Calculate the radius of a “super- Earth” with 10 times the mass of Earth, using Earth data as a starting point.
Assuming the mass distribution and average density of the super-Earth does not depart from our planet, then the radius will be increased by a factor of 10^(1/3)=2.154 approximately.
Use whatever radius of Earth that is given to you or you can find.
Use whatever radius of Earth that is given to you or you can find.
- what do you mean by this ?
Question says "using Earth data as a starting point" but does not give the radius of the earth.
It could be 6371km (from Google) or any other value with different accuracies.
So you will decide which value to use, you just have to state the source.
why is it increased by a factor of 10^(1/3)
To calculate the radius of a "super-Earth" with 10 times the mass of Earth, we can make use of the concept of the mass-radius relation. This relation is often described using the term "mass-radius relationship" or "mass-radius equation."
To begin, we need to consider the mass-radius relationship for Earth-like planets. It is important to note that this relationship depends on various factors, such as composition, density, and internal structure. For simplicity, let's use a basic approximation of the Earth's mass-radius relation.
The mass-radius relationship for Earth-like planets can be represented by the following equation:
(M / M_earth) = (R / R_earth)^(3/5)
Here,
- M represents the mass of the planet in question.
- M_earth is the mass of Earth, approximately 5.97 × 10^24 kilograms.
- R represents the radius of the planet in question.
- R_earth is the radius of Earth, approximately 6,371 kilometers.
Using this equation, we can solve for the radius, R, of the "super-Earth" with 10 times the mass of Earth.
Let's substitute the known values into the equation:
(10M_earth / M_earth) = (R / R_earth)^(3/5)
Cross-multiplying the equation:
10^(5/3) = (R / R_earth)^(3/5)
To solve for R, we need to raise both sides of the equation to the power of 5/3:
(R / R_earth)^(3/5) = (10^(5/3))^(5/3)
(R / R_earth)^(3/5) = 10^(25/9)
Now, we can simplify the equation:
(R / R_earth) = 10^(25/9)^(5/3)
(R / R_earth) = 10^(125/27)
Taking the fifth root of both sides:
(R / R_earth)^(1/5) = (10^(125/27))^(1/5)
(R / R_earth)^(1/5) = 10^(125/135)
Now, we solve for R by multiplying both sides by R_earth:
R = R_earth * (10^(125/135))^(1/5)
Finally, substitute the value of R_earth (approximately 6,371 kilometers) into the equation to calculate the radius of the "super-Earth" with 10 times the mass of Earth.
R = 6,371 km * (10^(125/135))^(1/5)
Evaluating the expression will give you the approximate radius of the "super-Earth" in kilometers.