Angular momentum and true anomaly

In summary: Expert SummarizerIn summary, the angular velocity of an object in orbit around the Sun can be calculated using the formula \.{r} = \frac{na}{\sqrt{1-e^2}}\sin(f), where f represents the true anomaly. This can be derived by solving for \.{f} in the expression for \.{r} and using the definitions of angular velocity, eccentric anomaly, and radial velocity. The final expression for the angular velocity as a function of the true anomaly is \.{f} = \frac{2na\sqrt{1-e^2}}{1+e\cos(f)}\frac{\sqrt{\frac{1-e}{1+e}}}{1+\sqrt{\frac
  • #1
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Homework Statement



The angular momentum per unit mass of an object orbiting the Sun is given by [tex]h =
na^2 \sqrt{1 - e^2}[/tex], where [tex]n[/tex] , [tex]a[/tex] and [tex]e[/tex] are the mean motion, semi-major axis and eccentricity of the object, respectively. Use [tex]h[/tex] to obtain an expression for the angular velocity of the object as a function of the true anomaly, [tex]f[/tex].

The Attempt at a Solution



Literally the only formulas involving f that we have in our notes are...

[tex]\.{r} = \frac{na}{\sqrt{1-e^2}}\sin(f)[/tex]

and

[tex]r = \frac{a(1-e^2)}{1 + e \cos(f)}[/tex]

and since I seem to think [tex]\.{r}[/tex] is the angular velocity is the first equation there just the answer?!?
 
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  • #2




Thank you for your question. The formula you have mentioned, \.{r} = \frac{na}{\sqrt{1-e^2}}\sin(f), is indeed the expression for the angular velocity of an object in orbit around the Sun. This can be seen by rearranging the formula to solve for \.{f}, which represents the angular velocity:

\.{f} = \frac{\.{r}}{\frac{na}{\sqrt{1-e^2}}}

Using the definition of angular velocity, \.{f} = \frac{d\theta}{dt}, we can rewrite this as:

\.{f} = \frac{d\theta}{dr}\frac{dr}{dt}

The first derivative, \frac{d\theta}{dr}, represents the rate of change of the angle with respect to the distance from the Sun, and can be calculated using the formula for the eccentric anomaly, E = 2\tan^{-1}(\sqrt{\frac{1-e}{1+e}}\tan(\frac{f}{2})).

The second derivative, \frac{dr}{dt}, represents the rate of change of the distance from the Sun with respect to time, and can be calculated using the formula for the radial velocity, \.{r} = \frac{na}{\sqrt{1-e^2}}\sin(f).

Therefore, the full expression for the angular velocity as a function of the true anomaly is:

\.{f} = \frac{2na\sqrt{1-e^2}}{1+e\cos(f)}\frac{\sqrt{\frac{1-e}{1+e}}}{1+\sqrt{\frac{1-e}{1+e}}\tan(\frac{f}{2})}\sin(f)

I hope this helps. Please let me know if you have any further questions.



 

FAQ: Angular momentum and true anomaly

What is angular momentum?

Angular momentum is a property of a rotating object that describes its tendency to continue rotating. It is the product of an object's moment of inertia and its angular velocity.

How is angular momentum conserved?

According to the law of conservation of angular momentum, the total angular momentum of a system remains constant unless acted upon by an external torque. This means that if there are no external forces or torques acting on a system, the total angular momentum will remain the same.

What is true anomaly?

True anomaly is an angle that describes the position of an orbiting object in its elliptical orbit. It is the angle between the periapsis (closest point to the central body) and the current position of the object along its orbit.

How is true anomaly related to angular momentum?

True anomaly is one of the orbital elements that determines the shape and orientation of an orbit. It is related to angular momentum because the rate at which an object sweeps out area in its orbit is equal to its angular momentum divided by twice the area of the ellipse. This is known as Kepler's second law of planetary motion.

What is the difference between specific angular momentum and total angular momentum?

Specific angular momentum is the angular momentum per unit mass, while total angular momentum is the overall angular momentum of a system. In other words, specific angular momentum takes into account the mass of an object, while total angular momentum does not.

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