# Calculating the distance a satellite traveled

1. Oct 22, 2011

### shemer77

1. The problem statement, all variables and given/known data
Im trying to figure out how to calculate how much a satellite in an elliptical orbit travels around in a day. The satellite is in a 12 day orbit with earth focused at (3,0). The equation i have for the ellipse is 16x^2 +25y^2 = 400. Other relevant information i have is the area that the satellite covered total every day, along with a and b coordinates for where it is every day, however I am having trouble cacluating the distance it travels day by day.

3. The attempt at a solution

My first intution was to use the ∫ of √(25*sin(t)^2 + 16*cos(t)^2) dt and then just change the limits of integration, but im not sure if that would be right and what would i set the limits too?

2. Oct 22, 2011

### dynamicsolo

Are you expected to compute the arclength along the ellipse analytically or numerically? I ask because I believe you are going to have little success getting a function that expresses the arclength. What you have on your hands is an "elliptic integral of the second kind": there is an exact value for the full perimeter of the ellipse, but the problem (a fairly old one) of finding the distance along an arbitrary arc along the ellipse does not produce an arclength integral with an antiderivative "in closed form".

Applying the Pythagorean Identity to your integral (which looks correct to me) gives us $\int_{t_{1}}^{t_{2}} \sqrt{25 - 9 \cos^{2} t } dt$ , which now has the typical form of such an elliptic integral. Unfortunately, not much can be done with this in exact form. You will need to use a numerical approximation method to get arclength values for each day, starting, say, from perigee.

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