Planet moving in elliptical orbit around a star

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    Centripetal Orbit
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Using centripetal force equations for a planet in an elliptical orbit is incorrect because these equations assume a constant radius, which contradicts the nature of elliptical motion. The discussion emphasizes that angular momentum is conserved, and the correct approach involves understanding the radial component of acceleration, which is not solely centripetal. The concept of central potential is introduced, indicating that the force acting on the planet depends on its distance from the star. The conversation concludes with a clarification on angular momentum and the need to avoid confusion with incorrect formulas.
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Homework Statement
Please see below
Relevant Equations
Not sure
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(a) I use centripetal force to solve this question, even though the question states elliptical orbit

$$\frac{GMm}{r^2}=m\omega^2 r$$
$$\omega^2 r^3=GM$$

So,
$$\omega_{A}^{2} r_{A}^{3}=\omega_{B}^{2} r_{B}^{3}$$
$$\frac{r_A}{r_B}=\left(\frac{\omega_{B}}{\omega_{A}}\right)^{\frac{2}{3}}$$

Is this correct?

Thanks
 
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I don't think you can use that formula in an elliptical orbit. There can be a radial component to the acceleration.
What do you know is conserved?
 
haruspex said:
I don't think you can use that formula in an elliptical orbit.
Just let me emphasise that you most definitely cannot use that formula. It assumes that the radius is constant, which directly contradicts the assumptions of the problem.

As an additional hint, all you need to know here is that the potential is central. The actual functional behaviour of the force magnitude is irrelevant.
 
haruspex said:
I don't think you can use that formula in an elliptical orbit. There can be a radial component to the acceleration.
Is the radial component of acceleration the centripetal acceleration? Or is it something else?

haruspex said:
What do you know is conserved?
I think it should be angular momentum, so:
$$m v_A r_A = m v_B r_B$$

I don't think I can use ##v=\omega r## because it is not circular motion so how to relate the equation to ##\omega##?

Orodruin said:
As an additional hint, all you need to know here is that the potential is central. The actual functional behaviour of the force magnitude is irrelevant.
Sorry, what do you mean by "potential is central"?

Thanks
 
songoku said:
I think it should be angular momentum, so:
$$m v_A r_A = m v_B r_B$$
Not exactly. The angular momentum depends only on the tangential component of the velocity.

songoku said:
I don't think I can use ##v=\omega r## because it is not circular motion so how to relate the equation to ##\omega##?
As such, ##\omega r## is definitely equal to the tangential component of the velocity by definition.

songoku said:
Sorry, what do you mean by "potential is central"?
A central potential depends only on ##r##, meaning the force is always pointing to the center.
 
Orodruin said:
Just let me emphasise that you most definitely cannot use that formula. It assumes that the radius is constant, which directly contradicts the assumptions of the problem.
An apparent paradox is that, if one assumes that the planet moves in a straight line at constant velocity, one gets the correct answer for part (a).
 
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Seeing the question, the first part can be done using Kepler law which you have done correctly. For second part basic conservation of energy can be utilised to solve it.
 
songoku said:
Is the radial component of acceleration the centripetal acceleration? Or is it something else?
Sorry, I should have been clearer. A radial component other than centripetal acceleration, i.e. ## \ddot r##
 
Super Man said:
the first part can be done using Kepler law which you have done correctly
If you mean Kepler III, the radius in that law refers to the semi-major axis. It is a law relating all planets orbiting the same star, not all positions of a given planet. It does not apply to the first part of this question.
 
  • #10
Orodruin said:
Not exactly. The angular momentum depends only on the tangential component of the velocity.
Maybe you mean something like this?
$$mv_Ar_A\sin \theta=mv_Br_B\sin \beta$$

where ##\theta## and ##\beta## is the angle between ##v## and ##r## at point A and B. Using ##v=\omega r##:
$$\omega_A r_{A}^{2}\sin \theta=\omega_B r_{B}^{2}\sin \beta$$

The question does not give information about ##\theta## and ##\beta##. How to continue?

haruspex said:
Sorry, I should have been clearer. A radial component other than centripetal acceleration, i.e. ## \ddot r##
Is ## \ddot r## the angular acceleration?

Thanks
 
  • #11
songoku said:
Using ##v=\omega r##:
No, ##v_B\sin(\beta)=\omega r_B##, etc.
songoku said:
Is ## \ddot r## the angular acceleration?
No, it is the rate of increase of the rate of increase of the radius. In a circular orbit it is zero.
The component of the acceleration vector in the radial direction is ##\ddot r-r\omega^2##. See e.g. https://proofwiki.org/wiki/Acceleration_Vector_in_Polar_Coordinates

Note also that the centripetal acceleration is not ##\omega^2r##. Centripetal acceleration is that component of the total acceleration which is normal to the velocity. If the velocity has a radial component then the centripetal acceleration is not entirely radial. The full expression for it in polar coordinates is horrendous.
 
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  • #12
songoku said:
Maybe you mean something like this?
$$mv_Ar_A\sin \theta=mv_Br_B\sin \beta$$

where ##\theta## and ##\beta## is the angle between ##v## and ##r## at point A and B. Using ##v=\omega r##:
$$\omega_A r_{A}^{2}\sin \theta=\omega_B r_{B}^{2}\sin \beta$$
For angular momentum use ##~~\mathbf L =\mathbf r\times\mathbf p##.

Remember that ##~~\mathbf p=m\mathbf v## and ##\mathbf v=\mathbf \omega \times \mathbf r##. Put it together using the vector triple product rule.
 
  • #13
Super Man said:
Seeing the question, the first part can be done using Kepler law which you have done correctly.
This is incorrect.

Please refrain from confusing the OP by providing wrong answers.
 
  • #14
I understand.

Thank you very much for all the help and explanation haruspex, Orodruin, kuruman, Super Man
 
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