Final answer:
The minimum mass of a planet with a given Doppler shift and period can be calculated using Kepler's third law and considering the star's radial velocity change due to the planet's gravitational influence, assuming an edge-on orbit.
Step-by-step explanation:
To calculate the minimum mass of a planet with a Doppler shift of 30 m/s and a period of 2 days orbiting a star with the same mass as the Sun, we can use the formula derived from Kepler's third law and the equation that relates the radial velocity of the star due to the planet's gravitational influence.
The mass function can be given by:
f(m) = (mp sin i)3 / (ms + mp)2 = (P v3 ) / (2 π G)
Where:
- P is the orbital period
- v is the radial velocity
- G is the gravitational constant
- mp is the mass of the planet
- ms is the mass of the star
- i is the inclination of the orbit
Assuming an inclination i of 90 degrees (edge-on), the sin i term is 1, and the mass of the star ms is far greater than the mass of the planet mp (ms >> mp) which allows us to approximate (ms + mp)2 as ms2.
Plugging in the given values and solving for the mass of the planet mp yields a minimum mass for the planet. We use units where G is in m3 kg-1 s-2, P is in seconds, and v is in m/s. This calculation will reveal the minimum mass of the planet, which is the mass it would have if the orbit is indeed edge-on.