Angular and orbital speed at perihelion

In summary, the angular and linear (orbital) speeds in perihelion of an eliptical orbit are related the same way as in circular orbit (v = rw).
  • #1
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Homework Statement
I want to derive how are angular and orbital speeds related in perihelion of eliptical orbit
Relevant Equations
Angular momentum of reduced body in polar coordinates
Hello to all good people of physics forums. I just wanted to ask, whether the angular and linear (orbital) speed in perihelion of eliptical orbit are related the same way as in circular orbit (v = rw). If we take a look at the angular momentum (in polar coordinates) of reduced body moving in eliptical orbit
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if we equate the underlined factors and solve for v_theta, we get v_theta = rw. As I understand v_theta is a component of velocity perpendicular to position vector, and generaly is not equal to orbital velocity, but it is in perihelion/aphelion.

Is this reasoning correct? Thank you for taking the time to read and respond :).
 
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  • #2
It's simpler than that. At perihelion/aphelion the radial component of the velocity vector is zero by definition. So the velocity is perpendicular to the radial direction at these two points.
 
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  • #3
Thank you kuruman, I also derived a general expression
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where v is the orbital (tangential) velocity, and theta is the angle between velocity and position vector.

I derived it by using the magnitude of angular momentum of reduced body (one body problem)
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and equating that expression with general one for angular momentum L = μrvsinθ and solving for ω. In polar coordinates, vsinθ is actually speed in theta hat direction, and in perihelion/aphelion theta is 90 degrees and angular speed becomes ω = v/r. However, my professor claimed this was wrong, and if I remember correctly, that I can't equate those two expression for angular momentum, but I can't figure out why.:wideeyed:
 

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  • #4
The starting point for considerations of this kind is that angular momentum is conserved because the orbiting mass is moving under the influence of a central force which can exert no torque. This means that at all points on the orbit ##\vec L =\vec r \times \vec p=const.## The magnitude of the cross product is maximum when the linear momentum vector ##\vec p## is perpendicular to the position vector ##\vec r##. This occurs at perihelion and aphelion. The magnitude of the constant angular momentum is commonly calculated at either one of these points, ##|\vec L|=r_a~p_a=r_p~p_p## where subscripts "a" and "p" stand respectively for aphelion and perihelion. If you want to bring in ##\omega## through ##|\vec L|=\mu \omega r^2##, you would have to write ##\omega_a r_a^2=\omega_p r_p^2##. The expression ##\omega=v/r## is better written as ##\omega_a=v_a/r_a## at aphelion or ##\omega_p=v_p/r_p## at perihelion. At other points on the orbit a sine will be required.

I cannot speak for your professor, but I think his/her objection is that writing ##\omega=v/r## is meaningless and misleading because ##\omega## varies along the orbit and it is equal to the ratio of the speed to the distance only at aphelion and perihelion with the understanding that ##\omega_a \neq \omega_p##. So how useful is this expression?
 

1. What is the difference between angular speed and orbital speed at perihelion?

Angular speed refers to the rate at which an object rotates or moves around a central point, while orbital speed at perihelion specifically refers to the speed of an object at its closest point to the sun in its orbit around the sun.

2. How are angular speed and orbital speed at perihelion related?

Angular speed and orbital speed at perihelion are related by the radius of the orbit. As the radius decreases (as it does at perihelion), the orbital speed increases and the angular speed decreases. This is due to the conservation of angular momentum.

3. Can the angular speed and orbital speed at perihelion change over time?

Yes, both the angular speed and orbital speed at perihelion can change over time. This can be affected by factors such as changes in the mass or distance of the object, as well as the influence of other objects in the solar system.

4. How is the angular speed and orbital speed at perihelion calculated?

The angular speed at perihelion can be calculated by dividing the distance traveled by the object by the time it takes to travel that distance. The orbital speed at perihelion can be calculated using the equation v = √(GM/r), where G is the gravitational constant, M is the mass of the sun, and r is the distance between the object and the sun at perihelion.

5. Why is it important to study the angular speed and orbital speed at perihelion?

Studying the angular speed and orbital speed at perihelion can provide valuable information about the motion of objects in our solar system and how they interact with each other. This can help us better understand the mechanics of our solar system and make predictions about future events, such as the potential for collisions between objects.

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