Orbital Dynamics/Keppler's Law

[DRM126]

Homework Statement

An asteroid revolves around the sun with a mean orbital radius twice that of Earth's. Predict the period of the asteroid in Earth years

Homework Equations

I guess t^2/r^3 where t = period and r = mean orbital radius

The Attempt at a Solution

I missed the lecture in class, and I'm ultimately confused on the subject of Orbital Dynamics and Kepler's Law, any help/explanation would be great, or a link to a useful website, honestly I'm clueless. I think the fact that the r for the asteroid is twice as long as Earth's creates a ratio?

 

[fluidistic]

Check out http://en.wikipedia.org/wiki/Kepler's_laws. I cannot enter the website right now but if I remember well it had an example of the third law I think.

 

[thomate1]

Hello,

Orbital Dynamics and Kepler's Law are fundamental principles in the study of celestial mechanics. Kepler's Law states that the square of the orbital period (t) of a planet or object is directly proportional to the cube of its mean orbital radius (r). This can be expressed mathematically as t^2 = k*r^3, where k is a constant.

In this scenario, we are given that the mean orbital radius of the asteroid is twice that of Earth's. This means that the ratio of the asteroid's mean orbital radius (r_a) to Earth's mean orbital radius (r_e) is 2:1. We can use this information to set up a proportion:

t_a^2 / r_a^3 = t_e^2 / r_e^3

Where t_a and t_e are the periods of the asteroid and Earth, respectively. We can then rearrange this equation to solve for the asteroid's period:

t_a = (t_e * r_a^3 / r_e^3)^1/2

Plugging in the values for Earth's period (1 year) and the ratio of the mean orbital radii (2), we get:

t_a = (1 year * 2^3 / 1^3)^1/2 = 2 years

Therefore, the predicted period of the asteroid in Earth years is 2 years. I hope this helps clarify the concept of Kepler's Law and its application to orbital dynamics. For more information and practice problems, I recommend checking out online resources such as Khan Academy or MIT OpenCourseWare.