How Many Stars Can Detectable Wobbles Reveal Extrasolar Planets?

[nick21324neo]

Description: An extrasolar planet can be detected by observing the wobble it produces on the star around which it revolves. Suppose an extrasolar planet of mass revolves around its star of mass . If no external force acts on this simple two-object system, then its CM is stationary. Assume and are in circular orbits with radii and about the system's CM.

A) Find radius of the star's orbit about the system's CM. (answered)

ra= (mb)(rb)/ma

B)Now consider a Sun-like star and a single planet with the same characteristics as Jupiter. That is, and the planet has an orbital radius of 7.8×1011 . Determine the radius of the star's orbit about the system's CM. (answered)

*used formula above: ra=8.58E8

C)When viewed from Earth, the distant system appears to wobble over a distance of . If astronomers are able to detect angular displacements of about 1 milliarcsec (1/3600 of a degree), from what distance d (in light-years = 9.46E15) can the star's wobble be detected ? (answered)

d = 37 lightyears; I found the answer by using the ArcLength formula:

Arclength=Radius of Arc*theta

D) The star nearest to our Sun is about 4 lightyears away. Assuming stars are uniformly distributed throughout our region of the Milky Way Galaxy, about how many stars can this technique be applied to in the search for extrasolar planetary systems?

Attempt for part D): I attempt this problem by using basic geometry

I determined the circumference by using the the distance of 4 light year and divided it by 2ra

The answer is found: 1.4E8 stars which is wrong. What am I doing wrong? It just seems to be a lot of stars.

Thanks

 

[mfb]

An exact calculation will be more complicated but for a reasonable approximation we can assume that the star density is 1 star within the volume of up to 4 light years distance (4/3 pi (4 ly)³). We don't have to calculate that volume, just the r³ dependence is important: Up to 37 light years we have (37/4)³ = 791 times the volume. We expect about 800 stars with a distance of up to 37 light years.

Gaia is currently taking data, it will detect wobbles of the order of 0.1 milliarcseconds. It is expected to find thousands to tens of thousands of planets that way.