computationalphysics

exercise10:solar system

ABSTRACT

• 1. Note that Newton's law of gravitation is already widely accepted ,we're using euler-cromer method to verify kepler's laws in a numerical way
• 2.the inverse-square law anf the stability of plantary orbits
• 3.precession of the perihelion of mercury

BACKGROUND

• newton's law of gravitation: $$F_G=\frac{GM_S M_P}{r^2}$$
  we write it as two first-order differental equation so that we can use euler-cromer method: $$\frac{dv_x}{dt}=-\frac{GM_S x}{r^3}\\ \frac{dx}{dt}=v_x\\ \frac{dv_y}{dt}=-\frac{GM_S y}{r^3}\\ \frac{dy}{dt}=v_y$$
  for circular motion we know that the force must be equal to $M_P v^2 /r$,which lead to:
  $$\frac{M_P v^2}{r}=F_G=\frac{GM_S M_P}{r^2}$$
  where we chose proper units,thus:
  $$GM_S=v_p^2r_p=v_e^2r_e=4\pi r^2\space AU^3/yr^2$$
  then we can simplify the calculation and estimate $v$'s(the last column of the next table) of other planet:

planet	mass(kg)	radius(AU)	eccentricity	speed(AU/yr)
Mercury	$2.4\times10^{23}$	0.39	0.206	10.061
Venus	$4.9\times10^{24}$	0.72	0.007	7.405
Earth	$6.0\times10^{24}$	1.00	0.0017	$2\pi\approx$6.283
Mars	$6.6\times10^{23}$	1.52	0.093	5.096
Jupiter	$1.9\times10^{27}$	5.20	0.048	2.755
Saturn	$5.7\times10^{26}$	9.54	0.056	2.034
Uranus	$8.8\times10^{25}$	19.19	0.046	1.434
Neptune	$1.03\times10^{26}$	30.06	0.010	1.146

• precession of the perihelion of mercury
  
  Under Newtonian physics, a two-body system consisting of a lone object orbiting a spherical mass would trace out an ellipse with the spherical mass at a focus. The point of closest approach, called the periapsis (or, because the central body in the Solar System is the Sun, perihelion), is fixed. A number of effects in the Solar System cause the perihelia of planets to precess (rotate) around the Sun. The principal cause is the presence of other planets which perturb one another's orbit. Another (much less significant) effect is solar oblateness.
  Mercury deviates from the precession predicted from these Newtonian effects. This anomalous rate of precession of the perihelion of Mercury's orbit was first recognized in 1859 as a problem in celestial mechanics, by Urbain Le Verrier. His reanalysis of available timed observations of transits of Mercury over the Sun's disk from 1697 to 1848 showed that the actual rate of the precession disagreed from that predicted from Newton's theory by 38" (arc seconds) per tropical century (later re-estimated at 43").[5] A number of ad hoc and ultimately unsuccessful solutions were proposed, but they tended to introduce more problems. In general relativity, this remaining precession, or change of orientation of the orbital ellipse within its orbital plane, is explained by gravitation being mediated by the curvature of spacetime. Einstein showed that general relativity[2] agrees closely with the observed amount of perihelion shift. This was a powerful factor motivating the adoption of general
  
  the force law of predicted by general relativity is:
  $$F_G\approx \frac{GM_S M_M}{r^2}(1+\frac{\alpha}{r^2})\\ \alpha\approx1.1\times10^{-8} AU^2$$

MAIN BODY

here is the code

• 1.a simulation of the motion of solar system planets
  The next are pictures of trajectories of solar system planets that I draw with matplotlib.
  
  to zoom it up,we can see this is nearly a perfect simulation of the seven planets of soalr system.
  

• 2.the inverse square law and the stabiliity of planetary orbits
  
  as what we can see from the plots,this world won't be what it is if there is no such inverse-squrare law.

• 3.precession of the perihelon of mercury
  cause we already know the effect of $\alpha$ is so small that truncation error is comparable with it，so we have to calculate for some big $\alpha$ and do a linear fit.
  the blue points in the next plot is shown in the following table：

$\alpha$	$d\theta /dt$	$\alpha$	$d\theta /dt$
0.0001	2.1360957657207646	0.0004	3.79859101235552
0.0030	31.586827548752012	0.0035	35.67506491602629
0.0040	39.82020809760321	0.0060	64.21130592694833
0.0070	80.79924387065746

the slope of line in the plot is $1.106\times 10^4$
when $\alpha=\approx1.1\times10^{-8} AU^2$,the precession angular speed is about：

$$\frac{d\theta}{dt}\approx1.21\times10^{-4}$$
(ps:I have to say this is net a good approximation even though it is so similar to the result of computational pysics，because the linear equation is $\frac{d\theta}{dt}=1.106\times10^4 \alpha - 0.913$whose vertical intercept is not $0$

conclusion

the programme of precession of perihelon of mercury needs improvement.

acknowledgement