Astronomers determine masses of distant objects by observing their smaller companions. For example, they can measure the period of a planet orbiting around a distant star, measure the radius of its circular orbit, and then employ Newtonian physics to determine the mass of the star. Callisto is one of the Jupiter's moons. Its period of revolution is 16.7days, and the radius of its circular trajectory is . What is Jupiter's mass in 1.9*10^6kg?
To determine Jupiter's mass using the period and radius of Callisto's orbit, we can use Kepler's Third Law of Planetary Motion and Newton's law of universal gravitation.
Kepler's Third Law of Planetary Motion states that the square of a planet's orbital period is directly proportional to the cube of its mean distance from the sun (or in this case, the star it orbits). Mathematically, this can be expressed as:
T^2 = k * r^3
where T is the period of revolution, r is the radius of the circular orbit, and k is a constant of proportionality.
To establish the relationship between the period of Callisto's orbit around Jupiter and Jupiter's mass, we need to determine the value of k. Since we're interested in Jupiter's mass in units of kilograms, k will have different units than if we were using astronomical units (AU). In this case, we can use the value of 4π^2G, where G is the gravitational constant. Its approximate value is 6.67430 × 10^-11 N(m/kg)^2.
Let's plug in the known values into Kepler's third law equation and solve for k:
(16.7 days)^2 = k * (radius of Callisto's orbit)^3
Converting the period from days to seconds:
(16.7 days)^2 = k * (radius of Callisto's orbit)^3
(16.7*24*60*60 seconds)^2 = k * (radius of Callisto's orbit)^3
Calculating the periodic square:
2.82376*10^8 seconds^2 = k * (radius of Callisto's orbit)^3
Now, we can solve for k by dividing both sides of the equation by (radius of Callisto's orbit)^3:
k = (2.82376*10^8 seconds^2) / (radius of Callisto's orbit)^3
Substituting the given radius of Callisto's orbit (which is missing from the question), we can solve for k.
Once we have the value of k, we can use the known value of Callisto's period and the radius of its circular orbit to determine Jupiter's mass:
Jupiter's mass = k / (Callisto's period)^2 * (radius of Callisto's orbit)^3.
However, it seems that the radius of Callisto's orbit is missing from the question. To calculate Jupiter's mass, we need that information.