Can you use a planet’s density to predict its mass?

Barry Lienert, a geophysicist at the University of Hawaii, provides the following explanation.

We start by determining the mass of the Earth. Issac Newton's Law of Universal Gravitation tells us that the force of attraction between two objects is proportional the product of their masses divided by the square of the distance between their centers of mass. To obtain a reasonable approximation, we assume their geographical centers are their centers of mass.

Because we know the radius of the Earth, we can use the Law of Universal Gravitation to calculate the mass of the Earth in terms of the gravitational force on an object (its weight) at the Earth's surface, using the radius of the Earth as the distance. We also need the Constant of Proportionality in the Law of Universal Gravitation, G. This value was experimentally determined by Henry Cavendish in the 18th century to be the extemely small force of 6.67 x 10-11 Newtons between two objects weighing one kilogram each and separated by one meter. Cavendish determined this constant by accurately measuring the horizontal force between metal spheres in an experiment sometimes referred to as "weighing the earth."

Calculating the Sun's Mass

Knowing the mass and radius of the Earth and the distance of the Earth from the sun, we can calculate the mass of the sun (right), again by using the law of universal gravitation. The gravitational attraction between the Earth and the sun is G times the sun's mass times the Earth's mass, divided by the distance between the Earth and the sun squared. This attraction must be equal to the centripetal force needed to keep the earth in its (almost circular) orbit around the sun. The centripetal force is the Earth's mass times the square of its speed divided by its distance from the sun. By astronomically determining the distance to the sun, we can calculate the earth's speed around the sun and hence the sun's mass.

Once we have the sun's mass, we can similarly determine the mass of any planet by astronomically determining the planet's orbital radius and period, calculating the required centripetal force and equating this force to the force predicted by the law of universal gravitation using the sun's mass.

Additional details are provided by Gregory A. Lyzenga, a physicist at Harvey Mudd College in Claremont, Calif.

The weight (or the mass) of a planet is determined by its gravitational effect on other bodies. Newton's Law of Gravitation states that every bit of matter in the universe attracts every other with a gravitational force that is proportional to its mass. For objects of the size we encounter in everyday life, this force is so minuscule that we don't notice it. However for objects the size of planets or stars, it is of great importance.

In order to use gravity to find the mass of a planet, we must somehow measure the strength of its "tug" on another object. If the planet in question has a moon (a natural satellite), then nature has already done the work for us. By observing the time it takes for the satellite to orbit its primary planet, we can utilize Newton's equations to infer what the mass of the planet must be.

MASS of asteroid Mathilde was calculated by measuring gravitational perturbations in the course of the passing NEAR spacecraft.

For planets without observable natural satellites, we must be more clever. Although Mercury and Venus (for example) do not have moons, they do exert a small pull on one another, and on the other planets of the solar system. As a result, the planets follow paths that are subtly different than they would be without this perturbing effect. Although the mathematics is a bit more difficult, and the uncertainties are greater, astronomers can use these small deviations to determine how massive the moonless planets are.

Finally, what about those objects such as asteroids, whose masses are so small that they do not measurably perturb the orbits of the other planets? Until recent years, the masses of such objects were simply estimates, based upon the apparent diameters and assumptions about the possible mineral makeup of those bodies.

Now, however, several asteroids have been (or soon will be) visited by spacecraft. Just like a natural moon, a spacecraft flying by an asteroid has its path bent by an amount controlled by the mass of the asteroid. This "bending" is measured by careful tracking and Doppler radio measurement from Earth. Recently, the NEAR spacecraft flew by the asteroid Mathilde, determining for the first time its actual mass. It turned out to be considerably lighter and more "frothy" in structure than had been expected, a fact that is challenging planetary scientists for an explanation.

Originally published on March 16, 1998.

The most accurate way of measuring the mass of a planet is to send a spacecraft to it and measure the acceleration due to gravity as the spacecraft passes by it.

Alternatively, if the planet has a moon then its mass can be calculated from the moon's orbit.

First of all we need to know how far the planet is from the Earth. In the case of Venus we do this by bouncing radar signals off the planet and measuring the time it takes for the radar to return. Given the distance of Venus from Earth at its closest point, we can calculate the distance from the Earth to the Sun. Now if we measure the orbital period of any other planet we can calculate the distance using Kepler's third law.

To calculate the mass of the planet we need the distance of the planet form Earth #R#. We then need to measure the orbital period #T# of the moon and the largest angular separation #theta# of the planet and the moon as the moon orbits the planet.

We can now calculate the radius of the moon's orbit #r=R theta#. We now use Newton's form of Kepler's third law:

#T^2=(4 pi^2)/(G(M+m))r^3#

Where #G# is the gravitational constant, #M# is the mass of the planet and #m# is the mass of the moon.
Rearranging the equation gives:

#M+m=(4 pi^2 r^3)/(GT^2)#

We now have calculated the combined mass of the planet and the moon. If the moon is small compared to the planet then we can ignore the moon's mass and set #m=0#. This is true of most moons in the solar system.

If the moon is relatively large such as the Earth's Moon and Pluto and Charon, then we need to find the centre of mass which the planet and the moon are orbiting around. The distance #d# from the centre of the planet to the centre of mass of the planet and moon can be used to calculate the ratio of that planet and moon masses and hence the planet''s mass.

#Md=m(r-d)#

This gives the planet's mass as:

#M=(4 pi^2 r^2(r-d))/(GT^2)#

Updated April 24, 2017

By John Papiewski

The physical density of any object is simply its mass divided by its volume; density is measured in units such as pounds per cubic foot, grams per cubic centimeter or kilograms per cubic meter. When calculating the density of a planet, look up its mass and radius, the latter of which is the distance from the surface to the center. Because planets are roughly spherical, calculate the volume of a sphere using the radius. Then divide the mass by the volume of the sphere to get the density.

Find the planet’s mass and diameter. For example, Earth’s mass is about 6,000,000,000,000,000,000,000,000 kg and its radius measures 6,300 km.

Enter the radius in the calculator. Multiply by 1,000 to convert kilometers to meters. Cube this number by pressing the "x^3" key; alternatively, you can press the "x^y" key, enter the number three and then press "equals." Multiply by the number pi -- or 3.1416 -- multiply by four and then divide by three. Store the result by pressing the "M+" or other memory key. The figure you see is the planet’s volume in cubic meters. To continue the example, 6,300 km times 1,000 meters/km = 6,300,000 meters. Cubing it gives 250,000,000,000,000,000,000. Multiplying by pi times 4/3 yields 1,047,400,000,000,000,000,000 cubic meters.

Key the planet’s mass into the calculator. Press the divide key, then recall the volume figure stored in the calculator’s memory. Press the equals key. This result is the planet’s density in units of kilograms per cubic meter. In our example, dividing 6,000,000,000,000,000,000,000,000 kg by 1,047,400,000,000,000,000,000 cubic meters results in a density of about 5,730 kg per cubic meter.