Mass and Density In the solar system How do we know?

Isaac Newton Isaac Newton in 1689 July 5, 1687 Isaac Newton discovered the relationship between gravitational force, mass and distance that we call the “law of gravity”.

The strength of the gravitational force that keeps one object in orbit around another depends on two things. Gravity and Orbits The distance between them and their mass If we could determine the strength of the gravitational force and the distance we could calculate mass.

Distance Distances can be found using astronomical observations and trigonometry. R d  d = R sin 

How can we find the gravitational force? Thanks to Isaac Newton, there is a way around this problem.

Isaac Newton Isaac Newton in 1689 July 5, 1687 Newton also discovered three laws that describe how the motion of an object is changed by forces, including gravity. We call these “Newton’s Laws of Motion”.

Isaac Newton Combining Newton’s laws of motion with the law of gravity for two objects orbiting each other... F m = ma m F = - G Mm r 2 (G is the Universal constant of gravitation.) r m M F M = Ma M F M = - F m = - G (M + m) r 2 d v d t... and then with the aid of calculus (which Newton invented) and some algebra... we get an equation describing the motion of the objects relative to each other...

  r  G (M + m) Isaac Newton Combining Newton’s laws of motion with the law of gravity for two objects orbiting each other... r m M = - G (M + m) r 2 d v d t... and then with the aid of calculus (which Newton invented) and some algebra... We obtain a relationship between orbital period, distance and mass. P 2 = P = orbital period we get an equation describing the motion of the objects relative to each other...

  r  G (M + m) P 2 =   r  G M Isaac Newton r m M For a planet with an orbiting moon, the mass of the moon is so small compared to the planet that the sum of the moon’s mass and the planet’s mass is about the same as the planet’s mass alone. P = orbital period Ganymede, Jupiter’s largest moon and the largest moon in the solar system has only % the mass of Jupiter. The Moon has a mass only 1.2% of Earth. So, if Earth’s mass = 1.000, the mass of Earth + Moon = 1.012

Gravitational Force and Mass So, if a planet has a moon and we measure both the moon’s orbital period and the distance between the moon and planet, we can calculate the mass. Here is an example:

However, Io takes MUCH less time for one orbit than the Moon. Jupiter’s moon Io orbits Jupiter at about the same distance as the Moon orbits Earth. Earth Jupiter Orbital Period 27.3 days Orbital Period 1.77 days Moon Io

Using the orbital periods we can compare the mass of Jupiter and the mass of Earth. Jupiter has a mass over 300 times larger than Earth’s mass! Jupiter’s moon Io orbits Jupiter at about the same distance as the Moon orbits Earth. EarthMoon Jupiter Io Orbital Period 27.3 days Orbital Period 1.77 days M J = M E (27.3 / 1.77) 2 (1.10) 3

Isaac Newton Isaac Newton in 1702 In his book that announced his laws of motion and gravity, Newton used these laws to calculate the densities of four objects in the solar system.

Only three planets were known to have moons during Newton’s lifetime. Credit: NASA/JPL Credit: NASA/JPL/ Southwest Research Institute Credit: NASA/JPL/Malin Space Science Systems Earth Jupiter Saturn

Newton calculated the density of the these three planets and the Sun. Credit: NASA/JPL Credit: NASA/JPL/ Southwest Research Institute Credit: NASA/JPL/Malin Space Science Systems Earth Jupiter Saturn

Newton used the orbit of Venus to calculate the Sun’s density This photograph shows the Sun and Venus during the Venus transit of The big white circle is the Sun. Venus is the black dot on the Sun. Venus is near the top of the Sun, just left of center. Image courtesy the U.S. Naval Observatory Library.

Newton’s Density Calculations Newton wrote, “Thus from the periodic times [orbital periods] of Venus around the Sun,... the outermost satellite of Jupiter [Callisto] around Jupiter,... the Huygenian satellite [Titan] around Saturn,... and of the Moon around the Earth... compared with the mean distance of Venus from the Sun and with [the measured angles that would allow Newton to calculate the planet- moon distances]..., by entering into a computation... The quantity of matter [mass] in the individual planets is also found.”

Newton’s Cast of Characters Earth Callisto Saturn Jupiter Titan Sun Venus Moon

Newton’s Density Calculations Newton could only calculate the masses of the planets relative to each other because the gravitational constant in his law of gravity had not yet been determined. With the relative masses known, and by also calculating the relative volumes, Newton wrote, “The densities of the planets also become known.” Delicate experiments performed by Henry Cavendish in 1797 and 1798 measured Earth’s average density, allowing the determination of the gravitational constant.

Newton’s Density Calculations As he could only calculate relative densities, he assigned the Sun an arbitrary density of 100 and calculated the densities of Jupiter, Saturn and Earth relative to the Sun. Newton’sModern Calculation Value Sun Jupiter 94 ½ 94.4 Saturn Earth