mass and density in the solar system how do we know?

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


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 NewtonCombining Newton’s laws of motion with the law of gravity for two objects orbiting each other . . .

Fm = mam

F = - G Mm

r 2(G is the Universal constant of gravitation.)

r mM

FM = MaM

FM = - Fm

= - 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 NewtonCombining Newton’s laws of motion with the law of gravity for two objects orbiting each other . . .

r mM

= - 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 mM

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 0.0078% 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.However, Io takes MUCH less time for one orbit than the Moon.

Jupiter’s moon Io orbits Jupiter at about the Jupiter’s moon Io orbits Jupiter at about the same distance as the Moon orbits Earth.same distance as the Moon orbits Earth.

Earth

Jupiter

Orbital Period27.3 days


Moon

Io

Using the orbital periods we can compare the mass of Using the orbital periods we can compare the mass of Jupiter and the mass of Earth. Jupiter and the mass of Earth.

Jupiter has a mass over 300 times larger than Earth’s mass!Jupiter has a mass over 300 times larger than Earth’s mass!

Jupiter’s moon Io orbits Jupiter at about the Jupiter’s moon Io orbits Jupiter at about the same distance as the Moon orbits Earth.same distance as the Moon orbits Earth.

Earth Moon

JupiterIo



MJ = ME (27.3 / 1.77)2 (1.10)3

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.

Only three planets were known to have moons during Newton’s lifetime.

Credit: NASA/JPLCredit: 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/JPLCredit: 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 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

Earth

Callisto

Saturn

Jupiter

Titan

Sun

Venus

Moon


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.


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 100Jupiter 94 ½ 94.4Saturn 67 50.4Earth 400 390.1

mass and density in the solar system how do we know?

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