Averaging the cube of semimajor axis to position ratio wrt to time

In summary, the average of (a/r)^3 takes time as an independent variable. (where a is the semi major axis and r is the distance in an elliptical keplerian orbit). The 1-e^2 terms cancel and then again we will be left with a/r isn't it?!?
  • #1
antythingyani
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Thread moved from the technical forums to the schoolwork forums
Summary:: Averaging (a power of) semimajor axis to position ratio wrt to time - celestial mechanics

I evaluated it this far, but i don't know how to change the dt to d theta ... the final solution is

Sideways equation 01.jpg

supposedly (1-e^2)^-(3/2) . Any help will be appreciated.photo1638096644.jpeg

[Image re-inserted with correct orientation by Mentor]
 
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  • #2
It's not very easy to read your post or to understand precisely what question you are asking.
 
  • #3
PeroK said:
It's not very easy to read your post or to understand precisely what question you are asking.
The question is basically : derive the average of (a/r)^3 taking time as an independent variable. (where a is the semi major axis and r is the distance in an elliptical keplerian orbit.
 
  • #4
antythingyani said:
The question is basically : derive the average of (a/r)^3 taking time as an independent variable. (where a is the semi major axis and r is the distance in an elliptical keplerian orbit.
Given that we cannot easily express ##r## as a function of time, what is your strategy?
 
  • #5
we can express r as a function of semimajor axis (a) and the true anomaly (theta). In that case maybe finding a way to turn dt into dtheta would be handy... or trying to express theta as a function of t.
 
  • #6
antythingyani said:
we can express r as a function of semimajor axis (a) and the true anomaly (theta). In that case maybe finding a way to turn dt into dtheta would be handy... or trying to express theta as a function of t.
I don't think you can get ##\theta## as a function of ##t## either. We have: $$\frac{d\theta}{dt} = \frac{L}{mr^2}$$But I don't know that solves the problem. There might be some trick using Kepler's law.
 
  • #7
... we also have $$\frac{a(1 - e^2)}{r} = 1 + e\cos \theta$$ Perhaps that does the trick?
 
  • #8
... which it does.
 
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  • #9
PeroK said:
... which it does.
The 1-e^2 terms cancel and then again we will be left with a/r isn't it?!?
 
  • #10
antythingyani said:
The 1-e^2 terms cancel and then again we will be left with a/r isn't it?!?
I'm not sure what you mean by that. I used my notes on the derivation of elliptical orbits to find the relevant equations. This looks like a tricky problem where you'll need to do the same. I've given you the two equations to get you started.

At this level, I think you need to learn a little Latex:

https://www.physicsforums.com/help/latexhelp/

If you reply to my posts you'll see what I've typed to render the mathematics.
 
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Related to Averaging the cube of semimajor axis to position ratio wrt to time

1. What is the purpose of averaging the cube of semimajor axis to position ratio with respect to time?

The purpose of this calculation is to determine the average rate of change of the semimajor axis to position ratio over a specific period of time. This can provide valuable information about the orbital dynamics of a celestial body.

2. How is the cube of semimajor axis to position ratio calculated?

The cube of semimajor axis to position ratio is calculated by cubing the semimajor axis and dividing it by the cube of the position. This value is then averaged over a period of time to determine the average rate of change.

3. What does the semimajor axis to position ratio represent?

The semimajor axis to position ratio represents the relationship between the distance of a celestial body from its primary body and the size of its orbit. It is a measure of the body's orbital eccentricity.

4. How is time factored into this calculation?

Time is factored into this calculation by taking measurements of the semimajor axis and position at different points in time. These values are then used to calculate the average rate of change over the specified time period.

5. What are some potential applications of this calculation?

This calculation can be used in various fields, such as astronomy and aerospace engineering, to study the orbital dynamics of celestial bodies. It can also be used to predict future positions of objects in orbit and to analyze the stability of orbits over time.

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