Question

The perihelion (Rp) of Mars’ orbit is 1.381 AU while its aphelion (Ra) is 1.666 AU. Using the relation Rp = a(1 - e) and Ra = a(1 + e),where a is the length of the semi-major axis and e is the eccentricity of the orbit, calculate a and e for Mars’ orbit around the Sun. Show all of your work

Answer 1

a = 1.523

e = 0.09353

Explanation:

From the statement we can establish the following two equations:

`\begin{gathered} R_p=a\mleft(1-e\mright)\rightarrow a=(R_p)/(1-e)\text{ (1)} \\ R_1=a\mleft(1+e\mright)\rightarrow a=(R_a)/(1+e)\text{ (2)} \end{gathered}`

We equate both equations and solve for e, like this:

`\begin{gathered} \frac{R_p_{}}{1-e}=(R_a)/(1+e) \\ (1.381)/(1-e)=(1.666)/(1+e) \\ (1.381)\cdot(1+e)=(1.666)\cdot(1-e) \\ 1.381+1.381e=1.666-1.666e \\ 1.666e+1.381e=1.666-1.381 \\ 3.047e=0.285 \\ e=(0.285)/(3.047) \\ e=0.09353 \\ \\ \text{now, for a, replacing in (1):} \\ a=(1.381)/(1-0.09353) \\ a=1.523 \end{gathered}`