Estimate the approximate (minimum) mass of an exoplanet in a 3.312 day orbit around a 1.3 solar-mass star, where it is known that the radial velocity caused by orbital motion is 471 metres per second?

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Answer 1 · 2017-05-28T14:52:48

The planet has about 4 Jupiter masses.

Start with Newton's form of Kepler's third law:

#a^3=(GM)/(4pi^2)p^2#

Where #a# is the semi-major axis, #G# is the gravitational constant, #M# is the mass of the star and #p# is the orbital period.

Now for the Sun #GM=1.327*10^(20)#, so for a 1.3 solar mass star #GM=1.725*10^(20)#.

The period is #p= 3.312 * 86400=886156.8# seconds.

Substituting the values gives the semi major #a=7.099*10^9# metres.

The radial velocity of the planet is #v_p=(2pia)/p#.

Rearranging the Kepler equation:

#(4pi^2a^2)/p_2=(GM)/a#

Taking the square root gives:

#(2pia)/p=v_p=sqrt((GM)/a)#

Substituting values gives #v_p=1.55*10^5# m/s.

The momentum equation relates the sun star mass and velocity to the planet mass and velocity.

#Mv_s=m_pv_p#

A solar mass is #1.99*10^(30)#kg. So:

#m_p=(1.3*1.99*10^(30)*471)/(1.55*10^5)=7.86*10^(27)#kg.

Jupiter's mass is #1.898*10^(27)# kg.#

This makes the exoplanet about 4 Jupiter masses.