You are the science officer on a visit to a distant solar system. Prior to landing on a planet you measure its diameter to be 2.070E+7 m and its rotation to be 19.8 hours. You have previously determined that the planet orbits 1.590E+11 m from its star with a period of 368.0 Earth days. Once on the surface you find that the free-fall acceleration is 13.00 m/s2. What is the mass of the planet (in kg)?
What is the mass of the star (in kg)?
To find the mass of the planet, we can use the following equation:
m = (g * r^2) / (G * T^2)
where
m is the mass of the planet,
g is the acceleration due to gravity on the planet's surface,
r is the radius of the planet,
G is the gravitational constant, and
T is the period of the planet's orbit.
Given:
g = 13.00 m/s^2 (acceleration due to gravity on the planet's surface)
r = diameter/2 = 2.070E+7 m / 2 = 1.035E+7 m
T = 368.0 Earth days = 368.0 * 24 * 60 * 60 seconds
The value of the gravitational constant, G, is approximately 6.6743 × 10^-11 N(m/kg)^2.
Calculating the mass of the planet:
m = (13.00 * (1.035E+7)^2) / (6.6743 × 10^-11 * (368.0 * 24 * 60 * 60)^2)
m ≈ 2.8095 × 10^24 kg
Therefore, the mass of the planet is approximately 2.8095 × 10^24 kg.
To find the mass of the star, we can use Newton's version of Kepler's Third Law:
M = (4π^2 * r^3) / (G * T^2)
where
M is the mass of the star,
r is the average distance between the planet and its star,
G is the gravitational constant, and
T is the period of the planet's orbit.
Given:
r = 1.590E+11 m
T = 368.0 Earth days = 368.0 * 24 * 60 * 60 seconds
Calculating the mass of the star:
M = (4π^2 * (1.590E+11)^3) / (6.6743 × 10^-11 * (368.0 * 24 * 60 * 60)^2)
M ≈ 1.2926 × 10^30 kg
Therefore, the mass of the star is approximately 1.2926 × 10^30 kg.
To find the mass of the planet, we can use the formula for the centripetal force exerted on the planet due to its orbit around the star:
F = (m * v^2) / r
Where:
F is the centripetal force,
m is the mass of the planet,
v is the orbital velocity of the planet,
r is the radius of the planet's orbit around the star.
We can find the orbital velocity of the planet using the formula:
v = (2 * π * r) / T
Where:
v is the orbital velocity of the planet,
π is a mathematical constant (approximately 3.14159265),
r is the radius of the planet's orbit around the star,
T is the period of the planet's orbit.
First, let's calculate the orbital velocity of the planet:
v = (2 * π * 1.590E+11 m) / (368.0 Earth days * 24 hours / Earth day * 3600 seconds / hour)
Simplifying the units, we have:
v = (2 * π * 1.590E+11 m) / (368.0 * 24 * 3600 seconds)
Now, we can calculate the centripetal force:
F = (m * v^2) / r
Since the centripetal force is provided by the gravitational force, we can equate it to the gravitational force:
F = G * (m * M) / r^2
Where:
G is the gravitational constant (approximately 6.67430E-11 N * m^2 / kg^2),
M is the mass of the star.
Equating these two equations, we can solve for the mass of the planet:
G * (m * M) / r^2 = (m * v^2) / r
Simplifying, we have:
G * M / r = v^2 / r
Now we can solve for the mass of the planet:
m = (v^2 * r) / (G * M)
Substituting the values we have:
m = (v^2 * 1.590E+11 m) / (G * M)
Let's calculate the mass of the planet.
G*Mp/R^2 = 13.00 m/s^2
G = 6.67*10^-11 N*m^2/kg^2
Solve for the planet mass, Mp.
The mass of the star can be deduced from Kepler's Third Law, using the period and radius of the orbit.
For the formula, see
http://spiff.rit.edu/classes/phys440/lectures/kepler3/kepler3.html
You do not need to use the period of rotation.