,O-Z)

REAL U,V

COMMON /PLACE/ X(100,3,14)

COMMON /SCOPE/ IU,IV,UMIN,VMIN,US,VS

DO 1 I = 1,N

U = (X(I,IU,K) - UMIN)*US

V = (X(I,IV,K) - VMIN)*VS

CALL PLOTXY(U,V,3)

U = (X(I,IU,L) - UMIN)*US

V = (X(I,IV,L) - VMIN)*VS

C CALL PLOTXY(U,V,2)

IT = MOD(I,14)

CALL SYMBXY(U,V,0.05,IT,0.0,-2)

1 CONTINUE

RETURN

END

SUBROUTINE PRINT(X,V,IST,LST)

IMPLICIT REAL*8 (A-H,O-Z)

REAL*8 X(100,3),V(100,3)

IF (IST.GT.LST) RETURN

DO 3 I = IST,LST

RXYZ = 0.0

VXYZ = 0.0

DO 1 J = 1,3

RXYZ = RXYZ + X(I,J)**2

1 VXYZ = VXYZ + V(I,J)**2

RXYZ = DSQRT(RXYZ)

VXYZ = DSQRT(VXYZ)

WRITE (6,2) I,(X(I,J),J = 1,3),RXYZ,(V(I,J),J = 1,3),VXYZ

2 FORMAT (I3,2(4F8.5,3X))

3 CONTINUE

RETURN

END

FUNCTION PE(N,L)

IMPLICIT REAL*8 (A-H,O-Z)

REAL*8 M(100)

COMMON /CONST/ G,CLOSE,M

COMMON /PLACE/ X(100,3,14)

T = 0

NL1 = 1

DO 2 I = 1,NL1

IP1 = I + 1

DO 2 J = IP1,N

R2 = 0.0

DO 1 K = 1,3

1 R2 = R2 + (X(I,K,L) - X(J,K,L))**2

2 T = T - M(I)*M(J)/DSQRT(R2)

PE = G*T

RETURN

END

REAL FUNCTION KE(N,L)

IMPLICIT REAL*8 (A-H,O-Z)

REAL*8 M(100)

COMMON /CONST/ G,CLOSE,M

COMMON /PLACE/ X(100,3,14),V(100,3,14)

T = 0

DO 2 I = 1,N

V2 = 0

DO 1 J = 1,3

1 V2 = V2 + V(I,J,L)**2

2 T = T + M(I)*V2

KE = T/2

RETURN

END

SUBROUTINE COM(N,L,TM,W,P)

Centre of mass.....

IMPLICIT REAL*8 (A-H,O-Z)

REAL*8 M(100),W(3),P(3)

COMMON /PLACE/ X(100,3,14),V(100,3,14)

COMMON /CONST/ G,CLOSE,M

DO 1 I = 1,3

W(I) = 0

1 P(I) = 0

TM = 0

DO 2 I = 1,N

TM = TM + M(I)

DO 2 J = 1,3

W(J) = W(J) + M(I)*X(I,J,L)

2 P(J) = P(J) + M(I)*V(I,J,L)

DO 3 J = 1,3

W(J) = W(J)/TM

3 P(J) = P(J)/TM

RETURN

END

IMPLICIT REAL*8 (A-H,O-Z)

LOGICAL PRANGE,ROTATE,DIVE

LOGICAL ASIS

INTEGER FANCY

INTEGER DEVICE(10),CARDS,BLANK

REAL*8 M(100)

REAL*8 XMIN(3),XMAX(3),W(3),P(3)

REAL XSIZE,YSIZE,B,BLOAT

COMMON /PLOTIT/ ROTATE,DIVE,DEVICE,BLOAT

COMMON /CONST/ G,CLOSE,M

COMMON /PLACE/ X(100,3,14),V(100,3,14),A(100,3,14)

COMMON /SCOPE/ IU,IV,UMIN,VMIN,US,VS

COMMON LINPR,CARDS

DATA BLANK/' '/

DIVE = .FALSE.

DIVE = .TRUE.

ASIS = .FALSE.

CARDS = 5

LINPR = 6

IN = 10

XSIZE = 9

YSIZE = 4.75

DEVICE(1) = BLANK

BLOAT = 10/(0.5 + 7.1)

WRITE (LINPR,1)

1 FORMAT (' Star Trails.')

IF (.NOT.DIVE) WRITE (LINPR,2)

2 FORMAT (' Enter size, (4.75 or 10.5)')

IF (.NOT.DIVE) READ (CARDS,*) XSIZE

YSIZE = XSIZE

IN = 10

OPEN (IN,FILE="TCL.dat")

READ (IN,*) N

WRITE (LINPR,*) N," bodies."

READ (IN,*) G

WRITE (LINPR,*) G," gravitational constant."

READ (IN,*) T1,DT

WRITE (LINPR,*) T1,DT," Run time, time step."

READ (IN,*) XMIN,XMAX

DO 3 I = 1,N

READ (IN,*) M(I),(X(I,J,1), J = 1,3),(V(I,J,1),J = 1,3)

DO 3 J = 1,3

A(I,J,1) = 0

3 CONTINUE

CLOSE (IN)

c WRITE (LINPR,4) N

IST = 1

LST = 3

4 FORMAT (' The first and last body to show (of',I3,')')

c READ (CARDS,*) IST,LST

FANCY = 1

WRITE (LINPR,5)

5 FORMAT (' Euler/2''nd/Runge/Milne (1/2/3/4)')

READ (CARDS,*) FANCY

c WRITE (LINPR,6)

c 6 FORMAT (' Centre of mass (T/F)')

c READ (CARDS,*) ASIS

ASIS = .NOT.ASIS

C

10 DO 11 I = 1,3

IF (XMIN(I) .GE. XMAX(I)) GO TO 12

11 CONTINUE

GO TO 15

12 DO 13 I = 1,3

XMIN(I) = X(1,I,1)

13 XMAX(I) = X(1,I,1)

DO 14 I = 1,N

DO 14 J = 1,3

XMIN(J) = DMIN1(XMIN(J),X(I,J,1))

14 XMAX(J) = DMAX1(XMAX(J),X(I,J,1))

15 CLOSE = 0

DO 16 I = 1,3

16 CLOSE = CLOSE + (XMAX(I) - XMIN(I))**2

IF (ASIS) GO TO 20

CALL COM(N,1,TM,W,P)

DO 17 I = 1,N

DO 17 J = 1,3

17 V(I,J,1) = V(I,J,1) - P(J)

C

20 IU = 1

IV = 2

UMIN = XMIN(IU)

VMIN = XMIN(IV)

B = 0.5

US = (XSIZE - 0)/(XMAX(IU) - UMIN)

VS = (YSIZE - 0)/(XMAX(IV) - VMIN)

IF (.NOT.PRANGE(-B,XSIZE + B,-B,YSIZE + B)) STOP

CALL DOBOX(0.0,0.0,XSIZE,YSIZE)

C

100 T0 = 0

T = T0

NSTEP = T1/DT + 0.5

IF (IST .LE. LST) WRITE (LINPR,101)

101 FORMAT (8X,'x y z R',

1 9X,'vx vy vz V')

IF (IST .LE. LST) CALL PRINT(X,V,IST,LST)

IF (FANCY .EQ. 4) GO TO 200

I1 = 1

I2 = 2

DO 110 I = 1,NSTEP

IF (FANCY .EQ. 1) CALL EULER(I1,I2,N,T,DT)

IF (FANCY .EQ. 2) CALL LUNGE(I1,I2,N,T,DT)

IF (FANCY .EQ. 3) CALL RUNGE(I1,I2,N,T,DT)

WRITE (LINPR,*) "T=",T

CALL JOIN(I1,I2,N)

CALL SWAPI(I1,I2)

IF (IST .LE. LST) CALL PRINT(X(1,1,I1),V(1,1,I1),1,N)

110 CONTINUE

GO TO 9000

C

200 EPS = 1

201 EPS = EPS/2

IF (1 + EPS .NE. 1) GO TO 201

EPS = EPS*2

CALL MILNE(N,T,T1,DT,EPS,NSTEP)

C

9000 IF (FANCY .NE. 4) CALL PLOTXY(0.0,0.0,999)

WRITE (LINPR,*) "Reached T=",T

WRITE (LINPR,*) "Target T=",T1

WRITE (LINPR,*) "Time step=",DT,"Nstep=",NSTEP

IF (FANCY .NE. 4) CALL PRINT(X(1,1,I1),V(1,1,I1),1,N)

CLOSE = DSQRT(CLOSE)

WRITE (LINPR,9001) CLOSE

9001 FORMAT (' Closest approach:',E15.6)

END

### Results[edit]

#### TCL matching?[edit]

I have been unable to follow the exact workings of the TCL example in the absence of annotations, in particular whether I am looking at x1 x2 x3 y1 y2 y3 z1 z2 z3 or, x1 y1 z1, etc. My file TCL.dat is as follows:

3 Bodies 0.01 Gravitational constant. 0.2, 0.01 Run time, time step. -2 -2 -2, +2 +2 +2 xyzmin, xyzmax for plot scaling 1, 0 0 0, 0.01 0 0 Mass, xyz Position, xyz Velocity 0.1, 1 1 0, 0 0 0.02 0.001 0 1 1, 0.01 -0.01 -0.01

This relies on Fortran's free-format reading looking only for the requested count of numbers on a line, so that what follows is ignored and so can be annotation. More generally, with this style of input a / character indicates an in-line comment and is treated as end-of-line for the reading of values. Output:

Star Trails. 3 bodies. 1.000000000000000E-002 gravitational constant. 0.200000000000000 1.000000000000000E-002 Run time, time step. Euler/2'nd/Runge/Milne (1/2/3/4) 1 x y z R vx vy vz V 1 0.00000 0.00000 0.00000 0.00000 0.01000 0.00000 0.00000 0.01000 2 1.00000 1.00000 0.00000 1.41421 0.00000 0.00000 0.02000 0.02000 3 0.00000 1.00000 1.00000 1.41421 0.01000-0.01000-0.01000 0.01732 T= 1.000000000000000E-002 1 0.00010 0.00000 0.00000 0.00010 0.01000 0.00000 0.00000 0.01000 2 1.00000 1.00000 0.00020 1.41421 -0.00004-0.00004 0.02000 0.02000 3 0.00010 0.99990 0.99990 1.41407 0.01000-0.01004-0.01004 0.01737 T= 2.000000000000000E-002 1 0.00020 0.00000 0.00000 0.00020 0.01001 0.00001 0.00000 0.01001 2 1.00000 1.00000 0.00040 1.41421 -0.00007-0.00007 0.02000 0.02000 3 0.00020 0.99980 0.99980 1.41393 0.01001-0.01007-0.01008 0.01741 T= 3.000000000000000E-002 1 0.00030 0.00000 0.00000 0.00030 0.01001 0.00001 0.00000 0.01001 2 1.00000 1.00000 0.00060 1.41421 -0.00011-0.00011 0.02000 0.02000 3 0.00030 0.99970 0.99970 1.41379 0.01001-0.01011-0.01012 0.01746 T= 4.000000000000000E-002 1 0.00040 0.00000 0.00000 0.00040 0.01001 0.00001 0.00000 0.01001 2 1.00000 1.00000 0.00080 1.41421 -0.00014-0.00014 0.02000 0.02000 3 0.00040 0.99960 0.99960 1.41364 0.01001-0.01014-0.01016 0.01750 T= 5.000000000000000E-002 1 0.00050 0.00000 0.00000 0.00050 0.01002 0.00002 0.00000 0.01002 2 1.00000 1.00000 0.00100 1.41421 -0.00018-0.00018 0.02000 0.02000 3 0.00050 0.99950 0.99950 1.41350 0.01002-0.01018-0.01019 0.01755 T= 6.000000000000000E-002 1 0.00060 0.00000 0.00000 0.00060 0.01002 0.00002 0.00000 0.01002 2 0.99999 0.99999 0.00120 1.41421 -0.00021-0.00021 0.02000 0.02000 3 0.00060 0.99939 0.99939 1.41336 0.01002-0.01021-0.01023 0.01759 T= 7.000000000000001E-002 1 0.00070 0.00000 0.00000 0.00070 0.01002 0.00003 0.00000 0.01002 2 0.99999 0.99999 0.00140 1.41420 -0.00025-0.00025 0.02000 0.02000 3 0.00070 0.99929 0.99929 1.41321 0.01002-0.01025-0.01027 0.01764 T= 8.000000000000000E-002 1 0.00080 0.00000 0.00000 0.00080 0.01003 0.00003 0.00000 0.01003 2 0.99999 0.99999 0.00160 1.41420 -0.00028-0.00028 0.02000 0.02000 3 0.00080 0.99919 0.99919 1.41307 0.01003-0.01028-0.01031 0.01768 T= 9.000000000000000E-002 1 0.00090 0.00000 0.00000 0.00090 0.01003 0.00003 0.00000 0.01003 2 0.99999 0.99999 0.00180 1.41419 -0.00032-0.00032 0.02000 0.02001 3 0.00090 0.99909 0.99908 1.41292 0.01003-0.01032-0.01035 0.01773 T= 0.100000000000000 1 0.00100 0.00000 0.00000 0.00100 0.01004 0.00004 0.00000 0.01004 2 0.99998 0.99998 0.00200 1.41419 -0.00035-0.00035 0.02000 0.02001 3 0.00100 0.99898 0.99898 1.41277 0.01004-0.01035-0.01039 0.01777 T= 0.110000000000000 1 0.00110 0.00000 0.00000 0.00110 0.01004 0.00004 0.00000 0.01004 2 0.99998 0.99998 0.00220 1.41418 -0.00039-0.00039 0.02000 0.02001 3 0.00110 0.99888 0.99888 1.41263 0.01004-0.01039-0.01043 0.01782 T= 0.120000000000000 1 0.00120 0.00000 0.00000 0.00120 0.01004 0.00004 0.00000 0.01004 2 0.99997 0.99997 0.00240 1.41418 -0.00042-0.00042 0.02000 0.02001 3 0.00120 0.99877 0.99877 1.41248 0.01004-0.01042-0.01047 0.01786 T= 0.130000000000000 1 0.00130 0.00000 0.00000 0.00130 0.01005 0.00005 0.00000 0.01005 2 0.99997 0.99997 0.00260 1.41417 -0.00046-0.00046 0.02000 0.02001 3 0.00130 0.99867 0.99867 1.41233 0.01005-0.01046-0.01051 0.01791 T= 0.140000000000000 1 0.00140 0.00000 0.00000 0.00140 0.01005 0.00005 0.00000 0.01005 2 0.99997 0.99997 0.00280 1.41417 -0.00050-0.00050 0.02000 0.02001 3 0.00140 0.99857 0.99856 1.41218 0.01005-0.01050-0.01055 0.01795 T= 0.150000000000000 1 0.00150 0.00000 0.00000 0.00150 0.01005 0.00005 0.00000 0.01005 2 0.99996 0.99996 0.00300 1.41416 -0.00053-0.00053 0.02000 0.02001 3 0.00150 0.99846 0.99846 1.41203 0.01005-0.01053-0.01058 0.01800 T= 0.160000000000000 1 0.00160 0.00000 0.00000 0.00160 0.01006 0.00006 0.00000 0.01006 2 0.99995 0.99995 0.00320 1.41415 -0.00057-0.00057 0.02000 0.02002 3 0.00160 0.99835 0.99835 1.41188 0.01006-0.01057-0.01062 0.01805 T= 0.170000000000000 1 0.00171 0.00001 0.00000 0.00171 0.01006 0.00006 0.00000 0.01006 2 0.99995 0.99995 0.00340 1.41415 -0.00060-0.00060 0.02000 0.02002 3 0.00171 0.99825 0.99824 1.41173 0.01006-0.01060-0.01066 0.01809 T= 0.180000000000000 1 0.00181 0.00001 0.00000 0.00181 0.01006 0.00006 0.00000 0.01006 2 0.99994 0.99994 0.00360 1.41414 -0.00064-0.00064 0.02000 0.02002 3 0.00181 0.99814 0.99814 1.41158 0.01006-0.01064-0.01070 0.01814 T= 0.190000000000000 1 0.00191 0.00001 0.00000 0.00191 0.01007 0.00007 0.00000 0.01007 2 0.99994 0.99994 0.00380 1.41413 -0.00067-0.00067 0.02000 0.02002 3 0.00191 0.99804 0.99803 1.41143 0.01007-0.01067-0.01074 0.01818 T= 0.200000000000000 1 0.00201 0.00001 0.00000 0.00201 0.01007 0.00007 0.00000 0.01007 2 0.99993 0.99993 0.00400 1.41412 -0.00071-0.00071 0.02000 0.02002 3 0.00201 0.99793 0.99792 1.41128 0.01007-0.01071-0.01078 0.01823 Reached T= 0.200000000000000 Target T= 0.200000000000000 Time step= 1.000000000000000E-002 Nstep= 20 1 0.00201 0.00001 0.00000 0.00201 0.01007 0.00007 0.00000 0.01007 2 0.99993 0.99993 0.00400 1.41412 -0.00071-0.00071 0.02000 0.02002 3 0.00201 0.99793 0.99792 1.41128 0.01007-0.01071-0.01078 0.01823 Closest approach: 0.140874E+01

#### Preparing parameters[edit]

Another possible run is for the earth's orbit. For a circular orbit, taking the unit of distance being its radius, the unit of mass being the sun, and the unit of time being a year, G comes out as though *not* as a pure number, its has the usual dimensions. Finding a coherent set of values for the Sun-Earth-Moon system was frustrating as the internet is littered with different values. Referring to the CRC handbook (87^{'th} edition, 2006-7), G = 6·6742(10)±0·0031 E-11, Ms = 1·98844E30 Kg, Me = 5·9742E24 Kg, Mm = 7·3483E22 Kg, Re = 1AU = 1·49597870E11 metres (actually the semi-major axis), Rm = 3·844E8 metres. However, the handbook also contains varying values, such as for the mass of the sun. Notoriously, the value of G is both difficult to measure and varying in result. My copy of Resnick & Halliday (1966) has 6·670±0·015 for example.

The idea was to start with an initial state having the Sun, Earth and Moon all in a straight line along the x-axis, but after some ad-hoc messing about, confusions escalated to the degree that an ad-hoc calculation prog. was in order. This turns out to need a cube root function, and a proper solution for this raises again the utility of palindromic functions. Later Fortran supplies intrinsic functions such as EXPONENT(x) which returns the exponent part of the floating-point number, *x* and extracting this would be useful in devising an initial value for an iterative refinement calculation. Clearly, one wants power/3 for this, and so being able to write something like `EXPONENT(t) = EXPONENT(x)/3`

would help, along with similar usage of the FRACTION(x) function - omitting details such as the remainder when the power is divided by three. The same ideas arise with the square root function, though here, SQRT is already supplied. Alas, only the SUBSTR intrinsic function of pl/i has palindromic usage.

Calculate some parameters for the solar system of Sun, Earth and Moon.

IMPLICIT REAL*8 (A-Z) !No integers need apply.

CBRT(X) = SIGN(EXP(LOG(ABS(X))/3),X) !Crude. Fails for zero.

PI = 4*ATAN(1D0)

G = 6.674210D-11 !Gravitational constant, MKS units.

MS = 1.98844D+30 !Mass of the sun.

ME = 5.9742D+24 !Mass of the earth.

MM = 7.3483D+22 !Mass of the moon.

RE = 1.49597870D+11 !Radius of the earth's orbit around the sun. 1 Astronomical Unit = semi-major axis.

RM = 3.844D+8 !Radius of the moon's orbit around the earth.

YS = 31556925.9747D0 !Earth's tropical year, in seconds. This includes precession.

Y = YS/24/3600 !In days. This is *not* the proper orbit-repetition time!

WRITE (6,*) Y,"Tropical year, days."

WRITE (6,*) "Earth's orbit taken as circular, but using Rmax."

WRITE (6,*) 2*PI*SQRT(RE**3/(G*MS))/3600/24,"Year, in days"

WRITE (6,*) 2*PI*SQRT(RE**3/(G*(MS + ME)))/3600/24,"Me included."

RC = CBRT(G*(MS + ME)*YS**2/(4*PI**2))

WRITE (6,*) "Circular orbit of period one tropical year."

WRITE (6,*) RC,"Rc: Earth's circular orbit radius."

WRITE (6,*) RE,"Re: Actual semi-major axis."

WRITE (6,*) RE - RC,"Re - Rc"

OS = RC*ME/MS

WRITE (6,*) OS ,"Os: Sun's offset from CoM due to Earth at Rc."

VE = 2*PI*RC/YS

WRITE (6,*) VE,"Ve: Earth's circular orbit velocity."

WS = 2*PI*OS/YS

WRITE (6,*) WS,"Ws: Sun's circular orbit velocity due to Earth."

WRITE (6,*)

WRITE (6,*) RM,"Rm: radius of the moon's orbit around the earth."

TM = 2*PI*SQRT(RM**3/(G*(ME + MM)))/3600/24

WRITE (6,*) TM,"Tm: time for the moon's circular orbit, days."

OE = RM*MM/ME

WRITE (6,*) OE,"Oe: Earth's offset from CoM due to Moon at Rm."

VM = 2*PI*RM/(TM*3600*24)

WRITE (6,*) VM,"Vm: Moon's circular orbit velocity."

WE = 2*PI*OE/(TM*3600*24)

WRITE (6,*) WE,"We: Earth's circular orbit velocity due to Moon."

Combine the offsets and wobbles for the Sun-----Earth-Moon in a straight line

WRITE (6,*)

WRITE (6,*) " (CoM)"

WRITE (6,*) "Sun--0-----------------------------Earth--Moon-->x"

RC = CBRT(G*(MS + ME + MM)*YS**2/(4*PI**2))

WRITE (6,*) RC," Rc: Earth's circular orbit for Sun+Earth+Moon."

SX = -(ME*(RC - OE) + MM*(RC + RM))/MS

WRITE (6,*) SX," Sx: Sun's x-position, offset by Earth+Moon."

SVY = 2*PI*SX/YS

WRITE (6,*) SVY,"SVy: Sun's y-velocity."

EX = RC - OE

WRITE (6,*) EX," Ex: Earth's x-position, offset by Moon."

EVY = VE - WE

WRITE (6,*) EVY,"EVy: Earth's y-velocity."

MX = RC + RM

WRITE (6,*) MX," Mx: Moon's x-position."

MVY = VE + VM

WRITE (6,*) MVY,"MVy: Moon's y-velocity."

Convert to 'AU', being one earth orbit radius for a circular orbit taking one year.

WRITE (6,10)

10 FORMAT (/," Time in years, masses in suns, distances in 'AU'",

1 /,12X,"Solar masses x-position y-velocity.")

WRITE (6,11) "Sun",1.0,SX/RC,SVY/RC*YS

WRITE (6,11) "Earth",ME/MS,EX/RC,EVY/RC*YS

WRITE (6,11) "Moon",MM/MS,MX/RC,MVY/RC*YS

11 FORMAT (A6,3F18.14)

WRITE (6,*)

WRITE (6,*) 4*PI**2,"4Pi^2: G in AU/year units."

WRITE (6,*) 2*PI,"2Pi: distance per year for R = 1."

END

Notably, wanting a circular orbit to ease inspection of the results meant that the actual semi-major axis of the earth's elliptical orbit could not be used as in the first trials. Instead, the orbital period was taken as a year (wrongly so, as this includes the precession of the earth's axis of revolution, with a period of about 26,000 years) and the radius of a circular orbit having that period determined. This however is affected by whether the calculation is for the sun alone (so, a massless Earth), or, the sun and the earth together, or the sun, earth and moon together. As well, there is no z-action: the moon's orbital plane is deemed the same as that of the earth and it isn't. Results:

365.242198781250 Tropical year, days. Earth's orbit taken as circular, but using Rmax. 365.256591252027 Year, in days 365.256042552793 Me included. Circular orbit of period one tropical year. 149594089981.644 Rc: Earth's circular orbit radius. 149597870000.000 Re: Actual semi-major axis. 3780018.35565186 Re - Rc 449450.329086289 Os: Sun's offset from CoM due to Earth at Rc. 29785.1377845590 Ve: Earth's circular orbit velocity. 8.948842819120138E-002 Ws: Sun's circular orbit velocity due to Earth. 384400000.000000 Rm: radius of the moon's orbit around the earth. 27.2801498336002 Tm: time for the moon's circular orbit, days. 4728141.87673663 Oe: Earth's offset from CoM due to Moon at Rm. 1024.71419780640 Vm: Moon's circular orbit velocity. 12.6040429509235 We: Earth's circular orbit velocity due to Moon. (CoM) Sun--0-----------------------------Earth--Moon-->x 149594091824.393 Rc: Earth's circular orbit for Sun+Earth+Moon. -454978.599317466 Sx: Sun's x-position, offset by Earth+Moon. -9.058914206677031E-002 SVy: Sun's y-velocity. 149589363682.517 Ex: Earth's x-position, offset by Moon. 29772.5337416081 EVy: Earth's y-velocity. 149978491824.393 Mx: Moon's x-position. 30809.8519823654 MVy: Moon's y-velocity. Time in years, masses in suns, distances in 'AU' Solar masses x-position y-velocity. Sun 1.00000000000000 -0.00000304142091 -0.00001910981119 Earth 0.00000300446581 0.99996839352531 6.28052640251386 Moon 0.00000003695510 1.00256962019898 6.49934904809141 39.4784176043574 4Pi^2: G in AU/year units. 6.28318530717959 2Pi: distance per year for R = 1.

#### Sun-Earth-Moon[edit]

This leads to file SEM.dat, as follows:

3 Bodies 39.478417604357434 G = 4Pi**2: Solar mass = 1, Earth's circular equivalent-time orbit radius = 1, time unit = 1 year. 1.0, 0.001 Run time, time step. (years) -2 -2 -2, +2 +2 +2 xyzmin, xyzmax for plot scaling 1, -0.00000304142091 0 0, 0 -0.00001910981119 0 Solar Mass, xyz Position, xyz Velocity 3.00446581E-6, 0.99996839352531 0 0, 0 6.28052640251386 0 Earth completes one circle of radius 1 in 1 year, so V = 2Pi AU/year. 3.69551E-8, 1.00256962019898 0 0, 0 6.49934904809141 0 Sun-Earth-Moon in a straight line along the x-axis.

The initial results...

Star Trails. 3 bodies. 39.4784176043574 gravitational constant. 1.00000000000000 1.000000000000000E-003 Run time, time step. Euler/2'nd/Runge/Milne (1/2/3/4) 3 x y z R vx vy vz V 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 0.99997 0.00000 0.00000 0.99997 0.00000 6.28053 0.00000 6.28053 3 1.00257 0.00000 0.00000 1.00257 0.00000 6.49935 0.00000 6.49935 T= 1.000000000000000E-003 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 0.99995 0.00628 0.00000 0.99997 -0.03927 6.28041 0.00000 6.28053 3 1.00254 0.00650 0.00000 1.00256 -0.05678 6.49849 0.00000 6.49873 T= 2.000000000000000E-003 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 0.99989 0.01256 0.00000 0.99997 -0.07853 6.28007 0.00000 6.28056 3 1.00246 0.01300 0.00000 1.00254 -0.11343 6.49590 0.00000 6.49689 T= 3.000000000000000E-003 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 0.99979 0.01884 0.00000 0.99997 -0.11780 6.27949 0.00000 6.28060 3 1.00231 0.01949 0.00000 1.00250 -0.16981 6.49162 0.00000 6.49384 T= 4.000000000000000E-003 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 0.99965 0.02512 0.00000 0.99997 -0.15706 6.27869 0.00000 6.28065 3 1.00212 0.02598 0.00000 1.00245 -0.22579 6.48567 0.00000 6.48960 T= 5.000000000000000E-003 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 0.99948 0.03140 0.00000 0.99997 -0.19633 6.27765 0.00000 6.28072 3 1.00186 0.03246 0.00000 1.00239 -0.28125 6.47811 0.00000 6.48422 T= 6.000000000000000E-003 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 0.99926 0.03767 0.00000 0.99997 -0.23559 6.27638 0.00000 6.28080 3 1.00155 0.03893 0.00000 1.00231 -0.33606 6.46900 0.00000 6.47772 T= 7.000000000000000E-003 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 0.99901 0.04395 0.00000 0.99997 -0.27486 6.27488 0.00000 6.28090 3 1.00119 0.04540 0.00000 1.00222 -0.39013 6.45840 0.00000 6.47017 T= 8.000000000000000E-003 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 0.99871 0.05022 0.00000 0.99997 -0.31412 6.27314 0.00000 6.28100 3 1.00077 0.05185 0.00000 1.00212 -0.44335 6.44639 0.00000 6.46162 T= 9.000000000000001E-003 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 0.99838 0.05650 0.00000 0.99998 -0.35339 6.27117 0.00000 6.28112 3 1.00031 0.05829 0.00000 1.00200 -0.49563 6.43307 0.00000 6.45214 T= 1.000000000000000E-002 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 0.99801 0.06277 0.00000 0.99998 -0.39265 6.26897 0.00000 6.28125 3 0.99978 0.06472 0.00000 1.00188 -0.54689 6.41853 0.00000 6.44179 etc... T= 0.998000000000001 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 0.99995-0.01262 0.00000 1.00003 0.07901 6.28523 0.00000 6.28573 3 0.99728-0.01296 0.00000 0.99736 0.10301 6.07537 0.00000 6.07625 T= 0.999000000000001 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 1.00001-0.00633 0.00000 1.00003 0.03933 6.28558 0.00000 6.28570 3 0.99737-0.00688 0.00000 0.99739 0.07940 6.07848 0.00000 6.07899 T= 1.00000000000000 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 1.00003-0.00005 0.00000 1.00003 -0.00034 6.28565 0.00000 6.28565 3 0.99744-0.00080 0.00000 0.99744 0.05550 6.08257 0.00000 6.08283 Reached T= 1.00000000000000 Target T= 1.00000000000000 Time step= 1.000000000000000E-003 Nstep= 1000 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 1.00003-0.00005 0.00000 1.00003 -0.00034 6.28565 0.00000 6.28565 3 0.99744-0.00080 0.00000 0.99744 0.05550 6.08257 0.00000 6.08283 Closest approach: 0.260123E-02

Using instead the first-order method, a year into the calculation the results are not so good:

T= 0.998000000000001 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 1.02057-0.19728 0.00000 1.03947 1.16628 6.04742 0.00000 6.15886 3 1.01296-0.09530 0.00000 1.01743 0.58893 6.20845 0.00000 6.23632 T= 0.999000000000001 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 1.02172-0.19123 0.00000 1.03947 1.13041 6.05436 0.00000 6.15898 3 1.01353-0.08909 0.00000 1.01744 0.55096 6.21201 0.00000 6.23640 T= 1.00000000000000 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 1.02284-0.18518 0.00000 1.03946 1.09449 6.06108 0.00000 6.15911 3 1.01406-0.08288 0.00000 1.01744 0.51297 6.21534 0.00000 6.23647 Reached T= 1.00000000000000 Target T= 1.00000000000000 Time step= 1.000000000000000E-003 Nstep= 1000 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 1.02284-0.18518 0.00000 1.03946 1.09449 6.06108 0.00000 6.15911 3 1.01406-0.08288 0.00000 1.01744 0.51297 6.21534 0.00000 6.23647 Closest approach: 0.163788E-02

The earth and moon go wandering. With too large a step size, the situation between steps can be very different from those at the start and stop of a step. The predictor-corrector scheme does better:

Star Trails. 3 bodies. 39.4784176043574 gravitational constant. 1.00000000000000 1.000000000000000E-003 Run time, time step. Euler/2'nd/Runge/Milne (1/2/3/4) x y z R vx vy vz V 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 0.99997 0.00000 0.00000 0.99997 0.00000 6.28053 0.00000 6.28053 3 1.00257 0.00000 0.00000 1.00257 0.00000 6.49935 0.00000 6.49935 Bmax= 8.092753583528147E-010 1 0.00000 0.00000 0.00000 0.00000 0.00000-0.00002 0.00000 0.00002 2 1.00003-0.00007 0.00000 1.00003 0.00061 6.28576 0.00000 6.28576 3 0.99733 0.00007 0.00000 0.99733 -0.01260 6.07555 0.00000 6.07556 DTmin= 3.906250000000000E-006 , DTmax= 2.000000000000000E-003 Reached T= 1.00001171874997 Target T= 1.00000000000000 Time step= 5.000000000000000E-004 Nstep= 1755 Closest approach: 0.260123E-02

Notice that it tries an increased time step but also a much smaller time step, in the end requiring 1,755 steps. Part of the project was to assess various schemes for changing the step size and on what basis. This also means that the calculation may well not produce results at desired times, not just because repeated addition of a floating-point number such as 0.001 may not produce a value such as 5.755 exactly, but also because the varying step size might hop past it.

More generally, when two bodies approach closely the resulting curvature forces a smaller step size, which is wasted when calculating details for widely-separated bodies. One ploy is to treat such pairs via two-body formulae (in their centre-of-mass coordinates) for the time of their closeness, retaining a larger time step for the rest of the calculation. This can be further generalised into clumping nearby bodies into a single mass when considering their effect on far-distant bodies. In all of this, the administration requirements become ever-more complex.

## J[edit]

Although in general the n-body problem doesn’t have an explicit solution, certain configurations do, and these 'nice' configurations can be used to validate and perform unbiased comparisons of numeric solvers. A fairly simple configuration with an exact explicit solution is a radially symmetric system of three bodies orbiting at constant velocity around a central axis:

- Given:
- n = 3
*-- number of bodies* - g = 1
*-- gravitational constant* - m = 1
*-- mass per body* - r = 1
*-- radius*

- Then:
- f = (1/3)^(1/2)
*-- centripetal force* - a = (1/3)^(1/2)
*-- centripetal acceleration* - v = (1/3)^(1/4)
*-- velocity*

This is a variation on the [Klemperer rosette[1]].

#### Implementing the physics

- The problem space is a collection of objects.
- Object is { id, mass, position, velocity }.
- Position is { x, y, z }.
- Velocity is { vx, vy, vz }.

g =: 1

I =: 0&{"1

M =: 1&{"1

IM =: 0 1&{"1

D =: 3 : 0"1

2 3 4{y

:

2 3 4{y-x

)

V =: 5 6 7&{"1

D3 =: 4 : '(%:+/*:x D y)^3'"1

F =: 3 : 0"1

y F/y

:

g*(M x)*(M y)*(y D x) % (x D3 y)

)

A =: 3 : 0

ff =. y F/y

f =. +/ff

f % (M y)

)

NEXT =: 4 : 0

dt =. x

im =. IM y

p0 =. D y

v0 =. V y

f =. +/(F/~y)

a =. f%M y

v1 =. v0 + dt * a

p1 =. p0 + dt * (v0 + v1)%2

z =. |: ((|: im),(|: p1),(|: v1))

out =: out,D z

z

)

#### Integrator

Iterate over a time interval. Plot the results (x and y coordinates over time).

dt =: 0.001

maxn =: 20000

require'plot'

ITER =: 3 : 0

maxn ITER y

:

out =: (0,(#y),3)$0

dt NEXT^:x y

plot 0 1 { |: out

)

#### Configurations

##### Configuration generator

Generate a radially symmetrical configuration for a given number of bodies and an initial rotational velocity

require'trig'

GEN =: 3 : 0

1 GEN y

:

m =. 1

r =. 1

p =. r*(|:(2,y)$(cos,sin)(i.y)*2*pi%y),.0

v =. x*(|:(2,y)$(cos,sin)(pi%2)+(i.y)*2*pi%y),.0

(i.y),.(y#m),.p,.v

)

##### Generate a few cases.

The 3-body static equilibrium case is proposed as the basis for validation, and for comparison of alternative methods.

cases =: 3 : 0

case =. 3 : 'y;".y'

r =. 0 2 $ a:

r =. r, case 'GEN 4'

r =. r, case '0.75 GEN 3'

r =. r, case '((1%3)^(1%4)) GEN 3'

)

<[cases''

+------------------------------------------------------------------------------+

¦GEN 4 ¦0 1 1 0 0 6.12323e_17 1 0¦

¦ ¦1 1 6.12323e_17 1 0 _1 1.22465e_16 0¦

¦ ¦2 1 _1 1.22465e_16 0 _1.83697e_16 _1 0¦

¦ ¦3 1 _1.83697e_16 _1 0 1 _2.44929e_16 0¦

+-------------------+----------------------------------------------------------¦

¦0.75 GEN 3 ¦0 1 1 0 0 4.59243e_17 0.75 0 ¦

¦ ¦1 1 _0.5 0.866025 0 _0.649519 _0.375 0 ¦

¦ ¦2 1 _0.5 _0.866025 0 0.649519 _0.375 0 ¦

+-------------------+----------------------------------------------------------¦

¦((1%3)^(1%4)) GEN 3¦0 1 1 0 0 4.65265e_17 0.759836 0 ¦

¦ ¦1 1 _0.5 0.866025 0 _0.658037 _0.379918 0 ¦

¦ ¦2 1 _0.5 _0.866025 0 0.658037 _0.379918 0 ¦

+------------------------------------------------------------------------------+

##### Static equilibrium case.

Bodies orbit with constant kinetic energy.

maxn ITER z0 =: ((1%3)^(1%4)) GEN 3

[*sinusoidal curves of constant amplitude*[2]]

Determine orbital period, and compare configuration after two rotations vs. initial position.

v =: (1%3)^(1%4)

dt =: 0.001

<[rot2 =: (>. (4 * pi) % (v * dt)) NB. two rotations

+-----+

¦16539¦

+-----+

rot2 ITER v GEN 3

<[({.out)-{:out

+------------------------+

¦_0.00559932 0.0897287 0¦

| _0.0749077 _0.0497135 0¦

¦ 0.080507 _0.0400152 0¦

+------------------------+

##### Dynamic equilibrium case.

Within a narrow range of initial velocities, the bodies orbit, maintaining symetry and swapping kinetic and potential energy back and forth. The ideal system is stable, but an integrating solver will not be, due to numerical precision.

maxn ITER z1 =: 0.6 GEN 3

[*sinusoidal curves of varying amplitudes* [3]]

##### Perturbed case.

Any perturbation from symmetry, however small, will eventually cause the bodies to escape. Note that the effect demonstrated here is due to the physics of the problem, independently of the numerical precision and increment size of the solver.

maxn ITER z2 =: (0.001+6{0{z0)(<0 6)}z0

[*curves start out nice then get jumbled and diverge* [4]]

##### n>3

We can generate and execute tests for radially symmetrical configurations of any number of bodies, run the simulations, and compare the results with an explicitly determined result:

r =. 4 : '%:+/*:(1-cos 2*pi*y%x),sin 2*pi*y%x' NB. distance to nth body

f0 =. 4 : '1%((x r y)^2)' NB. force due to nth body

f =. 4 : '(x f0 y)*(1-cos 2*pi*y%x)%x r y'"0 NB. centripetal component of force

a =. 3 : '+/(y f i. y)' NB. centripetal acceleration

GENM =: 3 : 0"0 NB. accumulated discrepancy over one rotation for n bodies

N =: y NB. number of bodies

VS =: (a N)^0.5 NB. velocity for static equilibrium

TS =: 2*pi%VS NB. rotational time

dt =: 0.001 NB. time increment for simulation

IS =: >.2*pi%VS*dt NB. number of steps for simulation

IS ITER VS GEN N NB. run the simulation

E =: >./>./({.out)-{:out NB. get max difference between initial and final state

N;VS;TS;E NB. return the results

)

smoutput ' n';'velocity';' period';' error'

smoutput GENM 3+i.17

+-----------------------------+

¦ n¦velocity¦ period¦ error¦

+--+--------+-------+---------¦

¦3 ¦0.759836¦8.26914¦0.0225829¦

+--+--------+-------+---------¦

¦4 ¦0.978318¦6.42243¦0.0293585¦

+--+--------+-------+---------¦

¦5 ¦1.17319 ¦5.35563¦0.0354237¦

+--+--------+-------+---------¦

¦6 ¦1.3518 ¦4.64803¦0.0400031¦

+--+--------+-------+---------¦

¦7 ¦1.51815 ¦4.13872¦0.0467118¦

+--+--------+-------+---------¦

¦8 ¦1.67477 ¦3.75166¦0.0505916¦

+--+--------+-------+---------¦

¦9 ¦1.82341 ¦3.44584¦0.0553926¦

+--+--------+-------+---------¦

¦10¦1.96531 ¦3.19704¦0.0586112¦

+--+--------+-------+---------¦

¦11¦2.10142 ¦2.98997¦0.0650293¦

+--+--------+-------+---------¦

¦12¦2.23248 ¦2.81445¦0.0669045¦

+--+--------+-------+---------¦

¦13¦2.35906 ¦2.66342¦0.0706303¦

+--+--------+-------+---------¦

¦14¦2.48166 ¦2.53185¦0.0763112¦

+--+--------+-------+---------¦

¦15¦2.60067 ¦2.41599¦0.0802535¦

+--+--------+-------+---------¦

¦16¦2.71641 ¦2.31305¦0.0802621¦

+--+--------+-------+---------¦

¦17¦2.82918 ¦2.22085¦0.0864611¦

+--+--------+-------+---------¦

¦18¦2.93922 ¦2.13771¦0.0898613¦

+--+--------+-------+---------¦

¦19¦3.04673 ¦2.06227¦0.0916438¦

+-----------------------------+

##### Sensitivity to discretization

Although the static-equilibrium configuration is stable, its simulation is not, due to discretization and numerical precision. We can examine the sensitivity to time increment, taking position error after one rotation as a stability metric:

GENDT =: 3 : 0"0

dt =: y

GENDT0 3

)

GENDT0 =: 3 : 0"0

N=:y

VS =: (a N)^0.5

IS=:>.2*pi%VS*dt

IS ITER VS GEN N

E =: >./>./({.out)-{:out

dt;E

)

smoutput 'dt ';'error '

smoutput (GENDT"0) 10^_4+i.5

+-----------------+

¦dt ¦error ¦

+-----------------+

¦0.0001¦0.00227764¦

+------+----------¦

¦0.001 ¦0.0225829 ¦

+------+----------¦

¦0.01 ¦0.228495 ¦

+------+----------¦

¦0.1 ¦1.54637 ¦

+------+----------¦

¦1 ¦6.0354 ¦

+-----------------+

## K[edit]

The simulation is meant to be viewed in-browser, using a Javascript implementation of (a nice subset of) K, plus graphics primitives, here: [http://johnearnest.github.io/ok/ike/ike.html]; just paste the code into the code window and push the go button. Playing with the parameters in this interactive and immediate environment gives a feel for the system beyond what you are likely to get from tables or graphs.

#### configuration generator

The configurations shown here are based on those in the J task.

sq:{pow[x;2]};sqrt:{pow[x;0.5]};pi:3.14159265 / math stuff

cs:{(cos 2*pi*x%y;sin 2*pi*x%y)} / positions

sc:{(sin 2*pi*x%y;-cos 2*pi*x%y)} / velocities

#### static equilibrium configuration

The system is stable for an initial velocity for which centrifugal and gravitational forces are in balance

n::3;m::n#1;p::cs\:[!n;n];v::sc\:[!n;n] / count, masses, positions, velocities

g::1;vs::pow[1%3;1%4];v*::vs / gravitational constant, initial velocity

t::0;dt::0.01;rot:2 / time, time increment, rotational period

#### dynamic equilibrium

Within a narrow range of initial velocities the system oscillates between more and less kinetic and potential energy.

/ as for static case, plus:

/ v*::0.7;rot::2

#### unstable

The slightest perturbation from symmetry causes the system to become unstable.

/ as for static case, plus:

/ v+::(0 0;0 0;0 0.0001);rot:4 / wait for it ... wait for it ...

#### physics

DV:{p[x]-p[y]};DS:{sqrt[+/sq'DV[x;y]]} / nth body position vector and magnitude

A1:{$[x=y;0;g*m[y]*DV[x;y]%pow[DS[x;y];3]]} / nth body centripetal acceleration component

A:{+/{A1/:[x;!n]}'!n} / total centripetal acceleration

tmax::2*pi*rot%vs / duration of simulation

N:{[dt] / increment

v0:v;v+::dt*A[];p+::dt*0.5*v+v0 / update velocities and positions

t+::dt;$[t>tmax;do[msg::,"[";tick::{}];0] / update time, detect end of time

}

#### events and graphics

s::n#3 / size of sprite bitmap

sc:1.25;P::{w*0.5*1+(1%sc)*x} / display units

wh::(w;h);C::wh%2 / viewbox

RGB:("#f00";"#0f0";"#00f") / color palette

BW:("#000";"#fff") / monochrome palette

msg::,":" / initial position marker

tick::{N[dt]} / time increment callback

draw::{ / display callback

r:(,:'P[p]-s%2),'`RGB,',:'(2#'s)#'!n / plot the bodies

r,:,(P[1 0];BW;~,/'[email protected]`i$msg) / plot the initial position marker

}

#### 3D

Although the animation is limited to two dimensions, we can verify that the implementation really does support three, by changing the axis of rotation in the initialization function:

cs:{(0;cos 2*pi*x%y;sin 2*pi*x%y)} / x-axis

sc:{(0;sin 2*pi*x%y;-cos 2*pi*x%y)} / objects on screen move up and down

cs:{(cos 2*pi*x%y;0;sin 2*pi*x%y)} / y-axis

sc:{(sin 2*pi*x%y;0;-cos 2*pi*x%y)} / objects on screen move left and right

cs:{(cos 2*pi*x%y;sin 2*pi*x%y;0)} / z-axis

sc:{(sin 2*pi*x%y;-cos 2*pi*x%y;0)} / objects on screen go round and round

## Octave[edit]

We'll show only the positions in the output.

global G = 6.674e-11; # gravitational constant

global au = 150e9; # astronomical unit

global ma = 2e30; # mass of the Sun

global mb = 6e24; # mass of Earth

global mc = 7.34e22; # mass of the Moon

function ret = ABCdot(ABC, t)

global G au ma mb mc

ret = ABC;

ret(1:9) = ABC(10:18);

a = ABC(1:3);

b = ABC(4:6);

c = ABC(7:9);

ab = norm(a - b);

bc = norm(b - c);

ca = norm(c - a);

ret(10:12) = G*(mb/ab^3*(b - a) + mc/ca^3*(c - a));

ret(13:15) = G*(ma/ab^3*(a - b) + mc/bc^3*(c - b));

ret(16:18) = G*(ma/ca^3*(a - c) + mb/bc^3*(b - c));

endfunction;

seconds_in_a_month = 60*60*24*27;

seconds_in_a_year = 60*60*24*365.25;

t = (0:0.05:1);

ABC = vec(

[

0 0 0 # Sun position

au 0 0 # Earth position

0.998*au 0 0 # Moon position

0 0 0 # Sun speed

0 2*pi*au/seconds_in_a_year 0 # Earth speed

0 2*pi*(au/seconds_in_a_year + 0.002*au/seconds_in_a_month) 0 # Moon speed

]'

);

disp(lsode('ABCdot',ABC, t)(1:20,1:9));

- Output:

0.0000e+00 0.0000e+00 0.0000e+00 1.5000e+11 0.0000e+00 0.0000e+00 1.4970e+11 0.0000e+00 0.0000e+00 2.2520e-11 1.8822e-19 0.0000e+00 1.5000e+11 1.4933e+03 0.0000e+00 1.4970e+11 1.5337e+03 0.0000e+00 9.0080e-11 1.3338e-18 0.0000e+00 1.5000e+11 2.9865e+03 0.0000e+00 1.4970e+11 3.0673e+03 0.0000e+00 2.0268e-10 3.1665e-18 0.0000e+00 1.5000e+11 4.4798e+03 0.0000e+00 1.4970e+11 4.6010e+03 0.0000e+00 3.6032e-10 6.7638e-18 0.0000e+00 1.5000e+11 5.9731e+03 0.0000e+00 1.4970e+11 6.1347e+03 0.0000e+00 5.6300e-10 1.2128e-17 0.0000e+00 1.5000e+11 7.4663e+03 0.0000e+00 1.4970e+11 7.6683e+03 0.0000e+00 8.1072e-10 1.9549e-17 0.0000e+00 1.5000e+11 8.9596e+03 0.0000e+00 1.4970e+11 9.2020e+03 0.0000e+00 1.1035e-09 3.0116e-17 0.0000e+00 1.5000e+11 1.0453e+04 0.0000e+00 1.4970e+11 1.0736e+04 0.0000e+00 1.4413e-09 4.3249e-17 0.0000e+00 1.5000e+11 1.1946e+04 0.0000e+00 1.4970e+11 1.2269e+04 0.0000e+00 1.8241e-09 1.0985e-16 0.0000e+00 1.5000e+11 1.3439e+04 0.0000e+00 1.4970e+11 1.3803e+04 0.0000e+00 2.2520e-09 1.8655e-16 0.0000e+00 1.5000e+11 1.4933e+04 0.0000e+00 1.4970e+11 1.5337e+04 0.0000e+00 2.7249e-09 2.7013e-16 0.0000e+00 1.5000e+11 1.6426e+04 0.0000e+00 1.4970e+11 1.6870e+04 0.0000e+00 3.2429e-09 3.6059e-16 0.0000e+00 1.5000e+11 1.7919e+04 0.0000e+00 1.4970e+11 1.8404e+04 0.0000e+00 3.8059e-09 4.5794e-16 0.0000e+00 1.5000e+11 1.9412e+04 0.0000e+00 1.4970e+11 1.9938e+04 0.0000e+00 4.4139e-09 5.6217e-16 0.0000e+00 1.5000e+11 2.0906e+04 0.0000e+00 1.4970e+11 2.1471e+04 0.0000e+00 5.0670e-09 6.7328e-16 0.0000e+00 1.5000e+11 2.2399e+04 0.0000e+00 1.4970e+11 2.3005e+04 0.0000e+00 5.7651e-09 7.9128e-16 0.0000e+00 1.5000e+11 2.3892e+04 0.0000e+00 1.4970e+11 2.4539e+04 0.0000e+00 6.5083e-09 9.1616e-16 0.0000e+00 1.5000e+11 2.5386e+04 0.0000e+00 1.4970e+11 2.6072e+04 0.0000e+00 7.2965e-09 1.0479e-15 0.0000e+00 1.5000e+11 2.6879e+04 0.0000e+00 1.4970e+11 2.7606e+04 0.0000e+00 8.1297e-09 1.1866e-15 0.0000e+00 1.5000e+11 2.8372e+04 0.0000e+00 1.4970e+11 2.9140e+04 0.0000e+00

## Pascal[edit]

This was written in Turbo Pascal, prompted by an article in an astronomy magazine about the orbit of an asteroid about a planet about a sun. It expects to receive command-line parameters for the run, and wants to use the screen graphics to animate the result. Alas, the turbo .bgi protocol no longer works on modern computers that have screens with many more dots. It contains options to select different methods for computation. An interesting point is that if the central mass is fixed in position (rather than wobbling about the centre of mass, as it does), then the Trojan orbit positions are not stable.

{$N+ Crunch only the better sort of numbers, thanks.}

{$M 50000,24000,24000] {Stack,minheap,maxheap.}

Program Swirl; Uses Graph, Crt;

{ Calculates and displays the orbit of an asteroid around a sun and a planet,

as described in the article by C.C. Foster, in Sky & Telescope, September 1994,

which included the source code of a programme written in basic, so some changes here.

The situation: the asteroid is at X = (AX,AY) with velocity V = (VAX,VAY)

and suffers acceleration from the central sun and the steadily orbiting planet.

If the acceleration at the start of a time step is A = (acX,acY)

then X = (A.dt/2 + V)dt + X at the end of a time step, A presumed constant.

and V = A.dt + V (so update X before altering V)

But that is a first-order method. The whole point is that A is not constant.

Accordingly, use this X as an estimate of the position at the end of a time step,

and calculate a fresh A at the new position. Use the average of this and the

first A to perform the actual step.

The next step in the escalation is the classic fourth-order Runge-Kutta method

that alas involves a lot of repetitious code, or else trickery with loops.

Whichever, there are four evaluations per step: at the start, then using that to

probe ahead a half step, then again a half step, then a full step probe, then

a weighted average to perform the step. While this gives fifth-order accuracy

for each step, and thus fourth order accuracy at the end of a sequence of steps,

this is rather painful. This is to say that halving the step size would

reduce the upper bound of the error in a single step by a factor of 2**5, and

that for the result after a sequence of steps by a factor of 2**4. However,

this requires an upper bound on the value of the higher order derivatives of

the function over the entirety of the area of interest, and for this problem,

it does not exist, for the acceleration is unbounded above as a particle approaches

a mass. It varies by 1/r**2 and as r approaches zero, there can be surprises.

Unfortunately, the next stages, say Milne's Predictor-Corrector (which has

the advantage of maintaining the recent history, just what procedure Apastron

needs), followed by the extrapolation to zero stepsize methods of Bulirsch and

Stoer require heavy administration.

This programme uses a few tricks to enable a step size of 1 so that no actual

multiplication need be made, and likewise takes G as being one. It is the work

of moments to change the step size with these methods, but the difficulty lies

in when to change the step size and by how much. Remember that with a large step,

singularities can be stepped over without noticing them, whereas a small step

involves not just more work but also accumulating round-off. The predictor-

corrector methods offer a good check on the step size (if you bother to compare

the difference between the predicted and corrected values) but require a lot

of administrative effort when changing it. However, this is not all to the bad

as you might as well add some checks that the step size is not being changed

too enthusiastically while you're at it. The problem is that the step size is

being changed according to the behaviour of the results, but the behaviour of

the results depends on the step size as well as the current region.

Rather than pursue this matter (and make the effort) I have retained the

fixed size step (but perhaps an option sometime later?) to allow comparison

with the original article in Sky & Telescope (and also to make things easy

for procedure Apastron). For close comparison you will need to fix the central

sun (i.e. Wobble is false) and use Euler's method. The article however uses

units convenient to its programme: the planet is at radius 200, the asteroid

at 100, both starting on the x-axis. The planet has a mass of 5 to the sun's

mass of 70 (which is more like that of a companion sun than a planet), with

a year of 920 steps and the asteroid has an initial velocity in the y-direction

of 0.7. (See why you shouldn't use full stops as decimal points!)

These conditions result in the asteroid having successive periods of 466,

452, 477, 437, etc steps in the original article, and the invocation

WANDER 920 0.0714285 0.5 0 0.0 1.1 r f

meaning

920 steps to the planet's orbit,

a planetary mass of 5/70 of the sun

asteroid's radial distance being half that of the planet

at zero degrees (i.e. on the x-axis)

zero radial velocity

circular velocity of 1.1 (an estimate from trials) times that of a circular

orbit at the asteroid's initial radius

Runge-Kutta method

Sun fixed at the centre of mass

Is about as close as I can get, giving an asteroid period of 390 or so.

But the orbit is very unstable. With Euler's method, the picture is even worse

with the asteroid spiralling outwards. I think that this is due to too large a

step size and a low-order method, as test runs with the planetary mass set zero

behave better, provided that there are many steps to an asteroid orbit and that it

doesn't pass close to a mass. StepSize control would help, but it would mean

more programming effort. Alternatively, I could have blundered, but I've stared

at this so long that if there is a blunder, I can't see it.

I prefer to believe that the step size is the problem. In a different programme

on a different machine, a circular orbit of 90 mins about the earth (just above

the atmosphere) with full units for G, etc, and a step size of one second (thus

no multiply) so that there were 5,400 steps to an orbit, Euler's method after

one orbit had the radial distance 2% high and increasing steadily, whereas the

2'nd order Euler's method with a two second step size (so the same number of

evaluations, as two per step) and also no multiplication (i.e. (a + a')/2 *dt) gave

an error of one part on 6,000,000 (the precision of the 32-bit floating point

numbers on an IBM1130) and oscillating.

On the other hand, a simple arrangement such as the asteroid in a Trojan

orbit with respect to the planet works well enough. Here the asteroid is in the

same orbit as the planet, but leading (or lagging) by sixty degrees, thus:

WANDER 920 0.001 1 +60 0 1

Test runs give an unstable orbit unless the planet is much smaller than 5/70

of the sun's mass, and also require that the sun not be fixed to the centre of

mass. Having lots more steps per orbit doesn't change this. Thus, even though

the sun's wobble is small, it is necessary and anyway, physically correct.

In the case of a more massive planet, the obvious extension is to three equal

masses in mutual orbit and invoking symmetry, a nice circular orbit would have

them spaced at a hundred and twenty degrees.

Incidentally, no attempt is made to spot occasions when the asteroid's position

is exactly that of some attracting mass, so zero divisions can cause dismay.

Except that they won't happen as a result of a computed step unless you are very

unlucky, especially if the floating-point hardware is available, with eighteen

digits that must all coincide. On the other hand, you can easily specify an

initial position that will provoke a division by zero so if that is what you

have asked for, then that is what you will receive.}

{Perpetrated by R.N.McLean (whom God preserve), Victoria University, Feb. VMMI.}

Type Grist = {$IFOPT N+} extended {$ELSE} real {$ENDIF};

Type Chaff = {$IFOPT N+} single {$ELSE} real {$ENDIF};

const esc = #27;

var colour,lastcolour,xmax,ymax,txtheight: integer;

var stretch: grist;

Var Asitwas: Word;

Type AnyString = string[80];

Function Min(i,j: integer): integer; BEGIN if i <= j then min:=i else min:=j; END;

Function Max(i,j: integer): integer; BEGIN if i >= j then max:=i else max:=j; END;

Procedure Beep(f,t: integer); BEGIN Sound(f); delay(t); NoSound; END;

Procedure Croak(Gasp: string); {A lethal word.}

BEGIN

TextMode(Asitwas);

WriteLn;

Writeln(Gasp);

HALT;

END;

Function MemGrab(Var p: pointer;s: word): boolean;

BEGIN

p:=nil;

if s > 0 then GetMem(p,s);

MemGrab:=p<>nil;

END;

Procedure MemDrop(Var p: pointer;s: word);

BEGIN

if (s > 0) and (p <> nil) then FreeMem(p,s);

p:=nil; {Oh that all usage would first check!}

END;

FUNCTION Trim(S : anystring) : anystring;

var L1,L2 : integer;

BEGIN

L1 := 1;

WHILE (L1 <= LENGTH(S)) AND (S[L1] = ' ') DO INC(L1);

L2 := LENGTH(S);

WHILE (S[L2] = ' ') AND (L2 > L1) DO DEC(L2);

IF L2 >= L1 THEN Trim := COPY(S,L1,L2 - L1 + 1) ELSE Trim := '';

END; {Of Trim.}

FUNCTION Ifmt(Digits : longint) : anystring;

var S : anystring;

BEGIN

STR(Digits,S);

Ifmt := Trim(S);

END; { Ifmt }

FUNCTION Ffmt(Digits: grist; Width,Decimals: integer): anystring;

var

S : anystring;

L : integer; { a finger }

BEGIN

if digits = 0 then begin Ffmt:='0'; exit; end; {Mumble.}

IF Abs(Digits) < 0.0000001 THEN STR(Digits,S)

ELSE STR(Digits:Width:Decimals,S);

s:=trim(s);

if copy(s,1,2) = '0.' then s:=copy(s,2,Length(s) - 1);

if copy(s,1,3) = '-0.' then s:=concat('-',copy(s,3,Length(s) - 2));

l:=Length(s);

if pos('.',s) > 0 then while (l > 0) and (s[l] = '0') do l:=l - 1;

if s[l] = '.' then l:=l - 1; {No non-zero fractional part!}

S := COPY(S,1,L); {Drop trailing boredom.}

l:=Pos('.',S); {Avoid sequences such as 3.1415.}

if l > 0 then s[l]:=chr(249); {By using decimal points, please.}

Ffmt := S;

END; { Ffmt }

Function Atan2(x,y: grist): grist;

var a: grist;

BEGIN

if x = 0 then

if y > 0 then atan2:=pi/2

else if y < 0 then atan2:=-pi/2

else atan2:=0

else if x > 0 then if y >= 0 then atan2:=arctan(y/x) else atan2:=2*pi + arctan(y/x)

else atan2:=pi - arctan(-y/x);

END;

Procedure PrepareTheCanvas;

Var mode: integer;

Var SavedResultCode: integer;

Var ax,ay: word;

Function BGIFound(Here: Anystring): boolean;

Var driver: integer;

BEGIN

Driver:=Detect;

InitGraph(driver,mode,Here);

SavedResultCode:=GraphResult; {Mumble!}

BGIFound:=SavedResultCode = 0; {Missing file gives -3.}

END; {Of BGIFound.}

Var aplace: anystring;

Var Palette: paletteType;

BEGIN

Mode:=0;

if not BGIFound('') then

if not BGIFound('C:\TP7\BGI') then

if not BGIFound('D:\TP4') then

begin

WriteLn;

WriteLn('Trouble with Borland''s Graphic Interface!');

WriteLn('Message: ',GraphErrorMsg(SavedResultCode));

WriteLn('I''ve tried "", "C:\TP4", and "D:\TP4"...');

Repeat

Write('Another place to look? (e.g. a:\bgiset)');

ReadLn(Aplace); {As opposed to Read. weird.}

If Aplace = '' then

begin

writeln;

write('No .bgi, no piccy.');

halt;

end;

until BGIFound(Aplace);

end;

SetViewPort(0,0,GetMaxX,GetMaxY,True); {Clipping on, thanks. No apparent slowdown.}

SetGraphMode(Mode);

GetPalette(Palette);

LastColour := Palette.Size - 1; {Colours are 0:15, 16 thereof.}

if LastColour <= 0 then LastColour:=1;

xmax:=GetMaxX; ymax:=GetMaxY;

txtheight:=TextHeight('I');

GetAspectRatio(ax,ay); stretch:=ay/ax; {I hope this handles the screen's physical dimensions}

END; {as well as the number of dots in the x and y directions.}

Procedure Splot(x,y:integer; bumf: anystring);

var l: integer;

BEGIN

l:=TextWidth(Bumf);

if x + l > xmax then x:=xmax - l;

SetFillStyle(EmptyFill,Black); Bar(x,y,x + l,y + txtheight - 1);

OutTextXY(x,y,bumf);

END;

Procedure FlashHint(msg: string);

var i,h,ph,pl,y0,rt: integer;

var l1,l2,nl: integer;

var board: pointer; BoardSize: word;

BEGIN

nl:=0; pl:=0; l2:=1;

repeat

l1:=l2;

while (l2 <= Length(msg)) and (msg[l2] <> '%') do l2:=l2 + 1;

nl:=nl + 1; pl:=max(pl,TextWidth(Copy(Msg,l1,l2 - l1)));

l2:=l2 + 1;

until l2 > Length(msg);

h:=TextHeight('Idiotic') + 1;

ph:=1 + nl*h;

y0:=GetMaxY div 6 + h;

BoardSize:=ImageSize(0,0,pl,ph);

If BoardSize <= 0 then begin Beep(6666,200); exit; end;

if KeyPressed then exit; {Last chance to cut and run.}

if not memgrab(board,BoardSize) then

begin

OutTextXY(0,y0,'Memory shortage...');

Beep(8888,200);

exit; {Thus save on too many "if"s below.}

end;

GetImage(0,y0,pl,y0 + ph,board^); {Doesn't check that Board is not nil!&^$#@!}

SetFillStyle(EmptyFill,Black); Bar(0,y0,pl,y0 + ph);

SetColor(White);

i:=0; l2:=1;

repeat

l1:=l2;

while (l2 <= Length(msg)) and (msg[l2] <> '%') do l2:=l2 + 1;

OutTextXY(0,y0 + 1 + i*h,Copy(Msg,l1,l2 - l1));

i:=i + 1; l2:=l2 + 1;

until l2 > Length(msg);

rt:=28 + Length(msg); {Read time, more for longer messages.}

i:=0; while not KeyPressed and (i < rt) do

begin

delay(100);

i:=i + 1;

end;

PutImage(0,y0,board^,NormalPut);

MemDrop(board,BoardSize);

END;

PROCEDURE LurkWith(Msg: string);

Var t,w: integer;

BEGIN

w:=111;

repeat

for t:=0 to w do

begin

if KeyPressed then exit;

delay(100);

end;

FlashHint(msg);

w:=222;

until KeyPressed;

END;

Function KeyFondle: char; {Equivalent to ReadKey, except...}

Var ticker: integer; { after a delay, it gives a hint.}

BEGIN {Screen and keyboard are connected by a computer...}

Ticker:=666; {A delay counter.}

While not KeyPressed do {If nothing has happened, }

begin { twiddle my thumbs.}

ticker:=ticker - 1; {My patience is being exhausted.}

if ticker > 0 then Delay(60) {Another irretrievable loss.}

else FlashHint('Press a key!'); {Hullo sailor!}

end; {Eventually, a key is pressed.}

KeyFondle:=ReadKey {Yum.}

END;

Var Wobble,Dotty: boolean;

Var Style: integer;

Var Year,PM,SM,AR,AA,ARV,ACV: grist;

Var n: longint;

Procedure WanderAlong;

const suncolour = Yellow;

const planetcolour = LightGreen;

const asteroidcolour = White;

const anglecolour = cyan;

const periodcolour = LightRed;

Const radiuscolour = LightBlue;

Const KeplerColour = Magenta;

Const EnergyColour = White;

Const dt = 1;

var

W,P,T,

SX,SY,RS,

PX,PY,RP,

AX,AY,VAX,VAY,acX,acY,

DSX,DSY,DS2,DS3,

DPX,DPY,DP2,DP3: grist;

var cxp,cyp,sxp,syp,pxp,pyp,axp,ayp: integer;

var bxp,txp,byp,typ: integer;

var nturn: longint;

var T1,T2: grist;

var pAX,pAY,ppAX,ppAY,D2,pD2,ppD2: grist;

Var nstep: longint;

var stepwise,bored: boolean;

Function AsteroidEnergy: grist; {The asteroid mass is considered divided out.}

BEGIN {Thus, 1/2 mv**2 becomes 1/2 v**2.}

AsteroidEnergy:=(VAX*VAX + VAY*VAY)/2

- SM/sqrt((AX - SX)*(AX - SX) + (AY - SY)*(AY - SY))

- PM/sqrt((AX - PX)*(AX - PX) + (AY - PY)*(AY - PY));

END; {Likewise, potential Energy = -G*M1*M2/r12 }

Procedure SplotStep;

BEGIN

SetColor(LightBlue);

Splot(0,0,'Steps: ' + ifmt(nstep));

Splot(4*cxp div 3,0,'R='+ffmt(sqrt(AX*AX + AY*AY),8,2));

SetColor(EnergyColour);

Splot(4*cxp div 3,Txtheight,'E/m: '+ ffmt(AsteroidEnergy,8,2) + ' ');

END;

const types = 5;

Const tummy = 666;

var v: array[1..types,1..tummy] of Chaff;

var vmax,vmin,vs: array[1..types] of Chaff;

var show,varies: array[1..types] of boolean;

const vcolour: array[1..types] of byte = (anglecolour,periodcolour,radiuscolour,KeplerColour,EnergyColour);

var nv: integer;

Procedure Splash;

var i,j: integer;

BEGIN

SetFillStyle(EmptyFill,Black); Bar(bxp,byp,txp,typ);

SetColor(Green);

Line(bxp,byp,txp,byp); line(txp,byp,txp,typ);

Line(txp,typ,bxp,typ); Line(bxp,typ,bxp,byp);

for i:=1 to types do

if show[i] and varies[i] then

if dotty then for j:=1 to nv-1 do putpixel(bxp + round((v[i,j] - vmin[i])*vs[i]),byp+j,vcolour[i])

else

begin

setcolor(vcolour[i]);

MoveTo(bxp + round((v[i,1] - vmin[i])*vs[i]),byp+1);

for j:=2 to nv-1 do LineTo(bxp + round((v[i,j] - vmin[i])*vs[i]),byp+j);

end;

END;

Procedure DealWith(ch: char);

var redraw: boolean;

BEGIN

Case upcase(ch) of

ESC: bored:=true;

'J': Dotty:=not Dotty;

'A': show[1]:=not show[1];

'P': show[2]:=not show[2];

'R': show[3]:=not show[3];

'K': show[4]:=not show[4];

'E': show[5]:=not show[5];

'S':stepwise:=not stepwise;

'?':FlashHint('Pressing a key may help.%APRK or E selects dots to plot%S for Stepwise%ESC to quit.');

end;

if pos(upcase(ch),'JAPRKE') > 0 then splash;

END;

Procedure Swallow(v1,v2,v3,v4,v5: grist);

var i,j,l: integer;

var hic,flux: boolean;

BEGIN

flux:=nv = 3;

if (nv >= typ - byp) or (nv >= tummy) then

begin

flux:=true;

l:=nv div 3;

for i:=1 to types do

for j:=l+1 to nv do v[i,j - l]:=v[i,j];

nv:=nv - l;

for i:=1 to types do begin vmin[i]:=v[i,1]; vmax[i]:=v[i,1]; end;

for i:=1 to types do

for j:=2 to nv do

begin

if v[i,j] > vmax[i] then vmax[i]:=v[i,j];

if v[i,j] < vmin[i] then vmin[i]:=v[i,j];

end;

for i:=1 to types do

begin

varies[i]:=vmax[i] <> vmin[i];

if varies[i] then vs[i]:=(txp - bxp)/(vmax[i] - vmin[i]);

end;

end;

nv:=nv + 1;

v[1,nv]:=v1; v[2,nv]:=v2; v[3,nv]:=v3; v[4,nv]:=v4; v[5,nv]:=v5;

if nv = 1 then for i:=1 to types do begin varies[i]:=false; vmax[i]:=v[i,1]; vmin[i]:=v[i,1]; end

else for i:=1 to types do

begin

hic:=false;

if v[i,nv] > vmax[i] then begin vmax[i]:=v[i,nv]; hic:=true; end;

if v[i,nv] < vmin[i] then begin vmin[i]:=v[i,nv]; hic:=true; end;

if hic then vs[i]:=(txp - bxp)/(vmax[i] - vmin[i]);

if not varies[i] then varies[i]:=vmax[i] <> vmin[i];

flux:=flux or hic;

end;

if nv > 3 then

begin

if flux then splash;

for i:=1 to types do

if show[i] and varies[i] then

if dotty then putpixel(bxp + round((v[i,nv] - vmin[i])*vs[i]),byp+nv,vcolour[i])

else

begin

setcolor(vcolour[i]);

Line(bxp + round((v[i,nv-1] - vmin[i])*vs[i]),byp+nv-1,bxp + round((v[i,nv] - vmin[i])*vs[i]),byp+nv);

end;

end;

END; {of Swallow.}

Procedure Apastron;

{Given: at the last three time steps, DS2, pD2, and ppD2, being (squares of)

the distances from the centre of mass, and further, DS2 < pD2, pD2 > ppD2.

Wanted: the time of greatest distance from the sun.

So, fit a parabola to the three equally spaced values ppD2, pD2, DS2,

differentiate and set to zero for the extremum, solving for x.

y = ax**2 + bx + c, and why not have x = -1, 0, +1

Thus c = y0, y-1 is ppD2 so c = pD2

b = (y1 - y-1)/2 y0 is pD2 b = (DS2 - ppD2)/2

a = (y1 + y-1 - 2y0)/2 y1 is DS2. a = (DS2 + ppD2 - 2pDS2)/2

The extremum is when 2ax + b = 0

so x = -b/2a when y is an extremum, call this h.

and x = 0 corresponds to the middle time, which was one back, so T-dt

Having found the time of greatest distance, a small adjustment from T

gives the location of the asteroid then (only a small change from time T)

and thus the direction of the apastron. Since the acceleration is at right

angles to the velocity it won't have much effect on the angle, and being

minimal, not much effect on the distance. But, I'm not doing the arithmetic.}

Var X,Y,a,r,h,k,e: grist;

BEGIN

{splot(0,20,'ppD2:'+ffmt(ppD2,15,2)+' ');}

{splot(0,30,' pD2:'+ffmt(pD2,15,3)+'> ');}

{splot(0,40,' D2:'+ffmt(D2,15,3)+' ');}

h:=(ppD2 - DS2)/(DS2 + ppD2 - 2*pD2) /2;

{splot(0,50,' h:'+ffmt(h,15,3+' '));}

t2:=T + (h - 1){*dt};

x:=((AX + ppAX - 2*pAX)/2*h + (AX - ppAX)/2)*h + pAX;

y:=((AY + ppAY - 2*pAY)/2*h + (AY - ppAY)/2)*h + pAY;

r:=sqrt(x*x + y*y);

SetColor(AngleColour); Line(cxp,cyp, cxp + round(x),cyp - round(y/stretch));

a:=atan2(x,y)*180/pi;

nturn:=nturn + 1;

h:=(t2 - t1)/P*Year; {Use the planet's 'year'.}

k:=8*r*r*r/h/h; {Kepler's law: G(m + m')/4pi = d**3/t**2}

e:=(VAX*VAX + VAY*VAY)/2

- SM/sqrt((x - SX)*(x - SX) + (y - SY)*(y - SY))

- PM/sqrt((x - PX)*(x - PX) + (y - PY)*(y - PY));

Splot(0,0 + 1*TxtHeight,'Orbits: ' + ifmt(nturn));

SetColor(AngleColour); Splot(0,ymax - 1*txtheight,'Apastron: ' + ffmt(a,8,3) + 'ø ');

SetColor(PeriodColour); Splot(0,ymax - 2*txtheight,'Period: ' + ffmt(h,12,3)+' ');

SetColor(RadiusColour); Splot(cxp*4 div 3,ymax - 1*txtheight,'Rmax: ' + ffmt(r,12,4)+' ');

SetColor(KeplerColour); Splot(cxp*4 div 3,ymax - 2*txtheight,'Kepler: ' + ffmt(k,12,3)+' ');

Swallow(a,h,r,k,e);

t1:=t2;

END; {of Apastron.}

Const Methodname: array[1..3] of string = ('Euler','Euler2','RK4');

var nSX,nSY,nPX,nPY,nAX,nAY: grist;

Procedure EulerStep; {First-order.}

BEGIN

DSX:=SX - AX; DSY:=SY - AY;

DS2:=DSX*DSX + DSY*DSY; DS3:=sqrt(DS2)*DS2;

DPX:=PX - AX; DPY:=PY - AY;

DP2:=DPX*DPX + DPY*DPY; DP3:=sqrt(DP2)*DP2;

acX:=SM*DSX/DS3 + PM*DPX/DP3; acY:=SM*DSY/DS3 + PM*DPY/DP3;

END;

Procedure SecondOrder; {Modified Euler, 2nd order predictor-corrector, etc..}

BEGIN

EulerStep;

nAX:=AX + (0.5*acX{*dt} + VAX){*dt}; nAY:=AY + (0.5*acY{*dt} + VAY){*dt};

DSX:=nSX - nAX; DSY:=nSY - nAY;

DS2:=DSX*DSX + DSY*DSY; DS3:=sqrt(DS2)*DS2;

DPX:=nPX - nAX; DPY:=nPY - nAY;

DP2:=DPX*DPX + DPY*DPY; DP3:=sqrt(DP2)*DP2;

acX:=(acX + SM*DSX/DS3 + PM*DPX/DP3)/2; acY:=(acY + SM*DSY/DS3 + PM*DPY/DP3)/2;

END;

Procedure RungeKutta4; {The classic.}

var h2,hsx,hsy,hpx,hpy,tax,tay,k1x,k1y,k2x,k2y,k3x,k3y: grist;

BEGIN

EulerStep; {The story at 0.}

h2:=dt/2; {Half a time step.}

hSX:=(SX + nSX)/2; hSY:=(SY + nSY)/2; {I'm not calling for a fresh sin and cos.}

hPX:=(PX + nPX)/2; hPY:=(PY + nPY)/2; {Halfway will do.}

tAX:=AX + (0.5*acX*h2 + VAX)*h2; tAY:=AY + (0.5*acY*h2 + VAY)*h2;

DSX:=hSX - tAX; DSY:=hSY - tAY;

DS2:=DSX*DSX + DSY*DSY; DS3:=sqrt(DS2)*DS2;

DPX:=hPX - tAX; DPY:=hPY - tAY;

DP2:=DPX*DPX + DPY*DPY; DP3:=sqrt(DP2)*DP2;

k1x:=SM*DSX/DS3 + PM*DPX/DP3; k1y:=SM*DSY/DS3 + PM*DPY/DP3;

tAX:=AX + (0.5*k1X*h2 + VAX)*h2; tAY:=AY + (0.5*k1Y*h2 + VAY)*h2;

DSX:=hSX - tAX; DSY:=hSY - tAY;

DS2:=DSX*DSX + DSY*DSY; DS3:=sqrt(DS2)*DS2;

DPX:=hPX - tAX; DPY:=hPY - tAY;

DP2:=DPX*DPX + DPY*DPY; DP3:=sqrt(DP2)*DP2;

k2x:=SM*DSX/DS3 + PM*DPX/DP3; k2y:=SM*DSY/DS3 + PM*DPY/DP3;

tAX:=AX + (0.5*k2X{*dt} + VAX){*dt}; tAY:=AY + (0.5*k2Y{*dt} + VAY){*dt};

DSX:=nSX - tAX; DSY:=nSY - tAY;

DS2:=DSX*DSX + DSY*DSY; DS3:=sqrt(DS2)*DS2;

DPX:=nPX - tAX; DPY:=nPY - tAY;

DP2:=DPX*DPX + DPY*DPY; DP3:=sqrt(DP2)*DP2;

k3x:=SM*DSX/DS3 + PM*DPX/DP3; k3y:=SM*DSY/DS3 + PM*DPY/DP3;

acX:=(acX + 2*(k1x + k2x) + k3x)/6; acY:=(acY + 2*(k1y + k2y) + k3y)/6;

END;

Const BlobSize = 8;

var r,vcirc: grist;

var d,m,distant2: grist;

var coswt,sinwt: grist;

var rsp,rpp: integer; {Radius of Sun in Pixels, and of Planet.}

var pspix,pppix,pswas,ppwas: pointer;{Points to Sun pix, Planet pix.}

var szspix,szppix: word; {Sizes of the pix map.}

var i1,i2: integer;

Var ch: char;

Var text: string;

BEGIN

bored:=false;

Dotty:=true;

Stepwise:=false;

for i1:=1 to types do show[i1]:=true;

bxp:=xmax; pyp:=round(ymax*stretch);

if pyp < bxp then bxp:=pyp; {Oh for square dots!##$%$#!}

cxp:=bxp div 2; cyp:=round(bxp/stretch/2); {The centre of mass on the screen.}

bxp:=bxp + 1; txp:=xmax; {Base and top of a side box.}

byp:=0; typ:=ymax; {Wherein will appear a plot.}

nv:=0; {No values saved for the box.}

R:=cxp - blobsize; {Available space on the screen, with safety.}

RP:=R; {All for the planet's radius.}

RS:=0; {If the sun is motionless.}

M:=SM + PM; {Total mass of the system.}

if pm = 0 then wobble:=false; {The asteroid is as if massless.}

if wobble then {But if the sun wobbles, }

begin { then a re-adjustment.}

RS:=R*PM/M; {Both orbit about the centre of mass.}

RP:=R*SM/M; {RS + RP = R.}

t:=RS; if RP > t then t:=RP; {Re-scale to fill the screen.}

RS:=RS*R/t; RP:=RP*R/t;

end;

D:=RS + RP; {Distance between planet and sun.}

Distant2:=6*D*D; {A long way out.}

M:=4*pi*pi*D*D*D/({G}year*year) /M; {Use the proper orbital period, with G = 1.}

SM:=SM*M; PM:=PM*M; M:=SM + PM; {Rescale so as to allow dt = 1.}

P:=year;

W:=2*Pi/P; {2Pi radians in one orbit, needing time P}

{dt:=P/year; {'Year' Timesteps complete one P.}

rsp:=Round(BlobSize*Sqrt(SM/M)); rpp:=Round(BlobSize*Sqrt(PM/M));

if rsp < 1 then rsp:=1; if rpp < 1 then rpp:=1;

{Prepare a solar disc.}

sxp:=cxp - round(RS); syp:=cyp - 0;

szspix:=ImageSize(sxp-rsp,syp-rsp,sxp+rsp,syp+rsp);

if not(MemGrab(pspix,szspix) and MemGrab(pswas,szspix)) then croak('Can''t grab memory for the solar disc.');

GetImage(sxp-rsp,syp-rsp,sxp+rsp,syp+rsp,pswas^);

SetColor(SunColour); Circle(sxp,syp,rsp);

SetFillStyle(SolidFill,SunColour); FloodFill(sxp,syp,SunColour);

GetImage(sxp-rsp,syp-rsp,sxp+rsp,syp+rsp,pspix^);

PutImage(sxp-rsp,syp-rsp,pswas^,0); {Remove the sun during preparation.}

{Prepare a planetary disc.}

pxp:=cxp + round(RP); pyp:=cyp + 0;

szppix:=ImageSize(pxp-rpp,pyp-rpp,pxp+rpp,pyp+rpp);

If not(MemGrab(pppix,szppix) and MemGrab(ppwas,szppix)) then Croak('Can''t grab memory for the planetary disc.');

GetImage(pxp-rpp,pyp-rpp,pxp+rpp,pyp+rpp,ppwas^);

SetColor(PlanetColour); Circle(pxp,pyp,rpp);

SetFillStyle(SolidFill,PlanetColour); {FloodFill(pxp,pyp,PlanetColour);}

GetImage(pxp-rpp,pyp-rpp,pxp+rpp,pyp+rpp,pppix^);

PutImage(pxp-rpp,pyp-rpp,ppwas^,0);

{Place some geometry.}

SetColor(1); Line(cxp - round(rp),cyp,cxp + round(rp),cyp);

Line(cxp,cyp + round(rp/stretch),cxp,cyp - round(rp/stretch));

Circle(cxp,cyp,Round(RP));

SetColor(5); Line(cxp-1,cyp,cxp+1,cyp); Line(cxp,cyp-1,cxp,cyp+1);

GetImage(sxp-rsp,syp-rsp,sxp+rsp,syp+rsp,pswas^); {+Geometry, no sun.}

GetImage(pxp-rpp,pyp-rpp,pxp+rpp,pyp+rpp,ppwas^); {+Geometry, no planet.}

PutImage(sxp-rsp,syp-rsp,pspix^,0);

SetColor(LightBlue); Splot(0,0+2*txtheight,'Method: ' + methodname[style]);

if wobble then text:='Wobble' else text:='Fixed';

setcolor(SunColour); Splot(0,0+3*txtheight,text);

SX:=0; SY:=0; {Place the sun at the centre of mass.}

PX:=RP; PY:=0; {Place the planet on the x-axis.}

if wobble then SX:=-RS; {Opposite sides.}

nSX:=SX; nSY:=SY; {In case it doesn't wobble.}

axp:=0; ayp:=0; {Previous asteroid place.}

D:=RP*AR; {Distance out from the C of M.}

AX:=D*Cos(AA); AY:=D*Sin(AA);

VCirc:=D*2*Pi/Sqrt(4*pi*pi*D*D*D/M); {Velocity of circular orbit.}

VAX:=VCirc*(-ACV*Sin(AA) + ARV*Cos(AA));

VAY:=VCirc*(+ACV*Cos(AA) + ARV*Sin(AA));

pAX:=AX; ppAX:=AX; pAY:=AY;ppAY:=AY;{Shouldn't be needed, but fear uninitialisation.}

D2:=AX*AX + AY*AY; {Distance from the centre of mass.}

pD2:=D2;ppD2:=D2; {We start at the Apastron.}

T1:=0;

nturn:=0;

nstep:=0;

{The story at time T, the start of a time step dt:

The sun's mass SM at (SX,SY) and (nSX,nSY) at the end of the time step.

The planet PM (PX,PY) (nPX,nPY)

The asteroid (AX,AY) velocity (VAX,VAY)

To compute its acceleration (acX,acY)

and therefrom a new value for (AX,AY) and (VAX,VAY).}

repeat

nstep:=nstep + 1; {Well, it is about to be made.}

t:=nstep{*dt}; {This time is after the step has been made.}

CosWT:=Cos(W*T); SinWT:=Sin(W*T); {The compiler probably does a poor job.}

nPX:=RP*Coswt;nPY:=RP*sinwt; {And at the end of the step.}

if wobble then {In proper centre-of-mass arrangements?}

begin {Yes. The sun is wobbling.}

nSX:=-RS*CosWT; nSY:=-RS*SinWT; {And at the end of the time step.}

end; {Enough fooling around. Now for the real task.}

Case style of

1:EulerStep;

2:SecondOrder;

3:RungeKutta4;

end;

AX:=AX + (0.5*acX{*dt} + VAX){*dt}; AY:=AY + (0.5*acY{*dt} + VAY){*dt};

VAX:=VAX + acX{*dt}; VAY:=VAY + acY{*dt};

PX:=nPX; PY:=nPY;

if wobble then begin SX:=nSX; SY:=nSY; end; {Thus attain the end of the time step.}

PutPixel(axp,ayp,0); {Scrub the old asteroidal spot.}

axp:=cxp + round(AX); ayp:=cyp - Round(AY/stretch);

PutPixel(axp,ayp,AsteroidColour); {Place the new asteroidal spot.}

PutImage(pxp-rpp,pyp-rpp,ppwas^,0); {Scrub the old planetary spot.}

pxp:=cxp + round(PX); pyp:=cyp - round(PY/stretch);

GetImage(pxp-rpp,pyp-rpp,pxp+rpp,pyp+rpp,ppwas^);

PutImage(pxp-rpp,pyp-rpp,pppix^,0); {Place the new planetary spot.}

if wobble then {The sun might be mobile, too.}

begin {So see if it has moved noticeably.}

i1:=cxp + round(SX); i2:=cyp - round(SY/stretch);

if (i1 <> sxp) or (i2 <> syp) then {At least a whole dot in some direction?}

begin {Yep.}

PutImage(sxp-rsp,syp-rsp,pswas^,0); {Restore what was there.}

sxp:=i1; syp:=i2; {And jump to the new position.}

GetImage(sxp-rsp,syp-rsp,sxp+rsp,syp+rsp,pswas^);

PutImage(sxp-rsp,syp-rsp,pspix^,0); {Splot.}

end; {So much for a noticeable move.}

end; {So much for sun movement.}

D2:=AX*AX + AY*AY; {The new position.}

if (D2 < pD2) and (pD2 > ppD2) then Apastron;

ppAX:=pAX; ppAY:=pAY; pAX:=AX; pAY:=AY; ppD2:=pD2; pD2:=D2;

if stepwise or (nstep mod 100 = 0) then splotstep;

if D2 > Distant2 then {Too distant for close interactions?}

if AsteroidEnergy > 0 then {Yeah, perhaps a getaway is in progress.}

begin

SplotStep;

Splot(cxp,cyp div 3,'Escaping!');

Bored:=true;

end;

if stepwise or keypressed then dealwith(KeyFondle);

until bored;

repeat

Text:='';

if nv > 3 then text:='Adjust dots, or%';

text:=text + 'ESC to quit.';

LurkWith(Text);

ch:=KeyFondle;

if ch <> esc then dealwith(ch);

until ch = esc;

MemDrop(pspix,szspix); MemDrop(pswas,szspix);

MemDrop(pppix,szppix); MemDrop(ppwas,szppix);

END; {Of WanderAlong.}

Procedure Grunt(n: integer);

Var i: integer;

Var Unflushed: boolean;

Procedure Z(Text: string); {Roll some text.}

BEGIN {With screen pauses.}

if unflushed then ClrEol; {Perhaps bumf lurks on this line.}

WriteLn(Text); i:=i + 1; {Writes only to the end of text, not eol.}

if i >= Hi(WindMax) then {Have we reached the bottom?}

begin {Yes, the display would soon scroll up.}

if unflushed then clreol; {A last remnant.}

unflushed:=false; {Once scrolling starts, new lines are blank.}

Write('(Press a key)'); {A hint, offering out-by-one possibilities.}

if ReadKey = #0 then if ReadKey = esc then; {Ignore a key.}

GoToXY(1,wherey); ClrEol; {Scrub the hint.}

i:=0; {Restart the count.}

end; {So much for a screen full.}

END; {So much for that line.}

BEGIN

i:=n; Unflushed:=true;

Z(' A massive planet is in a fixed circular orbit about its sun.');

Z('Between it and the sun is an asteroid that wanders as it can.');

Z('Although this problem is (just) within the scope of analytic methods,');

Z('the asteroid''s movement is computed by numerical integration, with one');

Z('step per ''day'' of the planet''s orbital period. As the simulation proceeds,');

Z('various attributes of its orbit are plotted in a box to the side of the screen');

Z('for each completed orbit. This is taken to be when it reaches a maximum');

Z('distance from the centre of mass, or in other words, starts to fall back.');

Z('No attempt is made to identify other orbital centres, such as the planet');

Z('or the Trojan points of its orbit or whatever other stabilities exist, nor');

Z('more baroque orbits such as about both the sun and the planet.');

Z(' The attributes plotted (radius, angle of apastron, etc) can be selected');

Z('or unselected by pressing the key for the first letter of the name, thus the');

Z('plot of the successive values for the Apastron may be suppressed by pressing');

Z('the A key, and so on. The plots appear as separate dots for each value, but if');

Z('you prefer Joined-up plots, pressing the J key will switch states.');

Z(' The S key switches between Stepwise and continual execution; pressing some');

Z('other key (e.g. Return) then advances the calculation one step each time.');

Z(' Everything depends on the initial state, and a distressingly long list of');

Z('parameters only begins to explore the opportunities, as follows:');

Z('');

Z(' YearLength: days in the planet''s year, e.g. 1000');

Z(' PlanetMass: as a fraction of the sun''s mass, e.g. 0.0125');

Z(' AsteroidR: as a fraction of the planet''s distance from the C of M');

Z(' AsteroidA: angle with the x-axis (in degrees)');

Z(' AsteroidRV: radial velocity as a factor of the circular orbit velocity');

Z(' AsteroidCV: circular orbit velocity factor');

Z(' Style: E = Euler, M = Euler 2''nd, R = Runge-Kutta.');

Z(' SunMotion: W = wobble about C of M, F = (improperly) fixed at C of M.');

Z('');

Z(' You need not supply all of these, but as they are not named, there can be');

Z('no gap in the list. Thus, if you supply the third parameter then you must');

Z('also supply the first and second, but need not supply those past the third.');

Z('If no parameters are offered, the default is equivalent to');

Z('');

Z(' WANDER 1000 0.15 0.5 0 0.0 -0.75 R W');

Z('');

Z(' This will use the classic fourth-order Runge-Kutta scheme for numerical');

Z('integration of a differential equation for the case where a massive planet');

Z('(the Earth''s mass is 1/300000 of the mass of the Sun) orbits in a fixed');

Z('circle having a thousand one-day steps, and a (massless) asteroid starts off');

Z('on the x-axis halfway between the planet and the centre of mass with an');

Z('initial velocity straight down of three quarters of the speed a circular orbit');

Z('of that radius would require in the absence of planetary perturbations.');

Z('');

Z(' Don''t forget the leading 0 in 0.15 that turbo pascal demands...');

Z(' ESC to quit.');

END;

Procedure Reject(Gripe: anystring);

BEGIN

Writeln;

Writeln('Unsavoury: ',Gripe);

Writeln;

Grunt(3);

Halt;

END;

Const Usage: array[1..8] of string = (

'Steps in a planetary year',

'Planetary mass ratio',

'Asteroid''s initial radial distance',

'Asteroid''s initial angular position',

'Asteroid''s initial outwards velocity',

'Asteroid''s initial orbital velocity',

'Numerical Integration Method',

'Fixed or Wobbling Sun position');

Procedure Scrog(i: integer; var v: grist);

var hic: integer;

BEGIN

hic:=-666;

if paramstr(i) <> '' then Val(Paramstr(i),v,hic);

if hic <> 0 then Reject(paramstr(i) + ' for ' + usage[i]);

END;

Function ScrogT(i: integer; var n: integer; List: string): boolean;

var c: char;

var stupid: string[1];

BEGIN

if paramstr(i) = '' then scrogt:=false

else

begin

stupid:=copy(Paramstr(i),1,1);

c:=char(hi(integer(stupid)));

n:=Pos(upcase(c),List);

if n <= 0 then Reject(paramstr(i) + ' not one of ' + list + ' for ' + usage[i]);

scrogt:=true;

end;

END;

{$F+} Function HeapFull(Size: word): integer; {$F-}

Begin HeapFull:=1; End; {Sez "If full, return a null pointer" to GetMem.}

{Damnit, Turbo pascal's pointer using procedures don't check for null pointers!}

var i: Integer;

BEGIN {The main programme at last.}

HeapError:=@HeapFull; {Memory shortages?}

AsItWas:=LastMode;

Writeln(' A Wanderer in an unstable orbit.');

Style:=3;

Wobble:=true;

Year:=1000;

SM:=1; PM:=0.15;

AR:=0.5; AA:=0;

ARV:=0; ACV:=-0.75;

if paramcount > 0 then

begin

if paramstr(1) = '?' then begin Grunt(1); exit; end;

Scrog(1,year); year:=round(year); {Whole steps only!}

Scrog(2,PM);

Scrog(3,AR);

Scrog(4,AA);

Scrog(5,ARV);

Scrog(6,ACV);

if ScrogT(7,Style,'EMR') then style:=max(1,min(style,3));

if ScrogT(8,i,'FW') then Wobble:=i = 2;

end;

AA:=Pi*AA/180;

PrepareTheCanvas;

WanderAlong;

TextMode(AsItWas);

END.

### Relativistic orbits[edit]

Another magazine article described a calculation for orbits twisted by the gravity around black holes. Only a two-body problem, and only just, because the orbiting body's mass is ignored.

{$N+ Crunch only the better sort of numbers, thanks.}

Program Swirl; Uses Graph, Crt;

{ Calculates and displays orbits strongly twisted by gravity, as described

in the article by J. Gallmeier et al, in Sky & Telescope, October 1995, which

included the source code of a programme written in basic, so some changes here.

An actual pulsar in orbit about another compact body has parameters

EC = .61713 and SG = .000025, resulting in a precession of .0037 degrees/orbit.

More fierce relativistic effects involve close passages by a black hole.

A dark disc appears corresponding to the event horizon diameter for a

non-rotating black hole (the screen is normally 'black' so the black hole is

not depicted as black) and a circle at twice its diameter is drawn; particles

venturing within that bound will not repeat their orbit as they are either on

the way down the drain, or else on a hyperbolic flypast.

This calculation is for orbits that continue indefinitely, and the authors

have used the classic Runge-Kutta fourth-order method to numerically integrate

the relativistic equations twisting the axes of the orbit.}

{Perpetrated by R.N.McLean (whom God preserve), Victoria University, Nov. VMM.}

Type Grist = {$IFOPT N+} extended {$ELSE} real {$ENDIF};

const esc = #27;

var colour,lastcolour,xmax,ymax,txtheight: integer;

var stretch: grist;

Type AnyString = string[80];

FUNCTION Trim(S : anystring) : anystring;

var L1,L2 : integer;

BEGIN

L1 := 1;

WHILE (L1 <= LENGTH(S)) AND (S[L1] = ' ') DO INC(L1);

L2 := LENGTH(S);

WHILE (S[L2] = ' ') AND (L2 > L1) DO DEC(L2);

IF L2 >= L1 THEN Trim := COPY(S,L1,L2 - L1 + 1) ELSE Trim := '';

END; {Of Trim.}

FUNCTION Ifmt(Digits : longint) : anystring;

var S : anystring;

BEGIN

STR(Digits,S);

Ifmt := Trim(S);

END; { Ifmt }

FUNCTION Ffmt(Digits: grist; Width,Decimals: integer): anystring;

var

S : anystring;

L : integer; { a finger }

BEGIN

if digits = 0 then begin Ffmt:='0'; exit; end; {Mumble.}

IF Abs(Digits) < 0.0000001 THEN STR(Digits,S)

ELSE STR(Digits:Width:Decimals,S);

s:=trim(s);

if copy(s,1,2) = '0.' then s:=copy(s,2,Length(s) - 1);

if copy(s,1,3) = '-0.' then s:=concat('-',copy(s,3,Length(s) - 2));

l:=Length(s);

if pos('.',s) > 0 then while (l > 0) and (s[l] = '0') do l:=l - 1;

if s[l] = '.' then l:=l - 1; {No non-zero fractional part!}

S := COPY(S,1,L); {Drop trailing boredom.}

l:=Pos('.',S); {Avoid sequences such as 3.1415.}

if l > 0 then s[l]:=chr(249); {By using decimal points, please.}

Ffmt := S;

END; { Ffmt }

Procedure Rogbiv;

BEGIN;

SetPalette( 0,Black);

SetPalette( 1,DarkGray); {Looks navy blue to me.}

SetPalette( 2,Blue);

SetPalette( 3,LightBlue);

SetPalette( 4,LightCyan);

SetPalette( 5,Cyan);

SetPalette( 6,LightGreen);

SetPalette( 7,Green);

SetPalette( 8,LightGray); {Looks light brown to me.}

SetPalette( 9,Yellow);

SetPalette(10,Brown); {Looks yellow to me.}

SetPalette(11,Red);

SetPalette(12,LightRed);

SetPalette(13,Magenta);

SetPalette(14,LightMagenta);

SetPalette(15,White);

END; {Of Rogbiv.}

Procedure PrepareTheCanvas;

Var mode: integer;

Var SavedResultCode: integer;

Var ax,ay: word;

Function BGIFound(Here: Anystring): boolean;

Var driver: integer;

BEGIN

Driver:=Detect;

InitGraph(driver,mode,Here);

SavedResultCode:=GraphResult; {Mumble!}

BGIFound:=SavedResultCode = 0; {Missing file gives -3.}

END; {Of BGIFound.}

Var aplace: anystring;

Var Palette: paletteType;

BEGIN

Mode:=0;

if not BGIFound('') then

if not BGIFound('C:\TP7\BGI') then

if not BGIFound('D:\TP4') then

begin

WriteLn;

WriteLn('Trouble with Borland''s Graphic Interface!');

WriteLn('Message: ',GraphErrorMsg(SavedResultCode));

WriteLn('I''ve tried "", "C:\TP4", and "D:\TP4"...');

Repeat

Write('Another place to look? (e.g. a:\bgiset)');

ReadLn(Aplace); {As opposed to Read. weird.}

If Aplace = '' then

begin

writeln;

write('No .bgi, no piccy.');

halt;

end;

until BGIFound(Aplace);

end;

SetViewPort(0,0,GetMaxX,GetMaxY,True); {Clipping on, thanks. No apparent slowdown.}

SetGraphMode(Mode);

GetPalette(Palette);

LastColour := Palette.Size - 1; {Colours are 0:15, 16 thereof.}

if LastColour <= 0 then LastColour:=1;

if LastColour = 15 then Rogbiv;

xmax:=GetMaxX; ymax:=GetMaxY;

txtheight:=TextHeight('I');

GetAspectRatio(ax,ay); stretch:=ay/ax; {I hope this handles the screen's physical dimensions}

END; {as well as the number of dots in the x and y directions.}

Procedure Splot(x,y:integer; bumf: anystring);

var l: integer;

BEGIN

l:=TextWidth(Bumf);

if x + l > xmax then x:=xmax - l;

SetFillStyle(EmptyFill,Black); Bar(x,y,x + l,y + txtheight - 1);

OutTextXY(x,y,bumf);

END;

Var nstep: longint;

Procedure SplotStep;

BEGIN

Splot(xmax,0 + TxtHeight,'Stepcount: ' + ifmt(nstep));

END;

Var EC,SG,SS,SA: grist;

Var n: longint;

Procedure PrecessRelativistically;

var

SX,SY,

C0,C2,RH,

Q,SN,QN,DN,HN,DP,SP,

K1,K2,K3,K4,L1,L2,L3,L4: grist;

var xc,yc,px,py: integer;

Procedure Fullturn;

BEGIN

Line(xc,yc, round(PX),round(PY));

n:=n + 1;

Splot(xmax,0,'Completed orbits ' + ifmt(n));

SplotStep;

if n = 1 then {First time round?}

begin {Yep. DP and DN straddle zero. Find S where D = 0.}

SA:=(SP - DP*(SN - SP)/(DN - DP))*180/Pi - 360;

Splot(xmax,ymax - txtheight,'Precession per orbit ' + ffmt(SA,15,5) + 'ø');

end; {y:x=0 given (x1,y1) and (x2,y2): y = y1 - x1*(y2 - y1)/(x2 - x1)}

colour:=colour + 1;

if colour > Lastcolour then colour:=3;

setcolor(colour);

END;

Var ch: char;

BEGIN

Splot(0,0,'Eccentricity ' + ffmt(EC,9,6));

Splot(0,ymax - txtheight,'Relativistic Factor ' + ffmt(SG,9,6));

Splot(xmax,0 + 2*txtheight,'StepSize ' + ffmt(SS,8,3));

xc:=xmax div 2; yc:=ymax div 2;

n:=0; nstep:=0;

SS:=0.0009*(SS - 0.9)*(1 - 0.9*exp(10*EC - 9)); {Step size stuff...}

SY:=yc/(1 + EC); SX:=SY*stretch;

C0:=(1 - SG*(3 + EC*EC)/(6 + 2*EC))/(1 - EC*EC);

RH:=SG*(1 - EC*EC)/(3 + EC);

C2:=1.5*RH;

SA:=0;

Colour:=1; SetColor(Colour); Circle(xc,yc,round(SX*RH));

SetFillStyle(SolidFill,Colour); FloodFill(xc,yc,Colour);

Circle(xc,yc,round(SX*RH*2));

Colour:=2; SetColor(Colour); Line(0,yc, xmax,yc); Line(xc,0, xc,ymax);

Colour:=3; SetColor(Colour);

SN:=0; QN:=1/(1 + EC); DN:=0;

repeat

DP:=DN; SP:=SN;

HN:=SS*QN*QN;

Q:=QN; K1:=(C0 - Q + C2*Q*Q)*HN; L1:=HN*DN;

Q:=QN + L1/2; K2:=(C0 - Q + C2*Q*Q)*HN; L2:=HN*(DN + K1/2);

Q:=QN + L2/2; K3:=(C0 - Q + C2*Q*Q)*HN; L3:=HN*(DN + K2/2);

Q:=QN + L3; K4:=(C0 - Q + C2*Q*Q)*HN; L4:=HN*(DN + K3);

QN:=QN + (L1 + (L2 + L3)*2 + L4)/6;

DN:=DN + (K1 + (K2 + K3)*2 + K4)/6;

SN:=SN + HN;

px:=xc + Round(SX*cos(SN)/QN); py:=yc - Round(SY*sin(SN)/QN);

PutPixel(px,py,Colour);

nstep:=nstep + 1;

if (DN > 0) and (DP < 0) then fullturn

else if nstep mod 250 = 0 then splotstep;

until keypressed;

if readkey <> esc then repeat until keypressed;

END; {Of PrecessRelativistically.}

Procedure Grunt;

BEGIN

WriteLn('Draws orbits under relativistic conditions.');

WriteLn('Activate with two parameters:');

WriteLn(' Orbital Eccentricity (0 to 0ù9),');

WriteLn(' Relativistic factor (0 to 0ù999)');

WriteLn('An optional third parameter is the step size (1 to 10, e.g. 5).');

WriteLn('(If no parameters are supplied, an example run results)');

WriteLn('Eg. Swirl 0.5 0.5 5');

WriteLn('Don''t forget the leading zeroes that turbo pascal demands...');

WriteLn('ESC to quit.');

END;

Procedure Reject(Gripe: anystring);

BEGIN

Writeln;

Writeln('Unsavoury: ',Gripe);

Writeln;

Grunt;

Halt;

END;

Var AsItWas: word;

Var i: integer;

Var Hic: array[1..3] of integer;

BEGIN

AsItWas:=LastMode;

Writeln(' Precession of Elliptical Orbits.');

EC:=0.8; SG:=0.3826057; SS:=5; {These values result in 72 degrees/orbit...}

if paramcount > 0 then

begin

if paramstr(1) = '?' then begin Grunt; exit; end;

Val(ParamStr(1),EC,hic[1]);

Val(ParamStr(2),SG,hic[2]);

hic[3]:=0; if paramcount >= 3 then Val(paramstr(3),SS,Hic[3]);

for i:=1 to 3 do if hic[i] > 0 then Reject('Parameter ' + ifmt(i) + ': ' + paramstr(i));

end;

if (ec < 0) or (ec >= 1) then Reject('an eccentricity of ' + ffmt(ec,15,5));

if (sg < 0) or (sg >= 1) then Reject('a relativistic factor of ' + ffmt(sg,15,5));

if ss <= 0 then Reject('a step size of ' + ffmt(ss,15,5));

PrepareTheCanvas;

PrecessRelativistically;

TextMode(AsItWas);

WriteLn('Orbital eccentricity:',EC:15:5);

WriteLn('Relativistic factor :',SG:15:5);

WriteLn('Precession per orbit:',SA:15:5);

WriteLn('Orbits completed: ',n);

WriteLn('Calculation steps:',nstep);

END.

## Perl 6[edit]

We'll try to simulate the Sun+Earth+Moon system, with plausible astronomical values.

We use a 18-dimension vector . The first nine dimensions are the positions of the three bodies. The other nine are the velocities. This allows us to write the dynamics as a first-temporal derivative equation, since

and thus

# Simple Vector implementation

multi infix:<+>(@a, @b) { @a Z+ @b }

multi infix:<->(@a, @b) { @a Z- @b }

multi infix:<*>($r, @a) { $r X* @a }

multi infix:</>(@a, $r) { @a X/ $r }

sub norm { sqrt [+] @_ X** 2 }

# Runge-Kutta stuff

sub runge-kutta(&yp) {

return -> \t, \y, \δt {

my $a = δt * yp( t, y );

my $b = δt * yp( t + δt/2, y + $a/2 );

my $c = δt * yp( t + δt/2, y + $b/2 );

my $d = δt * yp( t + δt, y + $c );

($a + 2*($b + $c) + $d) / 6;

}

}

# gravitational constant

constant G = 6.674e-11;

# astronomical unit

constant au = 150e9;

# time constants in seconds

constant year = 365.25*24*60*60;

constant month = 21*24*60*60;

# masses in kg

constant $ma = 2e30; # Sun

constant $mb = 6e24; # Earth

constant $mc = 7.34e22; # Moon

my &dABC = runge-kutta my &f = sub ( $t, @ABC ) {

my @a = @ABC[0..2];

my @b = @ABC[3..5];

my @c = @ABC[6..8];

my $ab = norm(@a - @b);

my $ac = norm(@a - @c);

my $bc = norm(@b - @c);

return [

flat

@ABC[@(9..17)],

map G * *,

$mb/$ab**3 * (@b - @a) + $mc/$ac**3 * (@c - @a),

$ma/$ab**3 * (@a - @b) + $mc/$bc**3 * (@c - @b),

$ma/$ac**3 * (@a - @c) + $mb/$bc**3 * (@b - @c);

];

}

loop (

my ($t, @ABC) = 0,

0, 0, 0, # Sun position

au, 0, 0, # Earth position

0.998*au, 0, 0, # Moon position

0, 0, 0, # Sun speed

0, 2*pi*au/year, 0, # Earth speed

0, 2*pi*(au/year + 0.002*au/month), 0 # Moon speed

;

$t < 1;

$t, @ABC »Z[+=]« .01, dABC($t, @ABC, .01)

) {

printf "t = %.02f : %s\n", $t, @ABC.fmt("%+.3e");

}

- Output:

t = 0.00 : +0.000e+00 +0.000e+00 +0.000e+00 +1.500e+11 +0.000e+00 +0.000e+00 +1.497e+11 +0.000e+00 +0.000e+00 +0.000e+00 +0.000e+00 +0.000e+00 +0.000e+00 +2.987e+04 +0.000e+00 +0.000e+00 +3.090e+04 +0.000e+00 t = 0.01 : +9.008e-13 +5.981e-22 +0.000e+00 +1.500e+11 +2.987e+02 +0.000e+00 +1.497e+11 +3.090e+02 +0.000e+00 +1.802e-10 +1.794e-19 +0.000e+00 -5.987e-05 +2.987e+04 +0.000e+00 -1.507e-05 +3.090e+04 +0.000e+00 t = 0.02 : +3.603e-12 +4.785e-21 +0.000e+00 +1.500e+11 +5.973e+02 +0.000e+00 +1.497e+11 +6.181e+02 +0.000e+00 +3.603e-10 +7.177e-19 +0.000e+00 -1.197e-04 +2.987e+04 +0.000e+00 -3.014e-05 +3.090e+04 +0.000e+00 t = 0.03 : +8.107e-12 +1.615e-20 +0.000e+00 +1.500e+11 +8.960e+02 +0.000e+00 +1.497e+11 +9.271e+02 +0.000e+00 +5.405e-10 +1.615e-18 +0.000e+00 -1.796e-04 +2.987e+04 +0.000e+00 -4.521e-05 +3.090e+04 +0.000e+00 t = 0.04 : +1.441e-11 +3.828e-20 +0.000e+00 +1.500e+11 +1.195e+03 +0.000e+00 +1.497e+11 +1.236e+03 +0.000e+00 +7.206e-10 +2.871e-18 +0.000e+00 -2.395e-04 +2.987e+04 +0.000e+00 -6.028e-05 +3.090e+04 +0.000e+00 t = 0.05 : +2.252e-11 +7.476e-20 +0.000e+00 +1.500e+11 +1.493e+03 +0.000e+00 +1.497e+11 +1.545e+03 +0.000e+00 +9.008e-10 +4.486e-18 +0.000e+00 -2.993e-04 +2.987e+04 +0.000e+00 -7.535e-05 +3.090e+04 +0.000e+00 t = 0.06 : +3.243e-11 +1.292e-19 +0.000e+00 +1.500e+11 +1.792e+03 +0.000e+00 +1.497e+11 +1.854e+03 +0.000e+00 +1.081e-09 +6.460e-18 +0.000e+00 -3.592e-04 +2.987e+04 +0.000e+00 -9.041e-05 +3.090e+04 +0.000e+00 t = 0.07 : +4.414e-11 +2.051e-19 +0.000e+00 +1.500e+11 +2.091e+03 +0.000e+00 +1.497e+11 +2.163e+03 +0.000e+00 +1.261e-09 +8.792e-18 +0.000e+00 -4.191e-04 +2.987e+04 +0.000e+00 -1.055e-04 +3.090e+04 +0.000e+00 t = 0.08 : +5.765e-11 +3.062e-19 +0.000e+00 +1.500e+11 +2.389e+03 +0.000e+00 +1.497e+11 +2.472e+03 +0.000e+00 +1.441e-09 +1.148e-17 +0.000e+00 -4.789e-04 +2.987e+04 +0.000e+00 -1.206e-04 +3.090e+04 +0.000e+00 t = 0.09 : +7.296e-11 +4.360e-19 +0.000e+00 +1.500e+11 +2.688e+03 +0.000e+00 +1.497e+11 +2.781e+03 +0.000e+00 +1.621e-09 +1.453e-17 +0.000e+00 -5.388e-04 +2.987e+04 +0.000e+00 -1.356e-04 +3.090e+04 +0.000e+00 t = 0.10 : +9.008e-11 +5.981e-19 +0.000e+00 +1.500e+11 +2.987e+03 +0.000e+00 +1.497e+11 +3.090e+03 +0.000e+00 +1.802e-09 +1.794e-17 +0.000e+00 -5.987e-04 +2.987e+04 +0.000e+00 -1.507e-04 +3.090e+04 +0.000e+00 t = 0.11 : +1.090e-10 +7.961e-19 +0.000e+00 +1.500e+11 +3.285e+03 +0.000e+00 +1.497e+11 +3.399e+03 +0.000e+00 +1.982e-09 +2.171e-17 +0.000e+00 -6.586e-04 +2.987e+04 +0.000e+00 -1.658e-04 +3.090e+04 +0.000e+00 t = 0.12 : +1.297e-10 +1.034e-18 +0.000e+00 +1.500e+11 +3.584e+03 +0.000e+00 +1.497e+11 +3.709e+03 +0.000e+00 +2.162e-09 +2.584e-17 +0.000e+00 -7.184e-04 +2.987e+04 +0.000e+00 -1.808e-04 +3.090e+04 +0.000e+00 t = 0.13 : +1.522e-10 +1.314e-18 +0.000e+00 +1.500e+11 +3.882e+03 +0.000e+00 +1.497e+11 +4.018e+03 +0.000e+00 +2.342e-09 +3.032e-17 +0.000e+00 -7.783e-04 +2.987e+04 +0.000e+00 -1.959e-04 +3.090e+04 +0.000e+00 t = 0.14 : +1.766e-10 +1.641e-18 +0.000e+00 +1.500e+11 +4.181e+03 +0.000e+00 +1.497e+11 +4.327e+03 +0.000e+00 +2.522e-09 +3.517e-17 +0.000e+00 -8.382e-04 +2.987e+04 +0.000e+00 -2.110e-04 +3.090e+04 +0.000e+00 t = 0.15 : +2.027e-10 +2.019e-18 +0.000e+00 +1.500e+11 +4.480e+03 +0.000e+00 +1.497e+11 +4.636e+03 +0.000e+00 +2.702e-09 +4.037e-17 +0.000e+00 -8.980e-04 +2.987e+04 +0.000e+00 -2.260e-04 +3.090e+04 +0.000e+00 t = 0.16 : +2.306e-10 +2.450e-18 +0.000e+00 +1.500e+11 +4.778e+03 +0.000e+00 +1.497e+11 +4.945e+03 +0.000e+00 +2.883e-09 +4.593e-17 +0.000e+00 -9.579e-04 +2.987e+04 +0.000e+00 -2.411e-04 +3.090e+04 +0.000e+00 t = 0.17 : +2.603e-10 +2.938e-18 +0.000e+00 +1.500e+11 +5.077e+03 +0.000e+00 +1.497e+11 +5.254e+03 +0.000e+00 +3.063e-09 +5.186e-17 +0.000e+00 -1.018e-03 +2.987e+04 +0.000e+00 -2.562e-04 +3.090e+04 +0.000e+00 t = 0.18 : +2.919e-10 +3.488e-18 +0.000e+00 +1.500e+11 +5.376e+03 +0.000e+00 +1.497e+11 +5.563e+03 +0.000e+00 +3.243e-09 +5.814e-17 +0.000e+00 -1.078e-03 +2.987e+04 +0.000e+00 -2.712e-04 +3.090e+04 +0.000e+00 t = 0.19 : +3.252e-10 +4.102e-18 +0.000e+00 +1.500e+11 +5.674e+03 +0.000e+00 +1.497e+11 +5.872e+03 +0.000e+00 +3.423e-09 +6.477e-17 +0.000e+00 -1.138e-03 +2.987e+04 +0.000e+00 -2.863e-04 +3.090e+04 +0.000e+00

## Tcl[edit]

package require Tcl 8.6

set G 0.01

set epsilon 1e-12

proc acceleration.gravity {positions masses} {

global G epsilon

set i -1

lmap position $positions mass $masses {

incr i

set dp2 [lmap p $position {expr 0.0}]

set j -1

foreach pj $positions mj $masses {

if {[incr j] == $i} continue

set dp [lmap p1 $position p2 $pj {expr {$p1-$p2}}]

set d3 [expr {

sqrt(

[tcl::mathop::+ {*}[lmap p $dp {expr {$p ** 2}}]]

+ $epsilon

) ** 3

}]

# Add epsilon here?

set dp2 [lmap a $dp2 b $dp {expr {$a - $G*$mj*$b/$d3}}]

}

set dp2

}

}

# The rest of the system; simple numeric solution of differential equations

proc velocity {velocities accelerations} {

lmap v $velocities a $accelerations {

lmap vi $v ai $a {expr {$vi + $ai}}

}

}

proc position {positions velocities} {

lmap p $positions v $velocities {

lmap pi $p vi $v {expr {$pi + $vi}}

}

}

proc timestep {masses positions velocities} {

set accelerations [acceleration.gravity $positions $masses]

set velocities [velocity $velocities $accelerations]

set positions [position $positions $velocities]

list $positions $velocities

}

# Combine to make a simulation engine

proc simulate {masses positions velocities {steps 10}} {

set p $positions

set v $velocities

for {set i 0} {$i < $steps} {incr i} {

lassign [timestep $masses $p $v] p v

puts [lmap pos $p {format (%.5f,%.5f,%.5f) {*}$pos}]

}

}

Demonstrating for 20 steps:

set m {1 .1 .001}

set p {{0 0 0} {1 1 0} {0 1 1}}

set v {{0.01 0 0} {0 0 0.02} {0.01 -0.01 -0.01}}

simulate $m $p $v 20

- Output:

(0.01035,0.00036,0.00000) (0.99646,0.99646,0.02000) (0.01035,0.98646,0.98611) (0.02107,0.00108,0.00002) (0.98934,0.98931,0.03994) (0.02108,0.96930,0.96822) (0.03214,0.00218,0.00005) (0.97856,0.97843,0.05973) (0.03221,0.94837,0.94617) (0.04359,0.00367,0.00011) (0.96402,0.96368,0.07928) (0.04377,0.92350,0.91977) (0.05544,0.00557,0.00021) (0.94559,0.94487,0.09851) (0.05581,0.89447,0.88875) (0.06768,0.00790,0.00036) (0.92308,0.92176,0.11729) (0.06837,0.86100,0.85279) (0.08036,0.01070,0.00057) (0.89626,0.89406,0.13548) (0.08152,0.82271,0.81146) (0.09350,0.01400,0.00086) (0.86482,0.86136,0.15292) (0.09535,0.77910,0.76420) (0.10714,0.01786,0.00126) (0.82839,0.82318,0.16939) (0.10996,0.72951,0.71025) (0.12133,0.02234,0.00179) (0.78643,0.77885,0.18457) (0.12552,0.67304,0.64858) (0.13614,0.02754,0.00251) (0.73827,0.72746,0.19806) (0.14224,0.60833,0.57768) (0.15167,0.03357,0.00346) (0.68293,0.66775,0.20924) (0.16048,0.53332,0.49521) (0.16805,0.04065,0.00476) (0.61899,0.59779,0.21710) (0.18075,0.44439,0.39722) (0.18551,0.04908,0.00659) (0.54422,0.51449,0.21990) (0.20386,0.33411,0.27581) (0.20445,0.05946,0.00934) (0.45470,0.41210,0.21398) (0.22971,0.17975,0.10892) (0.22573,0.07337,0.01421) (0.34228,0.27744,0.18934) (0.19746,-0.27259,-0.30685) (0.25165,0.09535,0.02601) (0.18355,0.06171,0.09512) (0.16823,-0.69092,-0.69109) (0.21467,0.08626,0.10162) (0.65371,0.15666,-0.63735) (0.13969,-1.10220,-1.06883) (0.17838,0.07727,0.17608) (1.11702,0.25051,-1.35831) (0.11149,-1.51049,-1.44390) (0.14224,0.06831,0.25028) (1.57874,0.34406,-2.07666) (0.08347,-1.91722,-1.81757)