Solving Kepler's Equation for Comet Orbit Time Around Sun

[susan__t]

Homework Statement

A comet is 10 times farther from the sun than the Earth. Find the time it take to make its orbit around the sun.

Homework Equations

T^2/R^3 = K
K= 3.36X10^18

The Attempt at a Solution

I tried to create a ratio, but I honestly have no idea where to start. I'm not even sure I need to know kepler's constant for this problem. and the answer is T=31.6

I don't know what to put as R, and that leaves me with two variables, R and T.
I also don't know if Kepler's constant is the same for the comet and the earth.

 

[Kurret]

since k is equal for both objects, you can set each objects time^2/distance^3 ratio equal to each other. Then use that $$R_{comet}=10R_{earth}$$.

 

[Moetasim]

I would approach this problem by first understanding the fundamental principles behind Kepler's equation and its application to celestial bodies in orbit around the sun. Kepler's equation is a mathematical formula that relates the orbital period (T) of a celestial body to its distance from the sun (R). The constant K in the equation represents the combined mass of the sun and the celestial body, and it is the same for all objects in orbit around the sun.

In this case, we are given that the comet is 10 times farther from the sun than the Earth. Therefore, we can express the distance of the comet from the sun as 10 times the distance of the Earth from the sun, or 10R (where R is the distance of the Earth from the sun). We can also express the orbital period of the comet as T (since that is what we are trying to find).

Substituting these values into Kepler's equation, we get:

T^2/(10R)^3 = K

Simplifying this equation, we get:

T^2/1000R^3 = K

Since we know the value of K (3.36x10^18), we can rearrange the equation to solve for T:

T^2 = 1000R^3 x 3.36x10^18

Taking the square root of both sides, we get:

T = √(1000R^3 x 3.36x10^18)

Now, we need to determine the value of R. We know that the distance of the Earth from the sun (R) is approximately 149.6 million kilometers. Therefore, the distance of the comet from the sun (10R) would be 149.6 million kilometers x 10, or 1.496 billion kilometers.

Plugging this value of R into the equation, we get:

T = √(1000 x (1.496 billion)^3 x 3.36x10^18)

Simplifying this, we get:

T = √(1.496^3 x 1000 x 3.36x10^18) x 10^9

T = √(7.113 x 10^27) x 10^9

T = 2.666 x 10^19 seconds

Converting this to years, we get:

T = 846,330,000 years