Dr. Evil's dynamics questions for the weekend

[Jonathan Whitehead]

While I am in the process of developing my death ridge L-A-S-E-R that I discuss in the "Bar & Grille" ,as you people call it, I have a few meteorological dynamics questions for you all.
These are not hard. No not hard at all. But I would appreciate it if anyone could explain the solution to these questions to me in a manner I can understand. It is all for my studies to became a profound meteorologist and of course as you all know...to take over the world.
So anyway without further adooooooo.. here are the questions.

1.1 Neglecting latitudinal variation in the radius of the earth, what is the calculated angle between the gravitational force and gravity vectors at the surface of the earth as a function of latitude? What is the maximum value of this angle?


1.2 What is the calculated altitude at which an artifical satellite orbiting in the equatorial plane can be a synchronous satellite (i.e., can remain above the same spot on the surface of the earth)?



Note: no, I did not think these questions up, I, of course, got them out of a book. But if anyone knows where i can get my hands on the solutions to questions in the book, I would greatly appreciate it.

 

[Ben Baranowski]

Chris,

I think I may have some help on #1, but having been a TA, I know the value of self discovery....that and I don't have time to work these out myself. I think this should help some.
See:
http://weather.ou.edu/~ashapiro/METR3123/lecture%207.pdf (Shapiro's Dyn II notes). At the bottom of the section there is a diagram and equation for gravity force and gravitational force and how they are related.

Hope this helps,

Ben

 

[John Hudson]

Hi Chris,

I believe you're talking about Geostationary Orbit (GEO), where an artificial satellite appears to "hover" over the earth's surface as it maintains a constant speed with respect to the ground below. This equatorial orbit, known as the Clarke Belt, is getting to be a rather crowded parking lot with its assortment of communications and observational satellites.

Simply stated, it's an orbit where centripetal acceleration provided by earth's gravity and centrifugal acceleration are in a state of equalibrium. This orbit is exclusively equatorial, and takes place at an altitude of approximately 22,240 statute miles (35,786 km).

There's an excellent paper by R.E. Tremblay that has all the arithmetic to calculate the answer at GEO Calculation

As an additional note, this orbit is independent of mass, so it doesn't matter how heavy the satellite is.

Hope this helps.

John
VE4 JTH

 

[Dan Dawson]

Simply stated, it's an orbit where centripetal acceleration provided by earth's gravity and centrifugal acceleration are in a state of equalibrium. This orbit is exclusively equatorial, and takes place at an altitude of approximately 22,240 statute miles (35,786 km).[/b]

Actually, any orbit is a balance between the Earth's gravitational force and the centrifugal force (as applied to the object itself in the object's frame of reference). It doesn't have to be geostationary, or even equatorial. The only thing special about a geostationary orbit is just that it happens to be at an altitude where the orbital velocity exactly matches the rotational velocity of Earth. There is no special balance of forces involved that isn't true of any other orbit. We are fortunate that the Earth rotates rather rapidly, so that the geostationary orbit isn't incredibly far from Earth. If Earth didn't rotate at all, for example, the geostationary orbit would be infinitely far away.

 

[John Hudson]

While I (and most of us here) understand that a balance between the centripedal acceleration caused by the Earth's gravity and centrifugal acceleration isn't the exclusive domain of a geostationary orbit, it is essential to this kind of orbit because your statement:

Actually, any orbit is a balance between the Earth's gravitational force and the centrifugal force
[/b]

is incorrect. A geostationary orbit requires constant monitoring and maintenance of its velocity and altitude, and satellite operators spend a great deal of money to achieve this balance (station keeping).

My intention was to emphasize this fact.

John
VE4 JTH

 

[Dan Dawson]

While I (and most of us here) understand that a balance between the centripedal acceleration caused by the Earth's gravity and centrifugal acceleration isn't the exclusive domain of a geostationary orbit, it is essential to this kind of orbit because your statement:

is incorrect. A geostationary orbit requires constant monitoring and maintenance of its velocity and altitude, and satellite operators spend a great deal of money to achieve this balance (station keeping).

My intention was to emphasize this fact.

John
VE4 JTH
[/b]

Well, if you really want to get technical, of course it is true that there are multiple perturbations caused by other bodies in space (i.e. the moon, sun, perhaps other planets to a small extent) to any orbiting body around the earth such that it is not always perfectly in balance. Hence, the need to continually monitor and adjust the orbits of geostationary satellites so that they don't "drift" out of place. I obviously wasn't arguing that. Nevertheless, it is still true that an at least approximate balance of the nature I stated before is common to every orbit, not just a geostationary one. Your original post seemed to imply that this was not the case, and that was what I was addressing. I apologize if I misunderstood you.

EDIT: I was thinking about this some more, and I realized my initial statement about the balance of forces really only applies to a circular orbit in the rotating frame of reference of the satellite. At different parts of an elliptical orbit, the forces will be out of balance.

 

[John Hudson]

Well, if you really want to get technical, of course it is true that there are multiple perturbations caused by other bodies in space (i.e. the moon, sun, perhaps other planets to a small extent) to any orbiting body around the earth such that it is not always perfectly in balance. Hence, the need to continually monitor and adjust the orbits of geostationary satellites so that they don't "drift" out of place. I obviously wasn't arguing that. Nevertheless, it is still true that an at least approximate balance of the nature I stated before is common to every orbit, not just a geostationary one. Your original post seemed to imply that this was not the case, and that was what I was addressing. I apologize if I misunderstood you.

EDIT: I was thinking about this some more, and I realized my initial statement about the balance of forces really only applies to a circular orbit in the rotating frame of reference of the satellite. At different parts of an elliptical orbit, the forces will be out of balance.
[/b]

I definitely could have been a bit clearer in my wording, and hey, no apologies are necessary or forthcoming. Spirited debate and discussion about what makes this "organized mess" of a universe tick is how everyone learns new ideas, and how there might not necessarily be one simple answer to every question. It is one of the things I truly love about these forums, and I've learned so much from being here.

John
VE4 JTH

 

[Dan Dawson]

I definitely could have been a bit clearer in my wording, and hey, no apologies are necessary or forthcoming. Spirited debate and discussion about what makes this "organized mess" of a universe tick is how everyone learns new ideas, and how there might not necessarily be one simple answer to every question. It is one of the things I truly love about these forums, and I've learned so much from being here.

John
VE4 JTH
[/b]

I completely agree . That's probably the main reason I post to and read forums like these: to learn and help others to learn about topics that interest me in a (usually) non-threatening environment with people from all walks of life. And, you're right, there is often not a single simple answer to every question, as I've found out time and time again