Radius of a planet, given the density

[s.dyseman]

Hello! I'd like to thank everyone in advance for any help I receive!

Homework Statement

In January 2006, astronomers reported the discovery of a planet comparable in size to the Earth orbiting another star and having a mass of about 5.5 times the Earth's mass. It is believed to consist of a mixture of rock and ice, similar to Neptune. Take mEarth=5.97×10^24kg and rEarth=6.38×10^6m.

If this planet has the same density as Neptune (1.76 g/cm3), what is its radius expressed in kilometers?

Homework Equations

density=m/v

volume(of a sphere)=4/3(pi)r^3

The Attempt at a Solution

I calculated the mass of the new planet (n) as (5.5)5.97*10^24kg=3.28*10^25kg

I set the given density of 1.76 g/cm3 equal to the found mass of 3.28*10^22g divided by the formula for the volume of a sphere, shown below:

1.76 g/cm3 = 3.28*10^22g/[4/3(pi)r^3]

I simplify until I reach r^3=4.46*10^21 cm3, then take the cubic root.

The answer I reach is 16456067.27 cm or 164.56 km (165 to three significant figures). I'm not sure why this answer is incorrect.

 

[BvU]

From 3.28*10^25kg to 3.28*10^22g is a bit strange. 1 kg = 1000 g, so I would expect 10^28 .
Everything else is just fine.

 

[nil1996]

Hi s.dyseman

welcome to PF

You have just gone wrong at conversions.

 

[s.dyseman]

Thanks everyone! I was able to correctly answer the problem after you pointed out my mistake.Thanks again

 

[HalfLostAndSearching]

I would like to clarify that the density of a planet is influenced by various factors such as composition, temperature, and pressure. Therefore, it is not always accurate to assume that two planets with the same density will have the same radius. However, for the purpose of this exercise, we can approximate the radius of this newly discovered planet using the given density and mass.

To calculate the radius, we can use the formula for density (ρ = m/V) and the formula for the volume of a sphere (V = 4/3πr^3). Rearranging the equations, we get r = (3m/4πρ)^1/3.

Substituting the given values, we get r = (3*5.5*5.97*10^24 kg)/(4*3.14*1.76 g/cm3)^1/3 = 3.05*10^7 m = 30,500 km.

Therefore, the approximate radius of this planet is 30,500 km. However, it is important to keep in mind that this is just an approximation and the actual radius could vary depending on the factors mentioned earlier.