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

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

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

Distance d q R d = R sinq Distances can be found using astronomical observations and trigonometry.

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

Isaac Newton 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 in 1689 July 5, 1687

Isaac Newton F = - G = - G FM = MaM Fm = mam FM = - Fm Combining Newton’s laws of motion with the law of gravity for two objects orbiting each other . . . we get an equation describing the motion of the objects relative to each other . . . . . . and then with the aid of calculus (which Newton invented) and some algebra . . . Mm F = - G (G is the Universal constant of gravitation.) r 2 d v (M + m) = - G r m d t r 2 M FM = MaM Fm = mam FM = - Fm 7

We obtain a relationship between orbital period, distance and mass. Isaac Newton Combining Newton’s laws of motion with the law of gravity for two objects orbiting each other . . . we get an equation describing the motion of the objects relative to each other . . . . . . and then with the aid of calculus (which Newton invented) and some algebra . . . d v (M + m) 4 p2 r 3 G (M + m)  = - G r P 2 =   m d t r 2 M P = orbital period We obtain a relationship between orbital period, distance and mass. 8

Isaac Newton 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. Ganymede, Jupiter’s largest moon and the largest moon in the solar system has only 0.0078% the mass of Jupiter. 4 p2 r 3 G M   4 p2 r 3 G (M + m)  r P 2 =   m M This relationship relates orbital period to the sum of the mass of both objects. However, if the objects are a planet and a moon, the mass of the moon is usually very small compared to the planet so we can take the resultant mass as the mass of the planet. For example, the Moon’s mass is only 1.2% the mass of Earth. So, if we take the calculated mass (using the orbit of the Moon) to be Earth’s mass, it will be accurate to about 1%. P = orbital period The Moon has a mass only 1.2% of Earth. So, if Earth’s mass = 1.000, the mass of Earth + Moon = 1.012 9

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:

Jupiter’s moon Io orbits Jupiter at about the same distance as the Moon orbits Earth. Orbital Period 27.3 days Earth Moon Orbital Period 1.77 days Io Io orbits Jupiter over 15 times in the time it takes Moon to orbit Earth once. Jupiter 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. Orbital Period 27.3 days Earth Moon MJ = ME (27.3 / 1.77)2 (1.10)3 Orbital Period 1.77 days Io Io is 1.10 times farther from Jupiter than Moon is from Earth The mass of Jupiter is 317.83 Earth masses Jupiter 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!

Isaac Newton 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. Isaac Newton in 1702

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

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

Newton used the orbit of Venus to calculate the Sun’s density This photograph shows the Sun and Venus during the Venus transit of 1882. 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 Venus Titan Moon Saturn Sun Earth Callisto Jupiter

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’s Modern Calculation Value Sun 100 100 Jupiter 94 ½ 94.4 Saturn 67 50.4 Earth 400 390.1