Stability of Orbits Homework: Analyzing Central Force Motion

[CAF123]

Homework Statement

A particle of mass m moves under a central force $\mathbf{F}(\mathbf{r}) = -\frac{\mu}{r^2} e^{-kr} \hat{r}$.
The particle undergoes motion in a circle if $h^2 = (a\mu/m)e^{-ka}$. I have shown that if $u(\theta) = 1/r,$ then the orbit eqn for $u(\theta)$ becomes $$u'' + u = \frac{1}{a}e^{-k(1/u - a)}.$$ This is a show that so I know it is correct.
I can't make further progress with the following:
Assume that the motion is slightly disturbed. Use the orbit eqn for $u(\theta)$ to show that in the perturbed orbit, the furthest distance from the centre occurs with a period of $2\pi/w$ where $w = \sqrt{1-ka}$

The Attempt at a Solution

Assume the perturbation, p(t), is small. That is p(t=0) << a. Let r = a + p(t) => u = 1/(a+p).
Differentiate twice wrt theta using the chain rule and I get $$u'' = -\frac{\ddot{p}}{a\mu e^{-ka}/m u^2}. $$ Sub into the orbit eqn: $$-\frac{\ddot{p} (a + p)^2 m}{a\mu e^{-ka}} + \frac{1}{a+p} = \frac{1}{a} e^{-k(1/(a+p) - a)}$$

Now Taylor expand the $(a+p)^2$, $(a+p)^{-1}$ and exp terms. Doing this, putting everything together, I get that $$\ddot{p} + \frac{\mu e^{-ka}}{ma^4} p = \frac{e^{1/a}}{ma^2}$$

I was thinking my next step would be to solve this for $p$, find the maximum of this (i.e the furthest distance it is perturbed) and then add this maximum to a. Does it sound good? (Can anyone confirm what I got above before I continue - it looks somewhat strange since when I find the solution, the arguments of sin/cos will contain an exponential). Actually something has had to have gone wrong since there will be a dimension associated with the arg of the trig terms.

MAny thanks.

 

[voko]

Differentiate twice wrt theta using the chain rule and I get $$u'' = -\frac{\ddot{p}}{a\mu e^{-ka}/m u^2}. $$

Seeing how you got that would be helpful. I am not saying this is incorrect, though.

Sub into the orbit eqn: $$-\frac{\ddot{p} (a + p)^2 m}{a\mu e^{-ka}} + \frac{1}{a+p} = \frac{1}{a} e^{-k(1/(a+p) - a)}$$

Not so fast. What does $$u'' = -\frac{\ddot{p}}{a\mu e^{-ka}/m u^2}$$ really mean? Is that $$u'' = -\frac{\ddot{p}}{\frac {a\mu e^{-ka}} {m u^2}} $$? If so, then $$-\frac{\ddot{p}}{a\mu e^{-ka}/m u^2} = -\frac{m u^2\ddot{p}}{a\mu e^{-ka}} = -\frac{\ddot{p} m}{a(a + p)^2\mu e^{-ka}} $$

 

[CAF123]

Seeing how you got that would be helpful. I am not saying this is incorrect, though.

First I computed u'. $$u' = -(a+p)^{-2} \dot{p} \frac{dt}{d\theta} = -\frac{\dot{p}}{(a+p)^2} \frac{1}{\sqrt{\frac{a\mu e^{-ka}}{m}}} u^2,$$ where $$\dot{\theta} = h/r^2 = \sqrt{\frac{a\mu e^{-ka}}{m}} \cdot \frac{1}{r^2}$$.

Then u'' = d/dθ (u') = $$-\frac{1}{\sqrt{\frac{a\mu e^{-ka}}{m}}} \frac{d}{d\theta} \frac{\dot{p}}{(a+p)^2} (a+p)^2 = -\frac{1}{\sqrt{\frac{a\mu e^{-ka}}{m}}} \frac{d}{dt} \dot{p} \frac{dt}{d\theta} = -\frac{\ddot{p}m}{a\mu e^{-ka} u^2}$$

I made a typo in the OP, it should be m/u^2 not /mu^2

Edit: is my method correct?

 

[haruspex]

$$e^{-k(1/(a+p) - a)}$$

Something wrong in the exponent. Should it just be -k(1/u - a) = -kp?

 

[voko]

How did you go from

$$u' = -(a+p)^{-2} \dot{p} \frac{dt}{d\theta} = -\frac{\dot{p}}{(a+p)^2} \frac{1}{\sqrt{\frac{a\mu e^{-ka}}{m}}} u^2,$$

to

$$u'' = -\frac{1}{\sqrt{\frac{a\mu e^{-ka}}{m}}} \frac{d}{d\theta} \frac{\dot{p}}{(a+p)^2} (a+p)^2$$

given that

$$ u^2 = \frac 1 {(a + p)^2} $$

?

 

[CAF123]

How did you go from

$$u' = -(a+p)^{-2} \dot{p} \frac{dt}{d\theta} = -\frac{\dot{p}}{(a+p)^2} \frac{1}{\sqrt{\frac{a\mu e^{-ka}}{m}}} \mathbf{u^2},$$

For some reason, I had a few initial problems writing the above out in latex, so a typo has crept in. u^2 should be on the denominator.

to

$$u'' = -\frac{1}{\sqrt{\frac{a\mu e^{-ka}}{m}}} \frac{d}{d\theta} \frac{\dot{p}}{(a+p)^2} (a+p)^2$$

given that

$$ u^2 = \frac 1 {(a + p)^2} $$

?

 

[CAF123]

Something wrong in the exponent. Should it just be -k(1/u - a) = -kp?

Indeed it should.

 

[voko]

I think this $\frac{\mu e^{-ka}}{ma^4} p$ should be $\frac{\mu e^{-ka}}{ma^3} p$. Then the units seem OK. What equation do you get when you fix the exponent and linearize?

 

[CAF123]

I think this $\frac{\mu e^{-ka}}{ma^4} p$ should be $\frac{\mu e^{-ka}}{ma^3} p$. Then the units seem OK. What equation do you get when you fix the exponent and linearize?

That is exactly the case. The final equation I attain after Taylor expanding is $$\ddot{p} + \frac{\mu e^{-ka}}{ma^3} p = \frac{k\mu e^{-ka}}{ma^2}p,$$ which I can subsequently write as $$\ddot{p} + p(\frac{\mu e^{-ka}}{ma^2} \left(\frac{1}{a} - k\right) = 0$$ Here I identify the harmonic solution, with $$w^2 = (\frac{\mu e^{-ka}}{ma^2} \left(\frac{1}{a} - k\right)$$

This can further be simplified using $h^2$ given in the problem statement. Hence $w^2 = \frac{h^2}{a^4} (1-ka)$ which is incorrect by a factor of $h/a^2$.

 

[voko]

$\exp -kp$, linearized, should have a constant term. Which is actually quite bizarre, because that means p = 0 is an equilibrium point for the perturbed system.

 

[CAF123]

$\exp -kp$, linearized, should have a constant term. Which is actually quite bizarre, because that means p = 0 is an equilibrium point for the perturbed system.

When I linearised $e^{-kp}$ I got $1 - kp$. Is this not correct?

 

[voko]

Yes, this is correct. But that means that the equation is $\ddot{p} + \lambda p = c$, which means that $p = c/\lambda$ is the equilibrium point. Not $p = 0$ as expected.

 

[CAF123]

Yes, this is correct. But that means that the equation is $\ddot{p} + \lambda p = c$, which means that $p = c/\lambda$ is the equilibrium point. Not $p = 0$ as expected.

I brought everything together on the LHS to get the familiar harmonic equation and then identified $w^2$. Is my final eqn correct that I posted in my last two posts? When I rearrange I get 0 on the RHS and so p = 0 would correspond to the equibilrium soln, no?

 

[voko]

You got the equation in the form $\ddot{p} + \lambda p = 0$. How is that possible if the original RHS had a constant term?

 

[CAF123]

The eqn I got before rearranging was:

$$\ddot{p} + \frac{\mu e^{-ka}}{ma^3} p = \frac{k\mu e^{-ka}}{ma^2}p,$$ which I can subsequently write as $$\ddot{p} + p(\frac{\mu e^{-ka}}{ma^2} \left(\frac{1}{a} - k\right) = 0$$

You got the equation in the form $\ddot{p} + \lambda p = 0$. How is that possible if the original RHS had a constant term?

The RHS depends on p, so when I bring it over I have the form $\ddot{p} + \lambda p = 0$. (Unless ofcourse I made an error). I am just off to a tutorial now.

 

[voko]

In #11 you said that the RHS should have a constant term. In #15, however, there is no such term.

 

[CAF123]

In #11 you said that the RHS should have a constant term. In #15, however, there is no such term.

I had the form $$-\frac{\ddot{p} (a+p)^2m}{a\mu e^{-ka}} + \frac{1}{a+p} = \frac{1}{a} e^{-kp}$$

Now linearise: $$-\frac{m}{a\mu e^{-ka}} \ddot{p} \left[a^2(1 + 2\frac{p}{a}) \right] + \frac{1}{a} (1-\frac{p}{a}) = \frac{1}{a}e^{-kp}$$

When I bring that single 1/a term on the LHS over to the RHS, I get $\frac{1}{a}(e^{-kp} - 1)$. Now expand the exp and the 1's cancel, leaving only a term in p.

Edit: I neglected the term containing $\ddot{p}p$

 

[voko]

I can't see any mistake in your derivation (assuming the original equations for u and h are correct).

Unless I am mistaken, $\frac {\mu e^{-ka}} {m a^3} = \Omega^2$, where $\Omega$ is the angular velocity of the circular motion. So $\omega = \Omega \sqrt {1 - ka}$. Can't this be what you are really asked about?

 

[CAF123]

I can't see any mistake in your derivation (assuming the original equations for u and h are correct).

Unless I am mistaken, $\frac {\mu e^{-ka}} {m a^3} = \Omega^2$, where $\Omega$ is the angular velocity of the circular motion. So $\omega = \Omega \sqrt {1 - ka}$. Can't this be what you are really asked about?

Yes, the question as stated in the OP is the one I need to answer. I see that $h/a^2 = \dot{\theta}$ so $\omega = \dot{\theta} \sqrt{1 - ka}$

I don't see how the dimensions of $\sqrt{1-ka}$ are correct anyway. I believe k has dimensions of 1/length. So $\sqrt{1-ka}$ would be dimensionless. $\dot{\theta} \sqrt{1-ka}$ has units of 1/s, as required. The question is a mistake maybe?

 

[voko]

They may have asked a bogus question to have you defend your "wrong" answer, they do that sometimes.

They might also have meant the "relative" period of the perturbed motion (in units such that the orbital period = 1), but then the answer is still off by a factor of $2 \pi$.

 

[CAF123]

They may have asked a bogus question to have you defend your "wrong" answer, they do that sometimes.

They might also have meant the "relative" period of the perturbed motion (in units such that the orbital period = 1), but then the answer is still off by a factor of $2 \pi$.

I do see they mention '...show that in the perturbed orbit, the furthest distance from the centre occurs with...'. Did I ever make use of this 'furthest distance'?

Do you agree with my comments about the dimensionality of the solutions?

 

[voko]

I can't see how that could change anything. $r = a + p = a + \sin (\omega t + \alpha )$, so the period between two maxima is still $2 \pi / \omega$.

Your dimensional analysis is quite correct, too.

 

[CAF123]

I spoke to some other people about this and they get the required $\sqrt{1-ka}$. Their argument was along the lines of say, for example, you have sin(2θ). The period of this is 2pi/2,
where the 2 is dimensionless.

This seems to make sense mathematically, but physically since $T = \frac{2\pi}{\omega}$, I don't really understand why it can't have units. (Unless of course there is intrinsic units associated with the '2', but that would contradict the statement of the 2 being dimensionless.

What are your thoughts?

 

[voko]

This is along the lines of the second sentence in #20. If the orbital period is taken to be the unit of time (just like we do with the "year"), then you could measure the features of the perturbed motion in this unit. When I thought about that yesterday, though, I think I was not getting the $2 \pi$ factor. Please give it a try and see where that gets you.

 

[CAF123]

This is along the lines of the second sentence in #20. If the orbital period is taken to be the unit of time (just like we do with the "year"), then you could measure the features of the perturbed motion in this unit. When I thought about that yesterday, though, I think I was not getting the $2 \pi$ factor. Please give it a try and see where that gets you.

What's the $2 \pi$ factor? Even if I were to assume that defintion, (i.e it is okay to have null dimensionality associated with the answer), it doesn't explain where the $\dot{\theta}$ comes from.

 

[voko]

As mentioned earlier, $\omega = \Omega \sqrt {1 - ka}$. Note I prefer $\Omega$ to $\dot{\theta}$ because the latter is not constant in a perturbed orbit.

Now we can always choose the time unit such that $\Omega = 1$. Using this time unit, $\omega = \sqrt {1 - ka }$ as required. Note also that $\sqrt {1 - ka }$ is independent of the time unit, because both $k$ and $a$ are strictly length-related.

 

[CAF123]

As mentioned earlier, $\omega = \Omega \sqrt {1 - ka}$. Note I prefer $\Omega$ to $\dot{\theta}$ because the latter is not constant in a perturbed orbit.

But if they represent the same thing, then surely the choice of symbol does not matter?
EDIT: I understand why: h/a^2 is a constant and is the velocity when r=a.

Now we can always choose the time unit such that $\Omega = 1$.

Could you maybe explain this step in a little more detail?
Thanks

 

[voko]

But if they represent the same thing, then surely the choice of symbol does not matter?

I denote by $\Omega$ the angular velocity of the circular orbit. Which is a constant. $\dot{\theta}$ is the angular velocity of any possible orbit, which does not have to be constant and which does not have to be equal to $\Omega$.

Could you maybe explain this step in a little more detail?
Thanks

Say you use seconds to measure $\Omega$ and it happens to be $5 s^{-1}$. You could then use the time unit which is $5^{-1} s$, let's call this unit a "quintasecond" and denote it as "q", then $\Omega = 1 q^{-1}$.

Or we could use dimensionless time, defined as $t* = \Omega t$, then the dimensionless angular frequency of the perturbation $\omega* = \frac {d\phi} {dt*} = \frac {d\phi} {dt} \frac {dt} {dt*} = \frac {\omega} {\Omega} = \sqrt {1 - ka}$ (where $\phi$ is the phase of the perturbation).

 

[CAF123]

Say you use seconds to measure $\Omega$ and it happens to be $5 s^{-1}$. You could then use the time unit which is $5^{-1} s$, let's call this unit a "quintasecond" and denote it as "q", then $\Omega = 1 q^{-1}$.

.
I think I see what you are doing, but to understand the physical interpretation: what does a time unit mean and physically what are you doing by defining this time unit?

 

[voko]

I just say that this shorter or longer duration is my time unit. They are completely arbitrary to begin with. But they do correspond to some physical process. A useful illustration is the year. This is basically a unit such that the orbital period of the Earth is 1. The month is the (roughly) orbital period of the Moon. The day is a unit such that the period of the Earth rotation is 1. We use all these units daily (oops, just did again), yet they were specifically chosen so that some particular celestial motions had periods = 1.

Note I am talking about periods here. In my previous post I was talking about the angular speed. There is no real difference, it all depends on what we consider more basic, the frequency or the period.

 

[CAF123]

I just say that this shorter or longer duration is my time unit. They are completely arbitrary to begin with. But they do correspond to some physical process. A useful illustration is the year. This is basically a unit such that the orbital period of the Earth is 1. The month is the (roughly) orbital period of the Moon. The day is a unit such that the period of the Earth rotation is 1. We use all these units daily (oops, just did again), yet they were specifically chosen so that some particular celestial motions had periods = 1.

Note I am talking about periods here. In my previous post I was talking about the angular speed. There is no real difference, it all depends on what we consider more basic, the frequency or the period.

Thanks voko,