Help with a Universal Gravitational Constant

[Atominate]

I am doing an Astronomy GCSE. I have to work out the orbital period for a satellite. I have got quite far with it, but I am really stuck. I have spent hours at it! This is the question:-
A space laboratory is in circular orbit around the Earth at a distance of 6000km from the Earth's centre; its period is 2hr 12min. An orbital telescope is to be placed into orbit at 12,000km. Calculate its expected period.
I am at the point where I need to square root 4(pi)²r³/GM = T
A few questions. Is r the radius of the Earth (from core to surface) or core to satellite? How do I calculate GM? I know that M is the mass of the Earth (5.972 × 10^24 kg) but how do I work out G (the Universal Gravitational Constant?). I really need help and it is very frustrating. Any help or advice would be very much appreciated. Thanks!

 

[faradayscat]

"r" is usually said to be the distance from the core of Earth to the satellite. So r should be something like:

$r = R + h$
Where R is the radius of Earth and h is the height of the satellite above Earth's surface.

I'm assuming you have to calculate G yourself, correct? You have sufficient information to find G with the information provided for the space laboratory.

 

[BvU]

G is usually a given. https://en.wikipedia.org/wiki/Gravitational_constant#Laws_and_constants

 

[SteamKing]

I am doing an Astronomy GCSE. I have to work out the orbital period for a satellite. I have got quite far with it, but I am really stuck. I have spent hours at it! This is the question:-
A space laboratory is in circular orbit around the Earth at a distance of 6000km from the Earth's centre; its period is 2hr 12min. An orbital telescope is to be placed into orbit at 12,000km. Calculate its expected period.
I am at the point where I need to square root 4(pi)²r³/GM = T
A few questions. Is r the radius of the Earth (from core to surface) or core to satellite? How do I calculate GM? I know that M is the mass of the Earth (5.972 × 10^24 kg) but how do I work out G (the Universal Gravitational Constant?). I really need help and it is very frustrating. Any help or advice would be very much appreciated. Thanks!

The universal gravitational constant G is one of those fundamental numbers which describes the universe. You don't calculate it; its value is determined experimentally.

For your satellite problem, you just look up the value of G, either in a book or online.

https://en.wikipedia.org/wiki/Gravitational_constant

 

[TonyS]

It is possible to solve your problem without knowing the value of G or the mass of the Earth.

 

[Atominate]

"r" is usually said to be the distance from the core of Earth to the satellite. So r should be something like:

$r = R + h$
Where R is the radius of Earth and h is the height of the satellite above Earth's surface.

I'm assuming you have to calculate G yourself, correct? You have sufficient information to find G with the information provided for the space laboratory.

Okay. That's very helpful. Thank you.

 

[Atominate]

I

The universal gravitational constant G is one of those fundamental numbers which describes the universe. You don't calculate it; its value is determined experimentally.

For your satellite problem, you just look up the value of G, either in a book or online.

https://en.wikipedia.org/wiki/Gravitational_constant

I looked it up online, but it looks like M, kg and s are required?

 

[Atominate]

It is possible to solve your problem without knowing the value of G or the mass of the Earth.

Oh. Could you please elaborate?

 

[TonyS]

Oh. Could you please elaborate?

The physics of the problem allows you to find an expression for the orbital period as a function of r, that involves your unknown constants.
You know the values of T and r for one particular case, namely the spacestation. The same relation between T and r also holds for the telescope, with the same bunch of constants, so you can eliminate the constants by dividing T(telescope) by T(spacestation).
(Sorry I could make this clearer if I'd learned to use latex)

 

[SteamKing]

I looked it up online, but it looks like M, kg and s are required?

Well, let's take a look at your formula:

A space laboratory is in circular orbit around the Earth at a distance of 6000km from the Earth's centre; its period is 2hr 12min. An orbital telescope is to be placed into orbit at 12,000km. Calculate its expected period.
I am at the point where I need to square root 4(pi)²r³/GM = T

G = 6.67408 × 10^-11 m³ / kg-s²

You want to find T, which is measured in seconds - check.
You'll have to look up the mass of the Earth in kilograms - check.
You know the distance from the center of the Earth to the satellite, but you need this in meters instead of kilometers - check.

Looks like G has all the right units.

From a practical standpoint, TonyS' post gives you a good tip for finding the new orbital period.

OTOH, you need to study up on fundamental constants, like G, if you want to do well in your astronomy course.

 

[Herman Trivilino]

A space laboratory is in circular orbit around the Earth at a distance of 6000km from the Earth's centre; its period is 2hr 12min. An orbital telescope is to be placed into orbit at 12,000km. Calculate its expected period.

Use ratio reasoning. The radius of the orbit is doubled, so what happens to the period?

 

[Atominate]

Use ratio reasoning. The radius of the orbit is doubled, so what happens to the period?

Is the period doubled? That would make everything a lot simpler! Thanks.

 

[Atominate]

Well, let's take a look at your formula:
G = 6.67408 × 10^-11 m³ / kg-s²

You want to find T, which is measured in seconds - check.
You'll have to look up the mass of the Earth in kilograms - check.
You know the distance from the center of the Earth to the satellite, but you need this in meters instead of kilometers - check.

Thanks very much. That helps a lot!
P.S. I completely agree. I am lacking with the more mathematical part of course. I will do further learning!

 

[Atominate]

(Sorry I could make this clearer if I'd learned to use latex)

It is very clear now! Thanks.

 

[ehild]

I am doing an Astronomy GCSE. I have to work out the orbital period for a satellite.
A space laboratory is in circular orbit around the Earth at a distance of 6000km from the Earth's centre; its period is 2hr 12min. An orbital telescope is to be placed into orbit at 12,000km. Calculate its expected period.
If you studied Astronomy, you must be familiar with Kepler's Laws of planetary motion. What does the third Law say?
http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html

 

[TonyS]

Is the period doubled? That would make everything a lot simpler! Thanks.

The square of the period is proportional to the cube of the radius, as you indicated in your original post, aside from a typo, I think. This is one of Kepler's laws, by the way.

 

[Atominate]

Kepler's third law says that T2/R3 = constant = 4π2 /GM. I was okay with that part, it was just that I was unsure about GM, but I'm okay now as everyone has helped a lot with their replies.

 

[Herman Trivilino]

Since 2 hours 12 minutes is 12.2 hours, we have

$\frac{T^2}{R^3}=\frac{(12.2 \ \mathrm{h})^2}{(6000 \ \mathrm{km})^3}$.

That is the constant of proportionality. So when $R$ doubles to 12 000 km, $T$ does not double to 24.4 hours.