How to get tangential velocity

[Bob Jones]

I am in the process of making a program that visually shows an elliptical orbit over time. I wish to find the tangential velocity of the satellite in the elliptical orbit based on the variables that I know.

Here is what I know:
a) The angle relative to the right focus with 0 radians being the positive x direction from that point
b) The angular velocity based on the angle in a
c) The instantaneous radius (distance between the center of mass of the satellite and right focus)
d) The semi-major and semi-minor axes
e) The mass of the planet

Is this enough information (assuming you can use constants) to calculate tangential velocity? If so, how?

 

[PeroK]

I am in the process of making a program that visually shows an elliptical orbit over time. I wish to find the tangential velocity of the satellite in the elliptical orbit based on the variables that I know.

Here is what I know:
a) The angle relative to the right focus with 0 radians being the positive x direction from that point
b) The angular velocity based on the angle in a
c) The instantaneous radius (distance between the center of mass of the satellite and right focus)
d) The semi-major and semi-minor axes
e) The mass of the planet

Is this enough information (assuming you can use constants) to calculate tangential velocity? If so, how?

Are you looking for a formula or a numerical method?

 

[Bob Jones]

Are you looking for a formula or a numerical method?

A formula that I can plug in whichever values of the above listed variables are needed to get the tangential radius.

Could I simply multiply the angular velocity by the radius, or are there more complex factors at hand?

 

[PeroK]

A formula that I can plug in whichever values of the above listed variables are needed to get the tangential radius.

Could I simply multiply the angular velocity by the radius, or are there more complex factors at hand?

An orbit is defined by the mass of the planet and the two axes. From that you can get the velocity at the turning points. Do you know how to do that?

That gives you the velocity at any point on the ellipse.

But, a formula for the velocity with respect to time may be more difficult! Do you know or can you see why?

 

[Bob Jones]

An orbit is defined by the mass of the planet and the two axes. From that you can get the velocity at the turning points. Do you know how to do that?

That gives you the velocity at any point on the ellipse.

But, a formula for the velocity with respect to time may be more difficult! Do you know or can you see why?

Well, I don't need it in relation to time but given the above variables. So, if there was a way to find it using the angle relative to the right focus, the angular velocity relative to the right focus, the radius measured from the right focus, and/or the rest of the variables, then that would work.

Could the momentum approach be used?
mr1v1 = mr1v2
r1 * v1 = r2 * v2

Well, I don't know the tangential velocity at any point, only the angular velocity.

Could the energy approach be used?

-GMm/r + 0.5mv^2 = TME

I don't know the total mechanical energy, so I couldn't get v at the locations in which the angle is 0 or 180 to use later.

These were the two approaches I considered when thinking about the problem.

 

[PeroK]

I am in the process of making a program that visually shows an elliptical orbit over time.

Well, I don't need it in relation to time

Which is it?

 

[Bob Jones]

Which is it?

Well, both...

What I meant by the first statement is that it shows the position the satellite it would be with the correct period by adding the angular velocity multiplied by the time it takes the program to loop to that part of the code to the angle. The radius is dependent on the angle. With both the radius and angle, the position can be shown.

So, if I have the tangential velocity with respect to the variables in the first post, that will work.

I hope that I was not unclear with what I said.

 

[PeroK]

Have you tried simulating an object falling directly towards a planet? This is a simpler problem and the formula for velocity in terms of distance is easy. But, the formula for distance in terms of time is relatively complicated.

For an elliptical orbit you may have to use a numerical method. For that you would need only an initial position - perigee perhaps - and initial velocity. That would be one approach.

 

[Bob Jones]

Have you tried simulating an object falling directly towards a planet? This is a simpler problem and the formula for velocity in terms of distance is easy. But, the formula for distance in terms of time is relatively complicated.

For an elliptical orbit you may have to use a numerical method.

It seems I was not being very clear with my intentions. The simulation functions well. The part where I mentioned that I was in progress of building the simulation was referring to how I was going to allow the user to specify the radii and mass of the planet being orbited instead of using hardcoded values as well as show some values like the tangential velocity.

So, the purpose of the tangential velocity is not for using it in the actual simulation of the orbit itself but just to show it in some of the data that the user will be able to view.

 

[PeroK]

It seems I was not being very clear with my intentions. The simulation functions well. The part where I mentioned that I was in progress of building the simulation was referring to how I was going to allow the user to specify the radii and mass of the planet being orbited instead of using hardcoded values as well as show some values like the tangential velocity.

So, the purpose of the tangential velocity is not for using it in the actual simulation of the orbit itself but just to show it in some of the data that the user will be able to view.

If you can simulate an orbit for a fixed set of values, why not replace those values with variables?

 

[Bob Jones]

If you can simulate an orbit for a fixed set of values, why not replace those values with variables?

Well, they are variables. You can change them in the code. I just meant that I was going to add the ability for the user to change it in the UI.

 

[Bob Jones]

Well, they are variables. You can change them in the code. I just meant that I was going to add the ability for the user to change it in the UI.

Well, I've been an idiot. All I needed to do was multiply the angular velocity by the radius to get the tangential velocity.

And I can get total velocity by v = sqrt(GM * (2/r - 1/a))

 

[nasu]

What is the difference between tangential velocity and "total velocity"?

 

[Bob Jones]

What is the difference between tangential velocity and "total velocity"?

By total velocity, I meant the total relative velocity of the satellite. As for tangential velocity, I meant the component of the total relative velocity that is perpendicular to the radius.

Also, is there a way to end a topic? I found what I was looking for.

 

[nasu]

So what would be the direction of the "total velocity"? Is not along the tangent to the trajectory?

 

[Bob Jones]

So what would be the direction of the "total velocity"? Is not along the tangent to the trajectory?

From my understanding, the total velocity is pointing slightly away from the ellipse at points in which it is not tangent. This image displays what I mean.

 

[nasu]

What is labeled there as "vt" is not along the tangent. What is labeled "v" is along the tangent.
The velocity is always tangent to the trajectory. Or the trajectory can be defined as the curve which is tangent to the velocity in every point.

 

[Bob Jones]

What is labeled there as "vt" is not along the tangent. What is labeleb "v" is along the tangent.
The velocity is always tangent to the trajectory. Or the trajectory can be defined as the curve which is tangent to the velocity in every point.

As said above, I meant perpendicular to the radius. You seem to be correct in that I was incorrectly labeling what I was looking for as 'tangential velocity,' so I apologize for that.

Thanks for helping me improve my understanding.

 

[Chestermiller]

Yes. I agree with nasu. The tangential velocity and the total velocity are one-and-the-same thing (as they must be).

 

[Bob Jones]

Yes. I agree with nasu. The tangential velocity and the total velocity are one-and-the-same thing (as they must be).

Yes, it seems I did not use the correct terminology, and I had a misunderstanding based on what I thought was correct, especially after seeing the last comment to the question here.

 

[nasu]

I suppose that in the image (post 16) the labels of the two components mean r -"radial" and t-"transverse". Maybe this is the source of confusion.
The transverse component is given by $v_t= \omega r$ where $\omega$ is the angular velocity and r is the distance from the rotation center.

 

[Bob Jones]

I suppose that in the image (post 16) the labels of the two components mean r -"radial" and t-"transverse". Maybe this is the source of confusion.
The transverse component is given by $v_t= \omega r$ where $\omega$ is the angular velocity and r is the distance from the rotation center.

While that is probably what was meant or at least what is actually correct, the source of the image says tangential.

 

[Bob Jones]

Since we are on the topic of Kepler's laws of planetary motion, I have another question.

One of the assumptions made when using Kepler's laws is that the orbited mass is much greater than the mass orbiting it, which is why it can be assumed to be 'fixed' without much error.

My question is what is the maximum ratio between the orbited mass and orbiting mass where Kepler's law is still reasonable (where you can assume the orbited mass is fixed)? I know the question is subjective, but if anyone could offer up a ballpark estimate with a bit of reasoning, I would highly appreciate it.

 

[PeroK]

Since we are on the topic of Kepler's laws of planetary motion, I have another question.

One of the assumptions made when using Kepler's laws is that the orbited mass is much greater than the mass orbiting it, which is why it can be assumed to be 'fixed' without much error.

My question is what is the maximum ratio between the orbited mass and orbiting mass where Kepler's law is still reasonable (where you can assume the orbited mass is fixed)? I know the question is subjective, but if anyone could offer up a ballpark estimate with a bit of reasoning, I would highly appreciate it.

The pair orbit their centre of mass, which is offset in proportion to the masses. At what stage do you want to consider this?

The other factor is that it's pointless to consider the motion of the Sun caused by the Earth, as it moves much more significantly in response to Jupiter.

 

[Bob Jones]

The pair orbit their centre of mass, which is offset in proportion to the masses. At what stage do you want to consider this?

The other factor is that it's pointless to consider the motion of the Sun caused by the Earth, as it moves much more significantly in response to Jupiter.

Basically, I wish to know what relative ratios of distances and/or masses can we assume that the effect of the gravitational pull of the satellite on the planet is small enough such that we can assume the planet is fixed with little error.

 

[PeroK]

Basically, I wish to know what relative ratios of distances and/or masses can we assume that the effect of the gravitational pull of the satellite on the planet is small enough such that we can assume the planet is fixed with little error.

That all depends how you define "little".

 

[Bob Jones]

That all depends how you define "little".

That's a good point. Also, I'll have to consider what values I am considering when I am thinking of the error.

All right. Let's say the distance between the planet and satellite is R. If I wanted the planet to move a maximum of R/1000 from the position where it would be considered 'static' from the view of Kepler's laws of planetary motion, what would be the requirements of the ratio of masses, the semi-minor and semi-major axes, and whatever else can be considered?

 

[PeroK]

That's a good point. Also, I'll have to consider what values I am considering when I am thinking of the error.

All right. Let's say the distance between the planet and satellite is R. If I wanted the planet to move a maximum of R/1000 from the position where it would be considered 'static' from the view of Kepler's laws of planetary motion, how would I go about checking that?

By Newton's laws, they move in proportion to their masses. So, the large mass must be 1000 larger than the satellite.

 

[Bob Jones]

By Newton's laws, they move in proportion to their masses. So, the large mass must be 1000 larger than the satellite.

Thanks for the help! :)