Homework 11

李尧（Neo lee）天眷班 2013301020048
problem 4.7 4.9
solar-system two-boding-system the-inverse-square-law

Abstract

In this paper, the solar system is studied by computer simulation. And more theoreticlly, we study the behaviors of two-star systems with different mass ratio and if the orbits are stable, if the inverse square law is not true.
skeyword:solar-system two-boding-system the-inverse-square-law

Background

Previously, astronomical study was observing, putting forward a hyperthesis, caculating and observing again. Now, computer science offers a new to study astronomy, which is not only that we prefer to caculate by computer, but also we can simulating the whole system with our computer. In this paper, we attempt to study our solar system，the behaviors of two-star systems with different mass ratio and if the orbits are stable, if the inverse square law is not true.

Physical mode and Mathematical method
Our university is very large and governed by gravity, which means that we don't need to consider about the uncertainty principle because the decoherence process is too fast to measure.And in weak field, such as the gravity field of the sun, the Newton gravity is good enough and the general relativity effort can be neglected.
Then we can write down the fundamental equations:
$v_{x,i+1}=v_{x,i}-\sum\frac{4\pi^2x_i}{r_{j}^3}\Delta t$
$v_{y,i+1}=v_{y,i}-\sum\frac{4\pi^2y_i}{r_j^3}\Delta t$
$x_{i+1}=x_i+v_{x,i+1}\Delta t$
$y_{i+1}=y_i+v_{y,i+1}\Delta t$
(if need to read the codes, please click here)

Two-star system

We choose the mass-center frame to see the behavior of the two-star system.In this paper, we choose units that $G=4\pi^2$,$M_{sun}=1$and$d_{sun-earth}=1$. Firstly,we try m=1, and result is showed in fig1.
fig1:

Then descrease m,we have
fig2:

fig3:

As descreasing m, the orbit of sun become smaller and smaller.In fig3,we just a big dot. If we use the real mass of earth,
fig4:

the solar orbit is a dot.

solar system

After study the two-star, I simulate the whole solar system.
fig5:

Amplify it....
fig6:

Go on.

The orbits are affected by each other a lot, and the sun is missing. Just amplify again.
fig7:

The sun is surounded by Mercury, and vibrating and moving because of the effect of all the planets.But comparing to the planets' motion, the sun is frozen.

the inverse square law

In the sense of quantum field theory, any field strengh at x interpret as the probability of field quantum show up at x. Such that the inverse square law means the conservation of energy in 3-D
space and the conservation of total probability

But now we supose it's false and forget the fundamental laws.Let's test the stablity of orbits.
Considering our numerial method unstable, we would better measure it firstly.
fig8:

It's showed that the unstablity causing by numerial method is no more than 0.000005.
let n=2.05,we get that
fig9:

the orbit become fat,which width is larger than 0.000005.the orbit is unstable.
How about n=3.0?
fig10：

Conclusion

1, As descresing the mass of earth, the two star system approach one star and one planet system.
2,the outside planet orbits is more stable than inner ones.
3,The orbits are nustable if the inverse square law falses.