Show Intro/Instructions We are going to look at how the radial velocity method c
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Question
Show Intro/Instructions We are going to look at how the radial velocity method can be used to determine the mass of a planet. The observations that get made look at two things: the wobble of the star (it is measured as a change in the speed the star traveling towards or away from us) and the amount of time it takes for the pattern to repeat itself Let's consider the planet beta Gemini b. The change in the speed of the star due to the planet's tug is approximately Ustar 40 m/sec. Knowing that the mass of the star is Mstar 2Mo and that the period of the orbit is approximately P-# 2 years, we can estimate the orbital separation a . Recall that Kepler's Third Law is MstaP a. Mo year AU where M is the mass of the star, P is the period of the orbit, and a is the separation. Notice the scaling. If M,P, and a are expressed in the right units then the calculation becomes much easier. What is the orbital separation between the star and the planet in astronomical units? Preview Given your answer above, let's estimate the speed of the planet in kilometers per second. This can be determined using the equation 2ma 6a for a circular orbit (and this planet is nearly on a circular orbit). What is the velocity of the planet in km/sec? Preview Now we can calculate the planet's mass by using the equation Mstar Ustarmplanet Vplanet What is the mass of the planet in Jupiter masses? A convenient conversion factor is that 1Mo 10 Miupiter PreviewExplanation / Answer
a) Here we know that from the Kepler’s law
(Mstar/M? )*(P/year)2 =(a/AU)3 -------------------(1)
Where P is the period of revolution in year and “a”is the orbital separation in Astronomical units
Given that Mstar=2* M? and P =2years
So substituting the values in equation (1) we get
The orbital separation a = (2*22)(1/3) =81/3 =2 AU
b) velocity of the plannet is given by the equation
vplannet = (2*?*a/T) =(6*a/T) ------------------------------(2)
here a=2 AU =(2* 149597871 km) -----------------------------(3)
T= 2 year =(2*31557600 s) -----------------------------(4)
Substitute (3) and (4) in (2)
so the vplannet = (6*2* 149597871)/( 2*31557600)
=28.44 km/s
c) the mass of the plannet can be calculated by the given relation
Mstar*Vstar = mplannet*vplannet ----------------------------------(5)
The Velcity of star is already given Vstar= 40 m/s = 0.04 km/s
velocity of the plannet vplannet=28.44 km/s
And the mass of the star is given that Mstar = 2* M? =2*103 Mjupiter
Substituting the values we get the mass of plannet
mplannet =(Mstar*Vstar)/ vplannet = (2*103 Mjupiter*0.04 km/s)/ (28.44 km/s)
=80 Mjupiter/28.44 = 2.816 Mjupiter