Orbital Speed: Earth vs Moon - How Does Altitude Affect Orbital Velocity?

In summary, the orbital speed for an object around the moon would be less compared to the speed required to maintain altitude around the Earth, due to the smaller mass and radius of the moon. This is because the orbital speed depends on the radius of the orbit and the mass of the object being orbited. Additionally, using the correct values for the mass and radius of the moon, the orbital velocity at a given altitude is reduced by a factor of approximately 4.71, increasing with altitude.
  • #1
robotnut
5
0
Just can't get my head around this one. If at 100Km high above Earth the orbital speed to maintain altitude is X, then what would the speed be for the same object around the moon?
Since gravity of the moon is smaller then earth's? I assume the orbital speed would need to be less?
 
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  • #2
orbital velocity might be less as the orbital velocity depends on the radius of the orbit in which they move and the mass of moon is less than Earth .
 
  • #3
Considering this mass orbiting around the Earth or around the moon means its covering a uniform circular motion.
Studying the sum of forces =0, where the forces are:
F=(G.M.m)/d^2 :Gravitational force done by the Earth on the object which is the centripetal force.
F1 :Centrifugal force which is also equal in magnitude to the centripetal force, but opposite direction.
applying Newtons 2nd law and knowing that the centripetal force is also given by the following fomula F=(m.V^2)/d
then [(G.M.m)/d^2]=[(m.V^2)/d]
calculate v and you will notice that it will be less than the speed on the earth.
 
  • #4
robotnut said:
...If at 100Km high above Earth the orbital speed to maintain altitude is X...

To avoid any confusion, we should all agree that these altitudes are above the center of the Earth and Moon, respectively; not above their surfaces.
 
  • #5
That is contrary to the meaning of the term "altitude", which invariably means height above some reference surface.
 
  • #6
This seems like a straightforward question. For a circular orbit (which is the only kind where you get a constant speed), we just set the gravitational field equal to the centripetal acceleration

[tex]\dfrac{v^2}{r} = \dfrac{GM}{r^2}[/tex]

[tex]v = \sqrt{\dfrac{GM}{r}}[/tex]

So if you're at some altitude h, just set the distance from the center equal to the radius of the planet R, plus the altitude

[tex]v = \sqrt{\dfrac{GM}{R + h}}[/tex]

The required speed is independent of the mass (this is for precisely the same reason as why all objects fall at the same acceleration without air resistance). Just set h to 100 km, and you can calculate the orbital speed for the Earth and the moon by substituting the appropriate values for the radius and mass of the Earth and the moon.
 
  • #7
arunma said:
The required speed is independent of the mass
of the orbiting body, of course (I know that's what you meant, just thought a clarification couldn't hurt)

If I remember correctly, the moon's mass is 1/6 that of Earth, and it's radius is about 1/3 of the Earth's, so I guess the orbital velocity at a given (small) altitude is reduced by about a factor of 2...
 
  • #8
The Moon's surface gravity is about 1/6 that of the Earth. The Moon's mass however is about 1/81 Earth masses.
 
  • #9
diazona said:
of the orbiting body, of course (I know that's what you meant, just thought a clarification couldn't hurt)

You're quite right, thanks!
 
  • #10
D H said:
That is contrary to the meaning of the term "altitude", which invariably means height above some reference surface.
Yes, I know. But what I'm asking is that we all refer to the distance above center of mass, because it removes an unnecessary complication, and avoids the need for secondary calculations such as...

diazona said:
If I remember correctly, the moon's mass is 1/6 that of Earth, and it's radius is about 1/3 of the Earth's, so I guess the orbital velocity at a given (small) altitude is reduced by about a factor of 2...

Distance above the surface does not effect orbital dynamics, and I believe this would help the OP gain the understanding he seeks.
 
  • #11
LURCH said:
D H said:
That is contrary to the meaning of the term "altitude", which invariably means height above some reference surface.
Yes, I know. But what I'm asking is that we all refer to the distance above center of mass, because it removes an unnecessary complication, and avoids the need for secondary calculations such as...
diazona said:
If I remember correctly, the moon's mass is 1/6 that of Earth, and it's radius is about 1/3 of the Earth's, so I guess the orbital velocity at a given (small) altitude is reduced by about a factor of 2...
Distance above the surface does not effect orbital dynamics, and I believe this would help the OP gain the understanding he seeks.
Whatever makes you think that? Distance above the surface most definitely does affect orbital dynamics. Moreover, distance below the surface most definitely hinders orbital mechanics. A satellite could not orbit the Earth at the orbital radius corresponding to an orbital altitude of 100 km above the surface of the Moon because that orbital radius is well inside the Earth. A satellite can however orbit 200 km above the surface of the Earth or the Moon.

The problem with diazona's analysis was incorrect values for the Moon's mass and radius. So, let's do it correctly.

The orbital velocity for a circular orbit at altitude h is

[tex]v=\sqrt{\frac{GM}{R+h}}[/tex]

The ratio of the orbital velocities for orbits at the same altitude about the Earth and Moon is thus

[tex]
\frac{v_e}{v_m} = \sqrt{\frac{M_e}{M_m}\,\frac{R_m+h}{R_e+h}}
\approx \sqrt{\frac{Me}{Mm}\,\frac{R_m}{R_e}}\,\left(1+\frac h 2\,\frac{R_e-R_m}{R_eR_m}\right)
[/tex]

Using the correct numbers, Mm/Me=0.0123 and Rm/Re=0.273, yields ve/vm=4.71 for h=0, increasing as altitude increases. The ratio is 5.15 for h=500 km, at which point the approximation is still valid to within about 1%.
 

1. What is the difference between the orbital speed of Earth and the Moon?

The Earth orbits around the Sun at an average speed of about 107,000 kilometers per hour, while the Moon orbits around the Earth at an average speed of about 3,600 kilometers per hour. This means that the Earth travels around the Sun much faster than the Moon travels around the Earth.

2. Why does the Earth have a faster orbital speed than the Moon?

The Earth is much larger and more massive than the Moon, which means it has a stronger gravitational pull. This strong pull allows the Earth to maintain a faster orbital speed as it travels around the Sun.

3. How does the orbital speed of the Earth and Moon affect their distance from each other?

The faster orbital speed of the Earth causes it to travel a longer distance in its orbit around the Sun compared to the Moon's orbit around the Earth. This results in the Earth and Moon being at varying distances from each other throughout their orbits.

4. Can the orbital speed of the Earth and Moon change?

Yes, the orbital speed of both the Earth and Moon can change over time due to various factors, such as gravitational interactions with other objects or changes in the Earth's rotation. However, these changes are very gradual and not noticeable in our lifetime.

5. How does the orbital speed of the Earth and Moon impact the tides?

The orbital speed of the Moon has a greater impact on tides compared to the Earth's orbital speed. This is because the Moon's gravitational pull is stronger on the side of the Earth facing the Moon, causing a bulge in the water and creating high tide. As the Earth rotates, this bulge moves, causing the tides to change. The Earth's orbital speed also plays a role in the tides, but it is a much smaller influence compared to the Moon's orbital speed.

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