(a) Use the orbital properties of the Moon to estimate the...

(a) Use the orbital properties of the Moon to estimate the mass of the Earth. (b) Use the orbital properties of the Earth to determine the mass of the Sun (neglect the mass of the Earth relative to that of the Sun). (c) Look up the orbital properties of the four Galilean moons of Jupiter (Io, Europa, Ganymede, and Callisto). Use this to determine the mass of Jupiter, assuming that it is much larger than the mass of a Galilean moon. Hint: Plot $\log P$ versus $\frac{3}{2} \log a$, where $P$ is the period and $a$ the semimajor axis, and apply Kepler's third law.

(a) Use the orbital properties of the Moon to estimate the mass of the Earth.
(b) Use the orbital properties of the Earth to determine the mass of the Sun (neglect the mass of the Earth relative to that of the Sun).
(c) Look up the orbital properties of the four Galilean moons of Jupiter (Io, Europa, Ganymede, and Callisto). Use this to determine the mass of Jupiter, assuming that it is much larger than the mass of a Galilean moon. Hint: Plot $\log P$ versus $\frac{3}{2} \log a$, where $P$ is the period and $a$ the semimajor axis, and apply Kepler's third law.

Key Concepts

Central Mass Approximation

This concept involves assuming that the mass of the orbiting body is much smaller than that of the central body, which simplifies the application of Kepler's Third Law and Newton's Law of Gravitation. By neglecting the satellite's mass, the calculations focus solely on the dominant gravitational source, making it feasible to accurately compute the central body’s mass using orbital data.

Logarithmic Transformation in Data Analysis

Transforming the power-law relationship of Kepler's Third Law into a linear relationship using logarithms (for example, by plotting log P versus (3/2) log a) is a powerful technique in data analysis. This approach simplifies the interpretation and analysis of the data, enabling researchers to verify the law's validity and extract the proportionality constant, which is directly related to the gravitational parameter and hence the mass of the central object.

Newton's Law of Universal Gravitation

This fundamental law states that every two masses in the universe attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. It underpins the method of determining a celestial body's mass by relating the orbital characteristics of its satellite to the gravitational force exerted by the body.

Kepler's Third Law

Kepler's Third Law establishes a precise relationship between the period of orbit (P) of a satellite and the semimajor axis (a) of its orbit, typically expressed as P² ? a³ for a system where the mass of the orbiting satellite is negligible compared to the central object's mass. The law is crucial for estimating the mass of the central body by linking observable orbital properties to the gravitational parameter in the equation.

Transcript

00:01 In this problem, we're given information data about the four largest moons of jupiter, and we're asked to find period with semi -major axes.

00:13 So we're going to go in turn to find each of those quantities.

00:16 A place to start is kepler's third law, which is 2 pi square root of g times the mass of jupiter.

00:25 We're looking at the jupiter system, not worrying about the sun, a to three halves.

00:30 So that's what we get for the period.

00:33 They might say, well, couldn't we go and find, using that data to find the mass of jupiter, and then just calculate directly every single one of those? yes, you certainly could do that.

00:43 But sometimes, sometimes that either is impossible to do it in the way you're thinking or is not very instructive or is very calculationally intensive.

00:55 So let's find another way of doing it.

00:57 So let's look at the ratio of two satellites, two moons, t -a over t -b, and that would be 2 pi over the square root of gmg, or m -j i should say, over again the square root, gmj.

01:20 Now on top we're going to have a for satellite a to three -haves over a satellite b, three -h halves on the bottom.

01:31 So this factor in here, there's two -pi over a square root, is a constant.

01:36 System so they cancel out.

01:39 So this becomes a ratio of the semi -major axes.

01:59 We sometimes thought we want to get an a out of this, so we can just reverse this and say that a for satellite a, a for satellite b is a ratio of the t's, put that to the two -thirds power.

02:23 So let's start filling in our data using those.

02:28 Or filling our table, i should say.

02:30 Let us find the period of europa.

02:33 We're going to use that ratio with io, and that will be a .e.

02:42 Over a .i.

02:44 Three -h halves, which gives me the t .e.

02:54 671034 kilometers.

02:56 Does not matter, kilometers, meters, miles, astronomical units.

02:59 It's a ratio.

03:02 4217 -00.

03:04 Kilometers and then the period of io 1 .769 days and this works out to be 3 .551 days.

03:29 So that is the period of europa and that fills out that column for periods.