HW5 Q1 (William Matzko)

1.  For radial velocities, we have the relation k = 28.4 (m/s) p^-(1/3)(M_p*sin(i)) (M_s)^(-2/3), where M_p is the planet mass in Jupiter masses, and M_s is the mass of the star in solar masses. For astrometry, we have α = (M_p/M_s)(a)(d)^(-1), where a is the semimajor axis in AU, and d is the distance to the system in parsecs. Just to make things easier to look at, we can make some simplifying assumptions. We are already told to assume sin(i) = 1 in all instances and M_s is the mass of the Sun. We might as well further assume that our target is one Jupiter mass. Further, we know P is proportional to a ^(3/2). The two above equations then reduce to
α = 0.001 (a) (d)^-1
k = 28.4 (m/s) (a)^(-1/2)

We are given that astrometry is sensitive to α = 10 micro arcseconds and above, and radial velocities are sensitive to k = 1 m/s and above. Note that only the astrometry method depends on d. We must now find when the astrometry method becomes more sensitive than the radial velocity method at this distance. To find this, I plotted the above equations for α and k for a wide range of "a" values. Below is the table I generated. 

As per my note in the above image, I made an "α" ratio and "k" ratio that can be thought of as the number of detections per observation. We see, then, that astrometry begins to surpass the radial velocity method at about a = 2 AU. So, for d = 10 pc, astrometry becomes more sensitive than RV at a = 2 AU. A similar approach is done for d = 100 pc. This yields a = 9.3 AU for when astrometry becomes more sensitive than RV. 

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