Time-weighted average distance in an elliptical orbit

[DJSedna]

Homework Statement

Using the polar formula for an ellipse, and Kepler's second law, find the time-weighted average distance in an elliptical orbit.

Homework Equations

The polar formula for an ellipse:

$$r = \frac { a(1-e^2)} {1 \pm e cos \theta},$$

Area of an ellipse:

$$ A = \pi a b $$

$$ b = \sqrt{a(1 - e^2)} $$

The Attempt at a Solution

I don't know if you'd call this much of an attempt, but I understand I need to be taking some sort of integral with respect to time. I have genuinely been staring at this for hours with no idea where to start, though, and I need some idea of how to get going.

I've messed with just about every algebraic combination of the three equations above, but I haven't found anything that pops out at me and says "oh, that's it."

Sorry if this is too vague, this is the first time I've ever really needed to post here. Let me know if I can add any more information.

Thanks!

 

[TSny]

Welcome to PF!

The problem mentions Kepler's second law. That will probably be very helpful.

Also, in general, if you have a function of time $f(t)$, how would you set up an integral to represent the time average of the function over a time interval from $t = 0$ to $t = T$?

 

[DJSedna]

Welcome to PF!

The problem mentions Kepler's second law. That will probably be very helpful.

Also, in general, if you have a function of time $f(t)$, how would you set up an integral to represent the time average of the function over a time interval from $t = 0$ to $t = T$?

Thanks!

Allegedly we have all of the information needed in those three equations, and does Kepler's Second Law have an actual mathematical form? If it does, I've gone through four years of undergrad, a year of research, and a year of grad school with misinformation, haha.

For such an integral, you'd need to do something with $\theta$ going from $0$ to $2\pi$, no?

 

[TSny]

does Kepler's Second Law have an actual mathematical form?

Yes.
https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion#Second_law

For such an integral, you'd need to do something with $\theta$ going from $0$ to $2\pi$, no?

Yes. But start with the general idea of setting up an integral for the time average of $r(t)$ for an orbit. In calculus, you probably covered finding the average of a function $f(x)$ over some interval $a < x < b$. If you need a review, try a web search for "average of a function integral". Then you can apply the general idea to this problem.

 

[DJSedna]

Yes.
https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion#Second_law

Yes. But start with the general idea of setting up an integral for the time average of $r(t)$ for an orbit. In calculus, you probably covered finding the average of a function $f(x)$ over some interval $a < x < b$. If you need a review, try a web search for "average of a function integral". Then you can apply the general idea to this problem.

Okay, is this what you're talking about? It does look vaguely familiar.

http://tutorial.math.lamar.edu/Classes/CalcI/AvgFcnValue.aspx

One thing, though---there's no time in the equations above. I might be too burnt out from a week of intense finals and missing something.

 

[TSny]

Okay, is this what you're talking about? It does look vaguely familiar.

http://tutorial.math.lamar.edu/Classes/CalcI/AvgFcnValue.aspx

Yes

One thing, though---there's no time in the equations above.

Once you set up the time integration that represents the time average of $r$, $\left<r\right>$, you can try a change of integration variable from $t$ to $\theta$. This will allow you to express $\left<r\right>$ as some integral with respect to $\theta$.

 

[DJSedna]

YesOnce you set up the time integration that represents the time average of $r$, $\left<r\right>$, you can try a change of integration variable from $t$ to $\theta$. This will allow you to express $\left<r\right>$ as some integral with respect to $\theta$.

I'm sorry, I guess I'm not fully picking up on what you're saying. How am I setting up a time integral with nothing that has time in it?

 

[TSny]

$r$ varies with time as the planet moves in its orbit. So, you can think of $r$ as a function of $t$, $r(t)$. If you knew the function $r(t)$, how would you set up an integral that represents the time average of $r(t)$?

 

[DJSedna]

$r$ varies with time as the planet moves in its orbit. So, you can think of $r$ as a function of $t$, $r(t)$. If you knew the function $r(t)$, how would you set up an integral that represents the time average of $r(t)$?

I'd do something like

$$ \frac{1}{t - 0} \int_{0}^{t} r(t) dr$$

But I don't know how to get an $r(t)$ with what I have right now.

 

[TSny]

I'd do something like

$$ \frac{1}{t - 0} \int_{0}^{t} r(t) dr$$

OK. You want the average over one orbit. So, what specific time should you use for the upper limit of the integral?

Also, you wrote the differential inside the integral as $dr$. Did you mean $dt$?

But I don't know how to get an $r(t)$ with what I have right now.

You'll be able to do a change of variable of integration from $t$ to $\theta$. This is where Kepler's 2nd law will be helpful.

 

[DJSedna]

OK. You want the average over one orbit. So, what specific time should you use for the upper limit of the integral?

Also, you wrote the differential inside the integral as $dr$. Did you mean $dt$?

You'll be able to do a change of variable of integration from $t$ to $\theta$. This is where Kepler's 2nd law will be helpful.

Yeah, I meant to write $dt$, my bad.

For the time of one orbit, you'd want to integrate from 0 to $P$? Am I intended to use an equation for $P$ as my upper-bound of integration?

 

[TSny]

Yeah, I meant to write $dt$, my bad.

For the time of one orbit, you'd want to integrate from 0 to $P$?

Yes, assuming $P$ is the period of the orbit. Or you could use $P/2$ since the shape of the orbit is symmetrical.

Am I intended to use an equation for $P$ as my upper-bound of integration?

You won't need an equation for $P$.