Mass of orbiting body calculations

[gtsinc]

Ok, so I am tryting to calcualte the mass of an orbiting body.

I am using M = 4 * PI^2 * r^3 / ( G * T^2 )

Lets use the Earth as an example.
However I am not getting what Nasa has for answers of 5.9 E24 kg.
http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html


Using this calculator: http://www.ajdesigner.com/phpgravity/keplers_law_equation_mass.php

I am using a Radius of 149,600,000,000 meters
Ref: Semimajor axis (106 km) 149.60

Period of 365.25 or 365.25*24*24 = 31,557,600 seconds

G = 6.67E-11 N-m^2 / kg^2

That calculator gets 1.98 E+21 kg. Not even close to 5.9E+24
I get 1.98E+30 because I am using meters. But still not matching 5.9E+24

Can someone help with my math - not sure what is wrong.

Thanks

 

[phyzguy]

You've made at least two mistakes. First, this formula will give you the mass of the central object being orbited around, not of the orbiting body. In this case, you are calculating the mass of the Sun, not the Earth. If the central body is much more massive than the satellite (which is the case here), the orbital parameters are independent of the mass of the satellite. You should review how that formula is derived. Second, you must have plugged in the numbers wrong. If you do it right, it will correctly give you the mass of the Sun.

 

[gtsinc]

Ok, so I did have the mass of the sun calculated correctly as 1.98 E+30 but I was thinking of the orbiting body.

It has been years since I did this math.

What formula would I use?

 

[phyzguy]

What formula would you use for what? You used the right formula to calculate the mass of the Sun. You can't calculate the mass of the Earth from the parameters of the Earth's orbit around the Sun. If you want to calculate the mass of the Earth, you need to use something orbiting the Earth. Is that your question?

 

[gtsinc]

Sorry, yes, that was my question.
I wanted to calcualte the weight of an orbiting body knowing its period and radius.

So if the body in question does not have a moon how else would it be calculated?

 

[phyzguy]

If the orbiting body is small compared to the body being orbited, there is no way to calculate the mass of the orbiting body from its period and radius. Look at the equations - the mass of the orbiting body cancels out. This means that objects of different mass follow the same orbit.

 

[gtsinc]

Now I understand - thank you.

So what would prevent a large planet like Jupiter from having an orbit as small as Mercury?

 

[Bandersnatch]

Now I understand - thank you.

So what would prevent a large planet like Jupiter from having an orbit as small as Mercury?

I'm guessing you haven't heard of Hot Jupiters?

http://en.wikipedia.org/wiki/Hot_Jupiter

 

[phyzguy]

Now I understand - thank you.

So what would prevent a large planet like Jupiter from having an orbit as small as Mercury?

Nothing. As Bandersnatch points out, planets like this exist. Conversely, there are asteroids much smaller than Mercury in orbits like Jupiter's.