Question

Suppose earth's orbital motion around the sun supposed to be in circular is suddenly stopped, what time will the earth take to fall into the sun?

• A. $65$ days
• B. $130$ days
• C. $183$ days
• D. $30$ days

Answer 1

The correct option is A $65$ days

Kepler's law applies to planetary orbits, whether they be of circular, or elliptical shape. It says that $\frac{T_2^2}{T_2^1}=\frac{R_3^2}{R_3^1}$ where T is the period of an orbit and R is its semi-major axis. The semi-major axis is the average of the planet's maximum and minimum distances from the sun.

Let the earth's mean radius be $R_1$..This straight fall can be considered 1/2 of a degenerate elliptical orbit with major axis equal to $R_1$. Its semi-major axis is $\frac{R_1}{2}$ (the average of R1 and zero). Its period will be designated $T_2$.

So: $\frac{T_2^2}{T_1^2}=\frac{{\left(\frac{R_1}{2}\right)}^3}{R_1^3}={\left(\frac{1}{2}\right)}^3$

And therefore, $T_2=0.353$ year, and the time to fall into the sun is the time to fall into the sun is ,64.52 days