Finding Mass Without a Scale

What is Mass?

The large asteroid Vesta, as photographed by the Dawn probe. Click for a much larger high-resolution version.

When I teach any introductory-level course, I always ask my students to list the important fundamental properties an object can have — properties that are unchanging under ordinary conditions. They always remember to list mass, though they don’t always remember what mass is. That’s perfectly understandable: mass is one of those fundamental concepts that’s a little slippery, not least because the ways we have to measure mass are often indirect. Here’s a 2-fold definition that works pretty well for the purposes of this post:

  1. Mass is the property of an object that measures how hard it is to change its velocity (i.e., start it moving if it’s at rest, change its direction of motion, etc.); the more it resists change, the greater its mass. Photons — particles of light — are massless because they move at the same speed all the time. Gravity still affects them, but that’s another story.
  2. Mass also dictates the strength of gravitational attraction: a larger mass will have a larger gravitational influence on other objects. (That this is the same mass as in definition 1 is called the equivalence principle, and is the starting point for Einstein’s general theory of relativity. The equivalence principle definitely deserves its own post at some point.)

We’re spoiled in our daily lives, in an odd way: we can stand on an inexpensive scale and determine our mass in kilograms. Sure, it’s not really mass that’s being measured — the scale is measuring how much we compress a spring (or strain gauge in the case of an electronic scale), which has a simple correspondence to mass. If you have any question about why the scale isn’t actually measuring mass, just locate an elevator, stand on the scale, and — taking care to ignore the weird looks you get — watch what happens to the reading as the elevator moves. Your mass isn’t changing, but the force you exert on the scale is. At the extreme end of this kind of behavior, you get free-fall, where your weight is zero even though gravity is still acting on you.

Even so, most of us aren’t going to be reading our weight on an elevator, so the scale you read in your bathroom is a good proxy for mass. (Kilograms are the standard for mass in the international system of units, but pounds are a unit of force. Mass in so-called “English” units is measured in slugs.) However, things get a lot more complicated when you can’t put an object on a scale — when the object in question is either too big or too small. In both of these cases, though, we have ways. Oh yes…we have our ways.

Big Stuff: Using Gravity to Measure Mass

Johannes Kepler led the way in understanding planetary motion as well as in astounding facial hair.

I had been intending to write a post on this subject for a while, but the specific impetus to write it today is the arrival of the Dawn probe at the asteroid Vesta. At present, Dawn is in orbit around the asteroid, but at a large altitude for safety — Vesta’s mass is not currently known very precisely and without that, the gravitational strength is unknown. By the very act of orbiting, Dawn will be able to measure the mass of Vesta, which in turn will tell us a lot about its composition.

The method for measuring mass through orbiting is also how we know the mass of Earth (through the orbit of the Moon), the mass of the Sun (through all the planets), the mass of Jupiter (through its moons), and also the mass of many stars (which are frequently in binary systems). The seed of this technique goes all the way back to Johannes Kepler, the 17th century astronomer who formulated three laws of planetary motion. We only need his third law, which in combination with Isaac Newton’s law of gravity yields a very simple relationship between the average distance between a satellite and the object it’s orbiting (usually labeled a), the length of time an orbit takes (labeled P), and the mass of the object being orbited (M).

A simple orbit to illustrate Kepler's third law. Images are adapted from http://dawn.jpl.nasa.gov/

In equation form, Kepler’s third law is which isn’t that hard to understand, even if your math allergies are strong:

  1. The larger the mass of the object being orbited, the less time it will take a satellite to complete an orbit of a certain size;
  2. If two satellites are orbiting the same object at different distances, the satellite that is farther away will take more time to complete its orbit;
  3. If you can measure the size of an orbit and the time to complete an orbit, you have the mass of the object being orbited!

In the case of most planets or other objects with natural satellites, Kepler’s third law is the best means we have of determining mass. In the case of moonless Mercury and Venus, the first truly accurate mass measurements were made by robotic probes, which played the role of artificial satellites; the Dawn mission will perform the same measurement for Vesta as it orbits the asteroid over the next year, then repeat the process for the largest asteroid, Ceres (also considered a dwarf planet, along with Pluto).

To summarize the story so far: without a direct way to take the mass of astronomical objects like planets, asteroids, and so forth, we rely on a detailed understanding of satellite motion to find the mass from motion. Keep that idea in mind as we turn our attention to….

Small Stuff: Using Magnetic Fields to Measure Mass

Gravity is the force of nature that holds the Solar System together, and keeps moons orbiting around their host body. On microscopic scales, other forces dominate, notably the electromagnetic force, which is responsible for holding atoms together. Mass is still going to play a role in resisting change of motion (definition 1 from above), but there won’t be a set of Kepler’s laws to guide us.

Electron orbit in a magnetic field. The orbit is circular -- we're seeing the setup at an angle for clarity. The arrow indicate direction of revolution; protons would orbit the opposite way.

Instead, let’s look at how an electrically-charged particle behaves in a magnetic field. The figure shows a schematic view of a large magnet, and the motion of an electron within that field: the electron follows a circular orbit! The diameter of the orbit depends on how strong the magnetic field is…and the mass of the electron. If you put a proton into this setup, you will also get a circular orbit, but because the proton is much more massive than an electron, it will have a larger orbit for the same magnetic field, since it’s that much harder to make it change its path of motion. It will also orbit in the opposite direction, since it’s a positive charge, as opposed to the negatively-charged electron, which is a simple way to distinguish positive from negative.

(Of course, you need an experiment to measure the electric charge independently of mass, but such things do exist. You may even have performed a classic version in high school or college: the Millikan oil-drop experiment.)

What I’ve described here is just a skeleton experiment; realistic experiments (carrying names like mass spectrometers and bubble chambers) necessarily have more detailed procedures to get everything right, just as I glossed over exactly how space probes measure distance and time. High-energy particle experiments have other ways of measuring mass as well, but things can be complicated if a particle is neutral — as with neutrinos, whose mass we still haven’t determined except to say that it’s much smaller than any other measured particle mass.

Farther Afield

Because mass can’t be determined directly, it’s a difficult physical properties to measure, no matter how fundamental it is. To make it worse, things can get tricky when interactions between objects are strong. The mass of a proton inside a nucleus is not the same as its mass when it is free, for example — part of the proton’s mass gets changed into energy (using Einstein’s famous E = m c2 equation) that is used to bind the nucleus together. Another challenge is that the Standard Model, the most widely-accepted theory for particles and interactions, has no way to predict the masses of elementary particles from first principles, so we don’t have a theoretical prediction with which to compare our experimental results. That’s all a subject for another day!

Mass dictates the evolutionary path of a star or a black hole on the astronomical scale, and relates to a lot of the interestingly strange quantum properties on the smallest size scale. Two of the most important questions anyone can ask of a scientist is “how do we know? how can we measure?” Think on that as you watch the news of the Dawn probe and the Large Hadron Collider.

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21 Responses to “Finding Mass Without a Scale”


  1. 1 Moises C.D. Marcon Rosado September 4, 2011 at 09:41

    Awesome read to start the day, thanks for it. I have a simpleton`s question, `p` in Kepler`s third law of motion is regularly meassured in seconds, am I wrong?

    • 2 Matthew R. Francis September 4, 2011 at 20:36

      Thanks for the question! What units you use in Kepler’s 3rd law depends on which form you are using. The form I teach in introductory astronomy is P2 = a3/M, where P is in years, a is in astronomical units, and M is mass in units of the Sun’s mass.

      If you want to derive Kepler’s 3rd law from Newton’s law of gravity, you will get a version with P in seconds, but you’ll pick up a few extra factors like Newton’s constant G and (inevitably) pi.

      Does this make sense?

  2. 3 Moises C.D. Marcon Rosado September 6, 2011 at 11:33

    Yes it does, since we are talking -in Kepler`s 3rd law- about the translation of Earth and other heavenly-bodies around the Sun it makes complete sense that it `P` is measured in years, which is the actual time of one full rotation around the Sun. I missed that one by lots! Hehe, now… since you touch the subject, how come one can derive Kepler`s principles for his third law of motion from Newtons equations, I mean… was it this way how Kepler himself arrived at his conclusions or did he, after making his conclusions, eventually saw fit to prove his theory by applying them to Newton`s equations?

    I apologize if my inquiry seems a tad squared… And also, would corroborate, G stands for gravity, right? Is that it? Or is it something completely different? If it is, would you be so kind as to explain to me this constant? Or share a link with me that does? Thanks.

    Pd,
    Your blog is great, keep up the awesome job of divulging the facts.

    • 4 Matthew R. Francis September 6, 2011 at 19:23

      Kepler died before Newton was born, actually. Kepler derived his three laws from careful astronomical observations and mathematical analysis. One of the early triumphs of Isaac Newton’s physics was that he could obtain Kepler’s laws from his newly-formulated law of gravity and the laws of motion (which bear his name, though a lot of credit is also owed to Galileo). In fact, there was a great debate in Newton’s day about whether planetary orbits were circular (Galileo’s view, following Copernicus) or elliptical (following Kepler), so Newton showing that his law of gravitation led directly to elliptical orbits was a vindication of Kepler. (And yes, G is for gravitation: it’s known as Newton’s gravitational constant, and its value is a measure of the strength of gravity.)

      So, here’s the chronology in brief: Copernicus proposed a Sun-centered Solar System in the early 16th century, with circular planetary orbits. Galileo endorsed the Copernican view when he wrote his classic works in the 17th century, but was in communication with his contemporary Kepler, who showed that elliptical orbits fit the data better and also could predict the size of an orbit based on how long it took. Newton was born the year Galileo died, and assembled the work of Kepler, Galileo, Descartes, and others into a coherent whole, showing how physics and astronomy were related subjects.

      I’m glad you’re enjoying the blog!

  3. 5 Moises C.D. Marcon Rosado September 7, 2011 at 11:06

    Man, you are the best, thanks for this thorough explanation of how these ideas came to fruition. I am grateful for your honesty and dedication. I salute your good-hearted and deeply intelligent mind.


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