Deriving Gauss' Variational Equation for True Anomaly

In summary, the student attempted to solve for the true anomaly, f, with respect to time, but was unable to do so due to a mistake in expressing the position and velocity vectors in terms of \hat x, \hat y, and \hat z. He was assisted by a hint from the teacher to factor p\cos f out of each term on the left hand side.
  • #1
jsandberg
7
0

Homework Statement


Derive the Gauss Variational differential equation for the true anomaly, f, with respect to time using components along the radius, angular velocity, and a unit vector orthogonal to those two (ir,itheta,ih).


Homework Equations


Sorry, I don't know how to use Latex. But I have attached the equations I need to start from and get to! ad is the perturbation, r_underline is the position vector, r is the norm of the position vector, v is the velocity vetor, h is the angular momentum, f is the tru anomaly, e is the eccentricity, p is the semilatus rectum.

The Attempt at a Solution


See attached handwritten solution- the first two lines are given in the assignment. I just can't seem to get the equation simplifed to the final equation.

View attachment df_dt.zip

View attachment Derivation df_dt.pdf
 
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  • #2


Your mistake is right at the start. Unfortunately this means everything you did was wrong.

Your mistake was in expressing the position and velocity vectors in terms of [itex]\hat x[/itex], [itex]\hat y[/itex], and [itex]\hat z[/itex]. You should have expressed these in the same coordinate system in which the perturbative acceleration is expressed -- in other words, [itex]\hat r[/itex], [itex]\hat \theta[/itex], and [itex]\hat h[/itex]. The position vector is simply [itex]\mathbf r = r \hat r[/itex]. I'll leave velocity up to you.

Hint: It does not take two pages of math to derive the result.
 
  • #3


Thank you for your quick response! Yes, changing the radius and velocity components helped a lot. I am still having trouble simplifyin the equation, however (see attached).

Thanks again for your time.
 

Attachments

  • df_dt simplified.pdf
    289.1 KB · Views: 326
  • #4


What's wrong? The last expression is exactly what you want to derive. Is your problem going from the penultimate expression to the last one? In other words, you are having a problem with showing

[tex]\left(1+\frac r p\right)re(1-\sin^2 f) + \frac{r^2} p \cos f = p\cos f[/tex]

Hint: All you need are [itex]1-\sin^2 f = \cos^2 f[/itex] and [itex]r=p/(1+e\cos f)[/itex].
 
  • #5


Yes, that is where my problem is. Do I substitute the equation for "r" every time I see an "r"?
 
  • #6


This is homework. I've given a couple of hints. I'll give one more: Factor [itex]p\cos f[/itex] out of each term on the left hand side. In other words, rewrite the left hand side as p cos(f) * (term1 + term2). Now show that term1+term2 is identically one.
 
  • #7


Thanks for all your help! Much appreciated.
 

FAQ: Deriving Gauss' Variational Equation for True Anomaly

1. What is the significance of Gauss' Variational Equation for True Anomaly?

Gauss' Variational Equation for True Anomaly is a fundamental equation in celestial mechanics that is used to describe the motion of a body in an elliptical orbit. It allows us to calculate the true anomaly, which is the angle between the periapsis (closest point to the central body) and the current position of the orbiting body.

2. How is Gauss' Variational Equation derived?

Gauss' Variational Equation is derived from the differential equations of motion for two-body systems, which take into account the gravitational force between the two bodies. By manipulating these equations and applying the law of conservation of angular momentum, we can arrive at the final form of the equation.

3. What are the assumptions made in deriving Gauss' Variational Equation?

The derivation of Gauss' Variational Equation makes several assumptions, including that the orbiting body is in a two-body system (i.e. the gravitational influence of other bodies can be ignored), and that the central body is much more massive than the orbiting body. Additionally, it assumes a circular or elliptical orbit with a constant gravitational force.

4. How is Gauss' Variational Equation used in practice?

Gauss' Variational Equation is used in a variety of applications, including orbit determination and mission planning for spacecraft, as well as in the study of celestial mechanics and planetary motion. It is also used in the development of numerical integration techniques for solving complex orbital problems.

5. Are there any limitations to Gauss' Variational Equation?

While Gauss' Variational Equation is a useful tool in celestial mechanics, it does have its limitations. It assumes a two-body system and does not take into account the effects of other bodies, such as gravitational perturbations. It also assumes a constant gravitational force, which may not hold true in all cases. Additionally, it is only applicable to elliptical orbits and cannot be used for other types of orbits, such as hyperbolic or parabolic.

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