Exercise 13

Question 1

Calculate the following quantities:

1. The speed of the Earth around the Sun, using Newton’s law, and assuming a circular orbit: $\( v = \sqrt{\frac{G M_{\odot}}{R}} \)$

   where \(M_{\odot}\) is the mass of the sun. Provide your answer in units of m/s and pc/yr.

2. The centripetal acceleration of the Earth: $\( a = \frac{v^2}{R} \)$

   Provide your answer in units of m/s\(^2\) and AU/yr\(^2\).

Question 2

The amount of energy from the Sun’s radiation that reaches the Earth over a year can be approximated by:

\[ E = 1\text{year } \times L_{\odot} \times \frac{\pi R_{E}^2}{4 \pi (1\text{AU})^2} \]

where \(L_{\odot}\) is the solar luminosity and \(R_{E}\) is the radius of the Earth. The assumptions that solar radiation near the Earth can be approximated as planar and that the Earth has a circular orbit of 1AU were used.

1. Calculate \(E\) using Astropy and print the result in Joules.

2. Print out the equivalent units Astropy has for Joules.