Suppose you want to find out how much you weigh. It’s easy enough – get a scale, stand on it, and read the number.
Now suppose you want to know how much the Earth weighs. You get out a scale and…hmm. What exactly do you do with it?
Let’s do a little physics class refresher. When a scale measures weight, that weight is a force, specifically the force of the gravitational attraction between what’s on top of the scale and what’s beneath it. If you stand on the scale and the scale is on the floor, it’s the attraction between you and the Earth.
Isaac Newton’s law of universal gravitation tells us that gravitational attraction depends on the mass of the two objects and the distance between them. The mass of the Earth is constant, and the distance from the scale to the center of the Earth is constant. So the only thing that affects the scale’s reading is your own mass – the amount of matter in your body, independent of Earth’s gravity.
Isaac Newton’s 3rd law of motion conveniently tells us that every action has an equal and opposite reaction. So if you weigh 180 pounds on the Earth, the Earth also weighs 180 pounds on you. Turn your bathroom scale upside down and try it!
As technically correct as this answer for the Earth’s weight is, it’s not super useful for finding the Earth’s mass – at least not by itself. But it could be a good first step. All we need is to know how much something else weighs on you to compare to the Earth.
So here’s what we need to know to find the mass the Earth:
- How much the Earth weighs on you
- Your distance to the center of the Earth
- How much some other object weighs on you
- The distance between you and that object
- That object’s total mass
- Some math to compare the results
Number 1 is easy to find – just use your upside-down scale. Number 2 is doable as well – we’ve known the radius of the Earth for thousands of years. Number 4 can be found out with a ruler so long as the object is nearby and stays in one place, and number 6 can be done with the math that Isaac Newton developed.
The real challenge is combining 3 and 5. Gravity is really weak, so it’s hard to find something that’s massive enough that we can detect its gravity but small enough that we can calculate its mass directly.
The first experiment to do this decently decided to use a mountain, specifically Schiehallion in Scotland, in 1774.
Image by Andrew2606, CC BY
Schiehallion is fairly symmetrical and isolated, and was conveniently accessible to the British Royal Society, who were running the experiment. Instead of weighing the mountain’s gravitational effects on a person, they measured its effect on a precision plumb line.
The Astronomer Royal at the time, Nevil Maskelyne, set up two astronomical observatories next to Schiehallion, one to the North and one to the South. Using the stars as a reference, Maskelyne’s team found that the plumb lines on either side of the mountain pointed just 0.0152 degrees apart. About 80 percent of that difference could be expected from the curvature of the Earth between the two sites, meaning the remainder – just 0.0032 degrees – was from the gravity of the mountain itself.
The final remaining step was to calculate the mass of Schiehallion directly. This was accomplished by a team of surveyors led by Charles Hutton, who mapped the mountain’s shape and the density of its rocks at different layers.
Putting together the results, Maskelyne and Hutton announced that the Earth was 1.8 times as dense as Schiehallion, or 4.5 metric tons per cubic meter. Using this value, we get that the mass of the Earth is 4.87×1021 metric tons. This is within 20% of the best modern measurements, which put the mass of the Earth at 5.91×1021 metric tons.
The Schiehallion experiment wasn’t the state of the art for long. A more precise result was achieved in 1798 by Henry Cavendish, who was on the committee for the Schiehallion experiment. Cavendish’s experiment measured the gravity of large lead spheres using an extremely precise torsion pendulum, and cut the error from 20% down to 1.2%. Still, Schiehallion was first, and it proved that gravity came from all kinds of objects, not just the Earth or the Sun. And that itself is a pretty massive achievement.
Coming soon: 1960: the Year of Africa