# Q&A: How We Know Earth’s Mass

July 3, 2017

**Question:** I was watching this science show where they said Earth weighed 6 zettatons. First, what the heck is a zettaton? Second, how can they possibly know what the Earth weighs? I mean, it’s not like they can put it on a scale as you would a sack of apples. — IN, Cambridge, England

**Answer:** The term *zettatons* (Zt) must refer to *metric tons* (t), where 1 t = 1000 kg. The metric prefix *zetta* = ×10^{21}, so they’re saying Earth weighs 6×10^{21} metric tons. It’s actually 5.972×10^{21} tons (to four significant figures), but they chose to round it up to 6. FYI, the metric ton is not a *base unit* in the metric system. Like the *radian*, *watt*, and *joule*, it’s one of many derived units in common use.

Also worth noting: Astronomers prefer to speak of the *mass* of astronomical bodies — not their *weight*. The two terms are used informally as synonyms, but their meanings are very different for scientists. If you want to get the technical explanation, go here. Suffice it to say that “weight” is a force that depends on the strength of gravity (and hence location), whereas “mass” is a constant that measures the amount of matter in a given body. So saying “Earth *weighs* 6 zetatons” is mixing terminology because the metric ton is a measure of mass.

Scientists use the International System of Units, (aka SI or “metric system”) wherein the base unit of force is newtons (N). The base unit of mass is kilograms (kg), though many variants (gram, milligram, zettagram) are also used. Thus, Earth’s mass is 5.972×0^{24} kg — or if you prefer metric prefixes, 5972 Yg (yottagrams). Non-standard notation must be used for the multiplier since there is currently no metric prefix higher than “yotta.”

In the British Imperial System weight is measured in units of ounces, pounds, and *imperial tons*, where 1 ton = 2000 pounds. Mass is measured in the obscure unit of *slugs*, where 1 slug = 14.5939 kg.

So we’ll just use 5.972×0^{24} kg for the mass of the Earth to answer your second question: How do we measure that number? There are two methods possible. The first method, which is more intuitive, is shown in the graphic. We simply measure the *weight* of an object with known *mass* (in this case standard kilogram **m _{2}**). From that we can calculate the value of

**m**(in this case the Earth). We’ll use the equation for gravitational force discovered by Sir Isaac Newton:

_{1}F_{g} = Gm_{1}m_{2}/r^{2} where:

**F _{g}** = force from gravity (in N)

**m**and

_{1}**m**= masses of the attracting bodies (in kg)

_{2}**r**= distance between the centers of the attracting bodies

**G**= the universal gravitational constant = 6.67408×10

^{-11}m

^{3}/kg•s

^{2}

You can see that finding the mass of the Earth involves solving that equation for **m _{1}**. Problem was, when Newton first published this equation in 1687 there was no way to measure

**G**(the proportionality constant). Doing so would require measuring the gravitational attraction between two standard masses in the lab, where the force of gravity would be on the order of 10

^{-10}N. Forces that small were well beyond the range of 1687 measuring instruments. It wasn’t until 1798, when Henry Cavendish first measured

**G**using a torsion balance, that calculation of Earth’s mass became possible.

The second (and less intuitive) method for finding the Earth’s mass requires combining Newton’s gravity equation with his equation for centripetal force: F_{c} = m(4π^{2}R/T^{2}). This allows you to calculate the mass of an orbited body from the parameters of any orbiting body. To find Earth’s mass we can use the Moon (or any artificial satellite). You can see how that calculation works in my Sep 28, 2015 post.

Next Week in Sky Lights ⇒ Images of Earth with Moon