Determining Mass from Orbital Period and Radius

In summary, a hot Jupiter is a planet that is too close to the sun. The increased heat makes the planet's atmosphere very hot, and the gas and water vapor in the atmosphere escape into space. This makes the planet very small and makes it difficult to see.
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procon4
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Homework Statement


What is the mass of a planet (in kg and in percent of the mass of the sun), if:

its period is 3.09 days,
the radius of the circular orbit is 6.43E9 m,
and the orbital velocity is 151 km/s.

Homework Equations



I'm unsure what formulas to use, though these seem relevant.

F= ma

accel. centripetal = v^2/r

Total Energy = -G*(mass of planet)*(mass of sun)/2*radius

The Attempt at a Solution



I thought I should use Force of gravity = mass of the planet times the centripetal acceleration, but the mass of the planet cancels out. I can't ID the right relationship between the period, radius and mass, so I'm not sure what to do.

Thanks for any help.

JS
 
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  • #2
Consider using vis viva equation as applied to circular orbits
 
  • #3
That's a really good suggestion--I'm surprised that equation isn't in our textbook. The problem is that the mass of the star around which the planet orbits is not given. So I guess there must be some relationship between period, orbital radius, and mass, but I'm not sure what it is.

I would have expected an energy-related equation could work, but I haven't found one that doesn't either require the star's mass or in which the mass of the planet doesn't cancel out.
 
  • #4
The mass of the sun is a known quantity which you can lookup. You could derive vis viva from what the question gives you though...

edit:I don't think you even need the period TBH
 
  • #5
Use Keplers law of period and the mass turns out to be 2.207610x1030
 
  • #6
110% mass of the sun.
 
  • #7
You can also use orbital velocity and work it out from there.
 
  • #8
So just to clarify the situation here, the star at the center of the planet's orbit is not the sun. But another problem was that I needed to find the mass of the star, not the planet. To do that, I just used the F=ma equation, with F being the force of gravity, m being the mass of the planet, and a =v^2/r. The mass of the planet cancels out and you're left with the mass of the star.

The answer fcb posted is correct. Thanks everyone.

This question was called "Hot Jupiter," from Mastering Physics Ch. 12.
 

1. How do you determine mass from orbital period and radius?

To determine mass from orbital period and radius, you can use the formula: M = (4π²R³)/(G*T²), where M is the mass, R is the radius, G is the gravitational constant, and T is the orbital period. This formula is based on Newton's form of Kepler's third law.

2. What is the relationship between mass, orbital period, and radius?

The relationship between mass, orbital period, and radius is described by Newton's form of Kepler's third law, which states that the square of the orbital period is directly proportional to the cube of the semi-major axis (radius) of an orbit. This means that as the mass increases, the orbital period increases, and as the radius increases, the orbital period also increases.

3. Can mass be determined from orbital period and radius for any object?

Yes, mass can be determined from orbital period and radius for any object that is in orbit around another object. This includes planets, moons, and artificial satellites.

4. How accurate is the mass determination from orbital period and radius?

The accuracy of the mass determination from orbital period and radius depends on the accuracy of the measurements used in the formula. If the orbital period and radius are measured with high precision, the calculated mass will also be more accurate. However, there may be other factors that can affect the accuracy, such as the presence of other objects in the orbit or non-uniform density of the object.

5. Are there any limitations to determining mass from orbital period and radius?

Yes, there are some limitations to determining mass from orbital period and radius. For example, if there are other objects in the orbit, their gravitational influence can affect the measurements and calculations. Additionally, the formula assumes that the orbit is circular, which may not always be the case. In some cases, other methods of determining mass, such as using gravitational lensing or spectroscopy, may be more accurate.

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