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?

1 Answer
May 28, 2017

The planet has about 4 Jupiter masses.

Explanation:

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.