Q. 53 - Calculus

Question

Measurements indicate that Earth’s orbital eccentricity is $0.0167$ and its semimajor axis is $1.00000011$ astronomical units.

(a) Write a Cartesian equation for Earth’s orbit. 

(b) Give a polar coordinate equation for Earth’s orbit, assuming that the sun is the focus of the elliptical orbit. 

Step-by-Step Solution

Answer

Part (a) The cartesian equation is $\frac{x^2}{(1.00000011{)}^2}+\frac{y^2}{(0.9998606552{)}^2}=1$.

Part (b) The polar equation is $r=\frac{1.00000011}{1+0.0167\cos \theta }$.

Part (a) Step 1. Given information.

The eccentricity is $0.0167 .$

The semi-major axis is  $1.00000011\text{ astronomical units.}$

Part (a) Step 2. Explanation.

Now,

$$\begin{gathered} e=\frac{\sqrt{A^2-B^2}}{A} \\ A=1.00000011, e=0.0167 \\ 0.0167=\frac{\sqrt{(1.00000011{)}^2-B^2}}{1.00000011} \\ 0.0167(1.00000011)=\sqrt{(1.00000011{)}^2-B^2} \\ 0.0167000018=\sqrt{1.00000022-B^2} \\ (0.0167000018{)}^2={\left(\sqrt{(1.00000022)-B^2}\right)}^2 \\ 0.00027889006=(1.00000022)-B^2 \\ {B}^2=0.99972133 \\ {A}^2=(1.00000011{)}^2=1.00000022 \\ Therefore, \\ \frac{x^2}{1.00000022}+\frac{y^2}{0.99972133}=1\text{ or }\frac{x^2}{(1.00000011{)}^2}+\frac{y^2}{(0.9998606552{)}^2}=1 \end{gathered}$$

Part (b) Step 1. Explanation.

Now, the standard form,

$$\begin{gathered} r=\frac{eu}{1+e\cos \theta } \\ Therefore, \\ r=\frac{1.00000011}{1+0.0167\cos \theta } \end{gathered}$$