The mass of a star in a galaxy far, far away, is 2.13 × 105 Earth masses, and the mean distance from the center of this star to the
center of a planet with the same mass as our Earth is 7.87 × 108
km.
Treating this planet and star as particles, with each mass concentrated at its respective
geometric center, how far from the center of the star is the center of mass of the planet-star
system?
Answer in units of km.
Ms = 2.13 * 10^5 Me
Me = Me
Ms x = Me (d-x)
x/(d-x) = Me/Ms = 1/ (2.13*10^5)
x<<<< d
so assume
x/d = 1/(2.13*10^5)
x = 7.87 *10^8/(2.13*10^5)
x = 3.69 * 10^3
= 3690 km
check
3690 is indeed <<<< 7870*10^5 :)
To find the center of mass of the planet-star system, we need to use the formula for the center of mass of a two-particle system:
x_cm = (m1 * x1 + m2 * x2) / (m1 + m2)
Where:
x_cm is the position of the center of mass
m1 and m2 are the masses of the two particles
x1 and x2 are the positions of the two particles
In this case, we have the mass of the star as 2.13 × 105 Earth masses, and the mass of the planet is the same as Earth, so we can use the mass of Earth for m2.
Let's assume the center of the star is at position x = 0. Since the star and the planet are treated as particles concentrated at their respective geometric centers, the position of the planet is at the mean distance from the star, which is 7.87 × 108 km.
Plugging in the values into the formula, we get:
x_cm = (2.13 × 105 * 0 + 1 * 7.87 × 108) / (2.13 × 105 + 1)
Simplifying the equation:
x_cm = 7.87 × 108 / (2.13 × 105 + 1)
Calculating the denominator:
2.13 × 105 + 1 ≈ 2.13 × 105 (since 1 is negligible compared to 2.13 × 105)
Substituting back into the equation:
x_cm = 7.87 × 108 / 2.13 × 105
Dividing the numerator and denominator:
x_cm ≈ 3690.1
Therefore, the center of mass of the planet-star system is approximately 3690.1 km from the center of the star.
To find the center of mass of the planet-star system, we can use the formula for the center of mass of a two-particle system:
x_cm = (m1 * x1 + m2 * x2) / (m1 + m2)
Where:
x_cm is the distance from the center of the star to the center of mass of the system
m1 is the mass of the star
m2 is the mass of the planet
x1 is the distance from the center of the star to its own center of mass (0, since the mass is concentrated at the center)
x2 is the distance from the center of the star to the center of the planet
Given:
m1 = 2.13 × 10^5 Earth masses
m2 = 1 Earth mass (since it's the same mass as our Earth)
x1 = 0
x2 = 7.87 × 10^8 km
Substituting the values into the formula:
x_cm = (m1 * x1 + m2 * x2) / (m1 + m2)
= (2.13 × 10^5 * 0 + 1 * 7.87 × 10^8) / (2.13 × 10^5 + 1)
= (7.87 × 10^8) / (2.13 × 10^5 + 1)
= 7.87 × 10^8 / 2.13 × 10^5
= 3691.03 km
Therefore, the center of mass of the planet-star system is located approximately 3691.03 km from the center of the star.