WS 10.2, Problem 2: Wobble, Baby, Wobble

a) Astronomers can detect planets orbiting other stars by detecting the motion of the star--its wobble|due to the planet's gravitational tug. Start with the relationship between $a_p$ and $a_*$  to find the relationship between speed of the planet and the star, $v_p$ and $v_*$ .

We started with the basic equation 

$a_*m_*=a_Pm_P$

We can also relate the semimajor axis and velocity in the following way

$v=\frac{2\pi a}{P}\Rightarrow a=\frac{vP}{2\pi}$

The first equation can now be rewritten and simplified as 

$\frac{v_*P}{2\pi}m_*=\frac{v_PP}{2\pi}m_P$

$v_*m_*=v_Pm_P$

b) Express the speed of the star, K, in terms of the orbital period P, the mass of the star $M_*$  and the mass of the planet.  Convert your units so that your expression is given as a speed in meters per second, with P measured in years, $M_*$ in solar masses and planet mass in Jupiter masses. 

Again let's start with the last equation from part (a)

$v_*m_*=v_Pm_P$

We can rearrange this to find the equation for the velocity of the star

$v_*= \frac{v_Pm_P}{m_*}$

We also know the following relationship between period and velocity

$v_P=\frac{2\pi a_P}{P}$

So now we can find K, the speed of the star

$K=v_*=\frac{m_P}{m_*}v_P=\frac{m_P}{m_*}\frac{2\pi a_P}{P}$

c) We can measure the velocity of a star along the line of sight using a technique similar to the way in which you measured the speed of the Sun's limb due to rotation. Specfi cally, we can measure the Doppler shift of stellar absorption lines to measure the velocity of the star in the radial direction, towards or away from the Earth, also known as the "radial velocity." What is the time variation of the line-of-sight velocity of the star as a planet orbits?

To someone who's grown up thinking that the solar system revolves around the sun, this problem might be a little bit conceptually difficult, because if the sun is truly the center of the solar system, it shouldn't move.  The point of this problem, however, is to show that the sun is not the exact center of the solar system (as I'm typing this, a bunch of dead white men are probably rolling in their graves because this problem debunks the work they devoted their lives to).  Instead, the sun wobbles--it moves away from the system's center of mass as the planet moves.  

If a planet is moving away from our viewpoint, the sun--assuming it only moves in our visual plane, and not up and down--is moving toward us.  If the planet is moving toward us, the sun has to compensate in its wobble and move away from the center of mass, or away from out viewpoint.  The sun's radial velocity would appear to be 0 (moving neither towards nor away from us) when the planet is at a point in its orbit when it's not moving towards or away from us, so when it's on the side of its orbital path.

d)  Sketch the velocity of a star orbited by a planet as a function of time. Denote the maximum velocity as K, and express the x-axis in terms of the number of periods, in intervals of P/4.

For this graph, we're starting at the point where the radial velocity is zero. The K is at it's peak hi and low at every other 1/4 of a period.  

e) What is the velocity amplitude, K, of an Earth-mass planet in a 1-year orbit around a Sun-like star?

This is another simple plug-and-chug problem where we take the equation from part (b) and substitute the sun's values where necessary.  

Thanks to WolframAlpha, I was able to find a K value of about 9.4 cm/s.  Earth's pretty freaking slow.