Double-star system

程熹 2013301020038 物基一班
Homework

Abstract

This article investigate the double-star system where the "sun" and the "earth" are similar in mass. We set different initial conditions and different ratios of the mass of the sun and the earth to see if there is some interesting results. To achieve in numberical way we use the Euler-Cromer method.

Key Word

double-star,Newton=law,euler-cromer method.

Introduction

A binary star is a star system consisting of two stars orbiting around their common barycenter. Systems of two, three, four, or even more stars are called multiple star systems. These systems, especially when more distant, often appear to the unaided eye as a single point of light, and are then revealed as double (or more) by other means. Research over the last two centuries suggests that half or more of visible stars are part of multiple star systems.
Binaries provide the best method for astronomers to determine the mass of a distant star. The gravitational pull between them causes them to orbit around their common center of mass. From the orbital pattern of a visual binary, or the time variation of the spectrum of a spectroscopic binary, the mass of its stars can be determined, for example with the binary mass function. In this way, the relation between a star's appearance (temperature and radius) and its mass can be found, which allows for the determination of the mass of non-binaries.
Because a large proportion of stars exist in binary systems, binaries are particularly important to our understanding of the processes by which stars form. In particular, the period and masses of the binary tell us about the amount of angular momentum in the system. Because this is a conserved quantity in physics, binaries give us important clues about the conditions under which the stars were formed.
It is estimated that approximately 1/3 of the star systems in the Milky Way are binary or multiple, with the remaining 2/3 consisting of single stars.
There is a direct correlation between the period of revolution of a binary star and the eccentricity of its orbit, with systems of short period having smaller eccentricity. Binary stars may be found with any conceivable separation, from pairs orbiting so closely that they are practically in contact with each other, to pairs so distantly separated that their connection is indicated only by their common proper motion through space. Among gravitationally bound binary star systems, there exists a so-called log normal distribution of periods, with the majority of these systems orbiting with a period of about 100 years. This is supporting evidence for the theory that binary systems are formed during star formation.
In pairs where the two stars are of equal brightness, they are also of the same spectral type. In systems where the brightnesses are different, the fainter star is bluer if the brighter star is a giant star, and redder if the brighter star belongs to the main sequence.

Main Features

According to Newton's law of gravitation, the magnitude of the gravitational force between two objects is given by

$$F_{12} = \frac{G M_{1 }M_{2}} {r _{12} ^{2}}$$
We assume that the orbits of the two planets are coplanar,and we take this to be the xy-plane.Writing F in terms of components we have $$F_{12,x} =\frac{-G M_{1 }M_{2}( x_{1}- x_{2} )}{ r _{12} ^{3} }$$ $$F_{12,y} =\frac{-G M_{1 }M_{2}( y_{1}- y_{2} )}{r _ {12} ^{3}}$$
we take account for the total force on the earth $$F_{e,x} =-\frac{ G M_{s}M_{e}( x_{e}- x_{s} )}{r _{es} ^{3}}$$ $$F_{e,y} =-\frac{ G M_{s }M_{e}( y_{e}- y_{s} )}{r _{es} ^{3}}$$
with $$r_{es} = \sqrt{ (x_{e}-x_{s}) ^{2}+(y_{e}-y_{s}) ^{2} }$$
combining with Newton's second law we get the acceleration of the earth in x and y direction.Then we use the Euler-Cromer method to translate the equations into equations suitable to design program $$v_{e,x}(i+1)= v_{e,x}(i)- \frac{4 \pi ^{2} x_{e}(i) }{ r_{e } ^{3} } \Delta t$$ $$v_{e,y}(i+1)= v_{e,y}(i)- \frac{4 \pi ^{2} y_{e}(i) }{ r_{e } ^{3} } \Delta t$$
where we use the astronomical unit here.
Using the same way, we can describle the change of the position and velocity of the Jupiter and the Saturn.
Then we can choose different initial conditions(different initial positions and velocity of these stars) to get different results.

Result

1.The mass of the sun is 10 times the mass of the earth with the sun and the earth on same axis,same direction of the velocity.

2.The mass of the sun is 2 times the mass of the earth with the sun and the earth on same axis, same direction of the velocity.

3.The mass of the sun is 10 times the mass of the earth with the sun and the earth on same axis but opposite direction of the velocity

4.The mass of the sun is 2 times the mass of the earth with the sun and the earth on same axis but opposite direction of the velocity

5. The mass of the sun is 10 times the mass of the earth with the sun and the earth on different axises, same direction of the velocity

6. The mass of the sun is 2 times the mass of the earth with the sun and the earth on different axises, same direciton of the velocity

Conclusion

1. From these diagram we will find that a double star system where the two stars are similar in mass will have do periodic motion
2. From the diagram 1 and diagram 2, we will find that 2 stars if their initial velocity is on the same direction, their future motion will be similar in shape, their mass ratio will influence their shape ratio and they will have a phase difference.
3. From the diagram 5 and diagram 6, we will find that if there is an angle between the initial velcocity and the two stars compara initial position, their future motion will be on the line of their compartable position.

Acknowledgement

This program is finished by myself.

Reference

Computational Physics(second edition) Nicholas J.Giordano and Hisao Nakanishi
Wikipedia Binary star system
https://en.wikipedia.org/wiki/Binary_star