Astonomers discover a new planet orbiting a fixed point in space, but for reasons unknown, they can't directly observe a star where one is expected. The radius of the orbit is measured to be 1.85 x 10^8 km, and the lone planet completes an orbit once every 530 days.
a.) Calculate the mass of the unseen star or other celestial object this planet is orbiting.
b.) For an orbit to be stable, the centripetal force must exactly equal the force of gravitational attraction between 2 bodies. If this planet orbited the object once every 580 days, would this be stable orbit? Explain. Assume this planet is about the same size as Earth, and use the mass of the hidden object found in part a.
Who still here in 2019
you know the velocity of the planet(r*2PI/530days)...change that to m/s
Then set force of gravity equal to centripetal acceleration
GMm/r^2=mv^2/r and solve for M
a.) To calculate the mass of the unseen star or celestial object, we can use the formula for centripetal force:
F_c = (m * v^2) / r
Where:
F_c is the centripetal force
m is the mass of the planet
v is the velocity of the planet
r is the radius of the orbit
Given:
r = 1.85 x 10^8 km = 1.85 x 10^11 m
v = (2 * π * r) / T
T = 530 days = 530 * 24 * 60 * 60 seconds
First, let's calculate the velocity v of the planet:
v = (2 * π * 1.85 x 10^11) / (530 * 24 * 60 * 60)
Next, we can rearrange the formula for centripetal force to solve for the mass of the celestial object:
m = (F_c * r) / v^2
Since the centripetal force is equal to the force of gravitational attraction between the planet and celestial object:
F_c = F_g
F_g = (G * m * M) / r^2
Where:
F_g is the gravitational force
G is the gravitational constant
M is the mass of the celestial object
Rearranging this equation, we get:
M = (F_g * r^2) / (G * m)
Plugging in the known values, we can calculate the mass of the unseen celestial object.
b.) For the stable orbit condition, the centripetal force must exactly equal the force of gravitational attraction between two bodies. Let's calculate the period T of the new orbit given:
T = 580 days = 580 * 24 * 60 * 60 seconds
With the new period T, we can calculate the new velocity v:
v = (2 * π * 1.85 x 10^11) / (580 * 24 * 60 * 60)
We can then use the same formula for the force of gravitational attraction:
F_g = (G * m * M) / r^2
With the estimated mass of the celestial object from part a, we can calculate the new force of gravitational attraction.
To determine if the new orbit would be stable, we need to compare the centripetal force to the gravitational force. If they are equal, the orbit will be stable. If not, the orbit will not be stable.
Now, let's calculate the answers step by step.