Planets outside our solar system are called exoplanets - Scottish Highers Physics - Question 5 - 2017

We apply Newton's law of universal gravitation, given by the formula:

$F = G \frac{m_1 m_2}{r^2}$

Where:

• $F$ is the gravitational force,
• $G$ is the gravitational constant ($6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2$),
• $m_1$ is the mass of the star ($3.83 \times 10^{30} \text{ kg}$),
• $m_2$ is the mass of the exoplanet ($5.69 \times 10^{27} \text{ kg}$),
• $r$ is the distance between them ($3.14 \times 10^{11} \text{ m}$).

Substituting in the values:

$F = 6.67 \times 10^{-11} \times \frac{(3.83 \times 10^{30})(5.69 \times 10^{27})}{(3.14 \times 10^{11})^2}$

Calculating this gives:

$F \approx 1.47 \times 10^{18} \text{ N}$

The redshift ($z$) can be calculated using the equation:

$z = \frac{v}{c}$

Where:

• $v$ is the velocity of the star,
• $c$ is the speed of light ($3.00 \times 10^8 \text{ m/s}$).

Given the acceleration of the star, we find its velocity over a unit of time (assuming it accelerates for 1 second):

$v = a \cdot t = 6.60 \times 10^{-12} \text{ m/s}^2 \cdot 1 \text{ s} = 6.60 \times 10^{-12} \text{ m/s}$

Now substituting this into the redshift formula:

$z = \frac{6.60 \times 10^{-12}}{3.00 \times 10^8} = 2.20 \times 10^{-20}$

For an exoplanet of greater mass orbiting the star, the gravitational influence on the star increases. This leads to a greater centripetal force requirement for maintaining the orbital motion. Hence, the resultant velocities of the star due to the gravitational interactions will be affected.

Therefore, the redshift will be greater for the greater mass exoplanet as compared to a smaller mass exoplanet, resulting from the differences in gravitational influence on the star.