The mass of a star in a galaxy far, far away,
is 3.12 × 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 2.46 × 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.
If we set the center of mass at 0, then we have
3.12*10^5 * 2.46*10^8 = m*d
where m is earth's mass and d is the distance to the center of the planet.
To find the center of mass of the planet-star system, we can use the formula:
Center of Mass = (m1 * x1 + m2 * x2) / (m1 + m2)
Where m1 and m2 are the masses of the star and the planet respectively, and x1 and x2 are their distances from the center of mass.
Given:
Mass of the star (m1) = 3.12 × 10^5 Earth masses
Distance from the center of the star to the center of the planet (x2) = 2.46 × 10^8 km
Since the mass of the planet (m2) is the same as the mass of the Earth, we can substitute the mass of the Earth for m2. The mass of the Earth is approximately 5.97 × 10^24 kg.
Now, we need to convert the mass of the star from Earth masses to kg. To do this, we multiply the mass of the star in Earth masses by the mass of the Earth:
m1 (in kg) = (3.12 × 10^5 Earth masses) * (5.97 × 10^24 kg/1 Earth mass)
Next, we can substitute the values into the center of mass formula:
Center of Mass = (m1 * x1 + m2 * x2) / (m1 + m2)
Since the star is considered to be at the origin (centered at (0,0)), the distance from the center of mass to the center of the star (x1) is 0.
Center of Mass = (m2 * x2) / (m1 + m2)
Now, we can substitute the values into the formula:
Center of Mass = ((3.12 × 10^5 Earth masses) * (5.97 × 10^24 kg/1 Earth mass) * (2.46 × 10^8 km)) / ((3.12 × 10^5 Earth masses) + (5.97 × 10^24 kg/1 Earth mass))
By multiplying and dividing the values, we can simplify the equation:
Center of Mass = (3.12 × 10^5) * (5.97 × 10^24) * (2.46 × 10^8) / (3.12 × 10^5 + 5.97 × 10^24)
Calculating this expression will give us the distance from the center of the star to the center of mass of the planet-star system in km.