Week 8: WS13 #1

1. a) We often say that planets orbit stars. But planets and their stars actually orbit their mutual center of mass, which in general is given by $x_{com}=\sum_im_ix_i\sum_im_i$. Set up the problem by drawing an x-axis with the star at $x=-a_\star$ with mass $M_\star$, and the planet at $x=a_p$ and $m_p$. Also, set $x_{com}=0$. How do $a_p$ and $a_\star$ depend on the masses of the star and planet? 

Since we know that  $x_{com}=\sum_im_ix_i\sum_im_i$, we can say that $x_{com}=\frac{-a_\star M_\star+a_pm_p}{M_\star+m_p}$. Setting $x_{com}=0$ allows us to determine that $M_\star a_\star=m_pa_p$. 

b) In a two-body orbital system the variable a is the mean semimajor axis, or the sum of the planet's and star's distances away from their mutual center of mass: $a=a_p+a_\star$. Label this on your diagram. Now derive the relationship between the total mass $M_\star+m_p\approx M_\star$, orbital period P, and the mean semimajor axis a, starting with the Virial Theorem for a two-body orbit. 

The Virial Theorem states that $-\frac{1}{2}U=K$ or $-\frac{1}{2}\frac{GMm}{r}=\frac{1}{2}mv^2$. Since the planet is much smaller than the star, the center of mass will be much closer to the star and therefore $a\approx a_p$. This allows us to find the velocity based on the planet's period: $v=\frac{2\pi a}{P}$.

$m_p(\frac{2\pi a}{P})^2=\frac{GM_\star m_p}{a}$

$\frac{4\pi a^2}{P^2}=\frac{GM}{a}$

$P^2=\frac{4\pi^2a^3}{GM}$ 

This is Kepler's third law! 

c) By how much is the Sun displaced from the Solar System's center of mass as a result of Jupiter's orbit? Express this displacement in a useful unit such as Solar radii. (Potentially useful numbers: $M_\odot\approx1000M_{Jup}$ and $a_{jup}\approx5.2AU$.)

$M_\star a_\star=m_pa_p$
$1000M_{Jup}a_\odot=M_{Jup}5.2AU$
$a_\odot=0.0052AU$
$\frac{0.0052AU}{1}\frac{1.5\times10^8km}{1AU}\frac{1R_\odot}{7.0\times10^5km}\approx1R_\odot$: Jupiter displaces the Sun by about one Solar radius.