Introduction

Having run out of precious metals to babble about, I’m going to change tacks. If you’ve been here a while, you might remember two postings I did on stars. These were independent posts, having nothing to do with politics (poly = many, ticks = blood sucking bugs) and at least some people enjoyed them. I wanted to go to the opposite end of the scale and talk about a certain sub-atomic particle, but then I realized that the best way to do that would be a very, very, very long post. (And yes, it’s a subatomic particle, but it has a lot to do with stars.) A huge part of it would be explaining where physics stood in 1895, and how four discoveries in the next four years basically overturned things, and eventually led to that subatomic particle, the real star (ahem) of the whole series.

So I decided to break this story up into pieces. And this is the first of those pieces.

And here is the caveat: I will be explaining, at first, what the scientific consensus was in 1895. So much of what I have to say is out of date, and I know it… but going past it would be a spoiler. So I’d appreciate not being “corrected” in the comments when I say things like “mass is conserved.” I know that that isn’t considered true any more, but the point is in 1895 we didn’t know that. I will get there in due time. (On the other hand, if I do misrepresent the state of understanding as it was in 1895, I do want to know it.)

Also, to avoid getting bogged down in Spockian numbers specified to nine decimal places, I’m going to round a lot of things off. I use 9.8, below, for a number that’s actually closer to 9.80665, for instance, similarly for the number 32.

OK so without further ado, mass.

What Mass Is, and Isn’t

Mass is not the same thing as weight, at least not when scientists are using the terms.

But the distinction between mass and weight didn’t become clear until Isaac Newton came along. He tried to imagine what objects would behave like if there were no gravity, and no friction. And he realized that an object under those circumstances would stay at rest unless a force acted on it, or, if it were already moving, it would continue to move in the same direction at the same speed, unless a force acted on it. It has inertia. But that doesn’t match what we actually see. You let go of an object that’s not on the ground, it falls (or if it’s a balloon, it might rise). An object that’s moving on a flat surface slows down and stops.

But the reason why we see things fall faster and faster, is that gravity exerts a force on them, a downward force, and the reason things moving horizontally slow down is friction, and that, too is a force.

For clarity, we have to ignore friction. Imagine these objects are wet ice on a hockey rink—or cars sliding on ice (yikes!). Or air hockey pucks. There’s still some friction in all of these cases, but not a whole lot. You can imagine, after watching these sorts of things, what it would be like with no friction.

Largely building upon what Galileo discovered (when he wasn’t looking through a telescope), Newton essentially defined the concept of inertia. It’s basically the resistance of an object to being shoved. And mass is essentially a measure of that. If object A is twice as massive as object B, it’s twice as hard to shove around and get the same effect. You need to exert twice as much force.

On the other hand, if you stick with the same object, and apply twice as much force, it reacts twice as much.

You can state this a little more precisely as, acceleration (a) is proportional to the force (F) and inversely proportional to the mass (m). You can increase or decrease any of the three items, decrease or increase one of the other two, and you will see the third item increase or decrease exactly in proportion.

Or to be even more concise, you can write the following:

a ∝ F/m.

That little Jesus fish-like thing means “is proportional to” and basically, it means that if you double one side, you double the other side, but they’re not equal.

You can rearrange to get:

F ∝ ma.

And this is the form you usually see this in, it’s Newton’s second law of motion. Well almost. There’s something we can do to get rid of that Jesus fishy thing and replace it with an equals sign. More on that shortly.

Mass is considered to be the ultimate measure of “how much matter” is in an object. Twice as much matter, will have twice as much inertia. And some object, say one of the big weights off of a weightlifting set, will have the same inertia even on the moon.

But the weight of the objects will change on the Moon, because weight is actually force. And the Moon’s gravity will pull on the same objects, with less force than on Earth.

The kilogram is actually a unit of mass. A chunk of metal massing a kilogram (think of it as a one kilogram gold bar if that will put a smile on your face) will still mass a kilogram on the moon, pick it up and swing it around, you will feel the same tugs as you would feel on earth, because now you are playing with its inertia, which doesn’t change—it will take the same force to keep the bar from flying out of your hand as it did on Earth.

But pounds are (usually) a unit of force. That kilogram of gold will weigh about 2.2 pounds here on Earth. That means that the earth’s gravity pulls on it with that amount of force. But take it to the moon, and it weighs about 5.9 ounces, that’s how much force the moon exerts on that one kilogram mass.

[Or—let’s be frank here—for me, and for most of you, it weighs exactly zero both on Earth and the Moon because your kilogram bar of gold doesn’t exist at all except in your dreams. Oh, OK, never mind, forget I said this and return to smiling.]

The one force we can’t get away from in our daily is gravity, and as such in the English system a lot of things are defined with respect to the amount of gravity Earth has. But when you’re doing engineering you have to deal with a lot of forces—the force exerted by a pile driver, the thrust of a jet engine, and so on. So you need a unit of force and a unit of mass, so you can figure out how much your masses will respond to your forces, or alternatively, how much force you’ll have to exert to make that mass move the way you want it to.

The metric system, which starts with unit of mass, has to derive a unit of force, and the English system which starts with a unit of force, has to derive a unit of mass. Metric invented a unit of force called the Newton—the amount of force needed to accelerate a kilogram, one meter per second, for every second it’s applied, and yes, it’s named after Sir Isaac. And the English system retro-invented the slug—it’s the amount of mass that, when acted upon by one pound of force, will accelerate one foot per second, every second.

Once you’ve defined your units, you can change that proportionality constant to an equals sign. But you may need a fudge factor, which I will call k. F = kma. (And yes, Biden can kma.) In metric, as long as you stick to meters, kilograms, and meters per second squared, the fudge factor is 1. The Newton was deliberately defined that way. And in the English system, as long as you stick to feet, slugs, and feet per second squared, the fudge factor is also 1. 1 will disappear if it’s in an algebraic multiplication, so now we’re dealing with

F = ma

Drop a kilogram of gold—it will accelerate at 9.8 meters per second per second, a = 9.8, and m is one. Plug it into the formula above. That means it’s being acted on by a force of 9.8 newtons.

Galileo showed that heavier objects fall at exactly the same rate as lighter ones (once you account for air resistance, which is a force and partially cancels out gravity). So a two kilogram mass of gold, falling at the same rate, which we call g, gives you F = 2 x 9.8 = 19.6 Newtons.

Switch to the English system now. Drop a pound of something else—bananas, say—it will accelerate at 32 feet per second, every second. (That’s the English equivalent of 9.8 meters per second per second… we’re just using a different measuring stick.) But this time we have a weight, not a mass, F, and we have a and are looking for m, so we need to do a bit of beginner’s algebra and come up with:

m = F/a

But this time F is 1 pound, and a is 32 feet per second squared. So our mass is 1/32. And indeed, if our answer is supposed to be in slugs, that’s the right answer. A slug, as it turns out, weighs 32 pounds here on earth, and 1/32 of a slug weighs one pound.

Engineers find it so useful to have a unit of mass, the ones working in the English system (poor sods) actually invented a “pound mass,” the mass of something that weighs a pound here on earth. But when they use the “pound mass” in their formulas they have to put a fudge factor of 32 in. With mass in pound mass, the formula becomes

m = 32 F/a

Or rearranging to the usual form

F = ma / 32

A pound mass will respond to a force 32 times as much as a slug would to the same force, as you can see when you solve for the acceleration (response), a = 32 Fm. Failure to properly account for this has doomed more than one rocket. Metric is cleaner, a kilogram is mass, and only mass.

I got my STEM education entirely in metric, that’s what I’m comfortable with, that’s what I’m going to use from here on out.

OK, so mass and weight are different. How do you measure them? If they are different things, you need different methods to measure them.

A force can be measured with a spring. Your typical bathroom scale, or your kitchen scale, will have a spring inside; the amount the spring is compressed by the stuff you put on the scale is a measure of the force exerted on the spring. Drag your kilogram of gold to the moon and bring your scale with it, it will push on the spring less and therefore weigh less.

A mass is best measured by a balance scale. Your doctor’s scale, for instance, is a balance scale. Take it to the moon, and if it read 100 kg on earth, it will read 100 kg on the moon.

But it’s probably marked off in pounds. If you weighed a hundred pounds on earth… that scale will read 100 pounds on the moon. It’s actually measuring your mass but is calibrated in pounds actually pounds mass. So it only looks like it’s measuring your weight.

A kilogram (mass) weighs 2.2 pounds on earth, the object that exerts a force of 2.2 pounds on earth, has a mass of one kilogram. That 2.2 will change on every different planet, however, since most of us never leave earth, we simply think of a kilogram as equaling or being the same as 2.2 pounds, when it really isn’t. It isn’t the same thing, any more than a gallon is the same as 8.33 pounds. (It weighs that much if it’s water and we’re on earth, but that doesn’t make a gallon the same thing as 8.33 pounds in any fundamental sense.)

With me still? I hope so.

Conservation of Mass

OK, I’ve shown you Newton’s Second Law by way of introducing you to the distinction between weight and mass. It’s called a “Law” not because some politician decreed it, but because the universe works this way.

F = ma

Always.

And when you have to use a fudge factor, the fudge factor is a constant. It’s a constant 1 in the case of metric and English slugs.

One could ask what would happen if the fudge factor were to change, and the answer is a bit surprising. Since the Newton is define as being the force necessary to accelerate a kilogram at one meter per second per second, if it suddenly, tomorrow, took twice as much force to do that, the Newton would simply get twice as big.

Since it’s awkward having your units of measure change (for exactly the same reason that inflation sucks), the fact that scientists set things up that way should be evidence enough that they are sure the fudge factor never changes.

How do we know that? It is an induction, not a deduction. It has always been true, we assume it’s just the way the universe works, until we find an exception, and believe me, people are always looking for the exception. And also for just outright blatant violation of the rule; such as objects suddenly and inexplicably changing their velocity. That would be an a without an F, or perhaps the m taking a vacation and reducing to almost nothing momentarily.

Our whole view of the universe wouldn’t make sense if Newton’s second law weren’t true. Car collisions on icy roads, air hockey… if someone made an animated movie where this rule were blatantly violated, it would look fake to us. Of course if the movie were almost spot on, with maybe the fudge factor changing by 1 percent at random, we’d have a hard time seeing it, but this has been measured in laboratories, and it’s always true. And the fudge factor doesn’t change.

One other thing turns out to be true, at least as of 1895. Mass never disappears into nothing, and it never appears from nowhere, either.

Sure, the mass of an object can change. It could decrease. But that mass always goes somewhere else, it never gets destroyed. Or similarly, if the mass of an object increases, that mass came from somewhere else.

This is one of those realizations that turned science into a form of bookkeeping. The books have to balance, the mass in has to equal the mass out.

Set a two kilogram log on fire. Weigh the ashes afterwards, they mass out to maybe 600 grams. Did the other 1400 grams just disappear? Nope. It can’t, it’s not allowed to. So that tells the scientists they didn’t account for something. In this case, they didn’t capture and weigh the smoke and the carbon dioxide given off by the burning log (whilst upsetting leftists). So if you add that in, are you okay?

Nope, because now the mass after is more than the mass before. That’s not allowed either; you can go back to the bench and re-run the experiment.

This time, count not just the wood, but also the oxygen used to burn the wood.

Once you do that, your mass before matches your mass after. Life is good.

The books balance.

And this was another thing that (as of 1895) was considered to be always true. Every time it had been tested, it was true. And a test isn’t just an explicit lab exercise like I just described, but literally everything done in a lab implicitly follows this rule.

In this case simple arithmetic is enough to do the bookkeeping. And nothing of negative mass has ever been seen, so you will see addition and subtraction, but never a negative result. Real accountants would find this dead easy. The trick, of course, is to account for everything, and measure carefully.

Gravity

There is one other thing about mass, though, that was (and still is) an important feature of the universe. And that is gravitation. I’ve mostly talked about gravity as something that exerts a force on a mass, so far, but I’ve not mentioned yet that mass actually exerts gravity. Every mass, exerts a force on every other mass. If you double the mass of the object, it exerts twice as much force. If you double the distance between the objects, however, you divide the force by 4. This is the square of the distance, and you’re dividing by it, so it’s called an inverse square relation. But one more thing. If you double the mass of the other object, the force you exert on it doubles too, and it responds with the same acceleration.

You are exerting a tiny gravitational force on the Andromeda galaxy. And vice versa. In fact, it’s the same amount of force in both directions.

The law of gravitation was also first noted by Sir Isaac Newton.

You can write this law as follows, at least as a first cut.

F = m₁ m₂ / d²

Multiply your masses together, divide by the square of the distances, and you get F.

Except, no you can’t. Meters, kilograms, seconds, and Newtons go together with F=ma, but they don’t go together in this equation. Two masses 1 kilogram each, at one meter’s distance? That equation says the force should be one Newton. It’s not. It’s a lot less. You need to plug in a fudge factor, and this one is named G. The law of gravitation properly reads

F = G m₁ m₂ / d²

And G is a very small number, because in Newton’s day you couldn’t even measure what F was. Without being able to measure F, we couldn’t figure out what G was. And for any scenario where we could measure F, we either couldn’t measure one of the masses, or d, or both. Either way, G was unmeasurable.

But even without that, Newton could see the law was good, because he could check the responses of things to earth’s gravity. An apple, and the moon.

To start this out, in fact, let’s assume we want to measure the acceleration, not the force. Both sides of the equation above are a force, dividing by the mass of one of the objects, the one we want to watch, gives:

a = G m₂ / d²

And you can substitute mass of the earth, m_e, for m₂. So the acceleration of, say, an apple dropped, is:

a = G m_e / d²

And if you’re standing on the surface of the earth, d is the radius of the earth. (You can treat the earth as if its entire mass were at the center, so long as it’s radially symmetrical (which it almost is). That’s one of many things Newton proved. He had to invent calculus to do that.)

In this particular case, we knew a, and we knew d, but we knew neither G or m_e. But we knew what their product had to be! This is called the earth’s gravitational parameter, and is usually written μ_e. (Greek letter mu, usually pronounced as “mew” in English, though logically it should be “moo.”) This is very handy, in fact, it’s so handy that even today people who work with orbital motion just use gravitational parameters; it saves them the bother of multiplying the same numbers over and over again.

a = μ_e / d²

Newton had pretty good information on how far away the moon was. He could compute how much it was accelerating as it orbits the earth (always downward, as if it were on a string being whirled around the earth), and his equation and the data matched. The moon responded to Earth’s gravitation exactly the same way as the apple did. This was the first time we had ever shown that something “up there” follows the same physical laws as something “down here.” And that’s why it’s called the universal law of gravitation.

So Newton knew the earth’s gravitational parameter, and the distance to the moon was known before his time. But what about the rest of the solar system? Well, life was rough for astronomers working out the solar system back then. Because we didn’t know the actual distance between the sun and any of the planets, nor between other planets and their moons. We did know the relative distances; we knew, for instance that Jupiter’s distance to the sun was 5.2 times that of Earth’s. Kepler had figured that out in the late 1500s. We could also see that the inverse square law worked: The acceleration Jupiter experienced was about 1/27th that of earth, though we couldn’t tell what it was because we didn’t know the scale. Newton, in fact, showed mathematically that any inverse square force will cause things to orbit in ellipses, thereby vindicating and strengthening Kepler, and using Kepler as evidence that an inverse square law was involved.

Measuring the Solar System, Massing the Earth

So astronomers didn’t know the gravitational parameter of the sun, much less G and the mass of the sun. And they didn’t know d, in this context the distance from anything to the sun, much less the earth-sun difference (but they named it: It’s an astronomical unit, and is still called that to this day). But if they could figure out what d was for Earth, they’d know it for everything else in the solar system, because we knew the proportions. And if they knew d, they could figure out the Sun’s gravitational parameter, because you can figure out the acceleration directly from d and the length of the year.

Astronomers got the first intimations of d when we were able to triangulate on Venus as it crossed between earth and the Sun, in 1761 and 1769. We could plot its motion and position on the sun’s disk from multiple places on earth, see how different it was, and determine how far away it was, just like when you move your head from side to side, a near object will move against the horizon more than a distant one will. If you measure that apparent shift, and know how much you moved your head, you can compute how far away it is. Multiple parties went to different places on earth, just to measure these transits, the more measurements the better. It was one of the first major examples of international scientific cooperation.

Similarly we now had a good number for the distance to Venus, and because we knew the proportions of the solar system, we instantly had the distance from earth to the sun. We therefore knew a, and… now we had the gravitational parameter of the sun, because it was equal to a times d².

The gravitational parameter of the sun is 333,000 times the amount of the gravitational parameter of the earth. Since both numbers are G times the mass, you can see that the sun’s mass, whatever it is, must be 333,000 times that of Earth. Whatever that is. We still didn’t know.

But even better. We now knew the exact distance to Jupiter. And we could therefore watch how far the moons of Jupiter got from it in the night sky, do a quick calculation and get their orbit sizes… and now we could figure out the gravitational parameter of Jupiter! It is 317.8 times that of the earth, so its mass is also 317.8 times ours (again, whatever that is). Saturn has moons, and we could do the same thing for it. Mars has moons too, but they hadn’t been discovered yet.

Then in 1781 the planet Uranus was discovered. And in 1787, two moons were discovered. Shazaam! We knew how many earth masses Uranus was, and six years earlier we hadn’t even known Uranus existed. And we eventually figured it out for Mars, when we finally did discover its moons a century later.

But Venus and Mercury don’t have moons, and we could only make educated guess at their masses… until the 1960s and 1970s when we sent probes to them and could see how they interacted with those planets.

One last piece of the puzzle. Henry Cavendish (1731–1810) was actually able to measure the force of gravity between two heavy lead balls in his laboratory in 1798. This was painstaking work, but he now had a situation where he knew every term in the law of gravity except for G; he had the force, the distance, and the masses. So from that he was able to compute the value of G, and (in metric) it is:
6.67 x 10^-11.

Now that we had that number, we could go back to every gravitational parameter we knew, divide by G and get the masses.

Now we knew the mass of the Earth. It’s 5.972×10²⁴ kilograms. And everything else proportionately.

We did this, but we did not have to go “out there” and weigh anything.

Problems with the Law of Gravitation?

The law of gravitation worked really, really well. We never saw anything inconsistent with it… well, almost!

When we tracked Uranus in its orbit about the sun, it was clear it was not following the law, not quite. Was something wrong with the law of gravitation? The law was so useful everywhere else, and I mean everywhere else, that it would make no sense for it to be broken here, so instead of assuming the law was broken, we figured that there was something unknown out there, pulling on Uranus. It was complicated work, but Le Verrier in France and John Couch Adams in England both did the calculations in 1845, and when they told Galle, another astronomer, one who used a telescope rather than being a theorist, where to look… well, Galle found the planet Neptune almost immediately.

Far from it being a problem for the theory of gravitation, the discrepancy with Uranus’ orbital motion turned into a triumph, for the theory of gravity had been used to discover a planet, and had predicted it so well that it took someone who knew where to look less than an hour to find it.

That gives you a really strong feeling that this is the truth!

But I mentioned two problems. What was the other one?

The other was the orbit of Mercury. It’s an elliptical orbit, and if Mercury and the Sun were alone in the universe, that ellipse would never, ever move. But it does move, the long axis shifts 574 arc seconds every century. And of course Mercury and the Sun aren’t alone in the universe. So what we should be able to see is that the planets—and the sun’s slight oblateness—explain Mercury’s orbit precessing.

But when you add up all those effects, there’s still a discrepancy. They don’t add up. There’s still 43 arcsecond per century left over. And this bothered scientists.

But really, this probably isn’t a problem. We know what the answer has to be. There’s an unknown planet pulling on Mercury, one so close to the sun we just couldn’t see it in all the glare. The same Le Verrier that predicted Neptune predicted this planet. We even gave it a name, Vulcan; a perfect name because he was the Roman god of the forge and it gets hot near forges and near the Sun. But despite what you may have heard, scientists usually want to square things away, they want to see that planet, then they’ll be confident they know why Mercury is misbehaving.

But Vulcan was never found, and in 1895, it remained an open question. They expected to find it, they just hadn’t, yet. In truth, it really would be hard to see something like that; it can only be done during solar eclipses.

Conclusion

Well, this turned out to be pretty long. And maybe hard to follow (I hope not). But as I wrote it I realized how much “hung off” the concept of mass, and the law of gravitation, and how much we were able to learn about the solar system and our own Earth, each bit of knowledge building on the prior, with theory used as a framework. And you saw some limitations… we could only estimate the mass of bodies with no moons. This wasn’t even where I wanted to go with this, but it was too good to pass up. (Next week we continue towards our final destination.)

But hopefully you saw some notion of how science is supposed to function. It’s full of humans with their own foibles, of course, but in the end the truth does out. It’s nice when you can use a theory to predict something unexpected; it gives you a very warm fuzzy sense that the theory is correct. But at the same time, there are implicit assumptions; that the generalizations we see will continue to hold true. Sometimes we discover otherwise, and have to adjust; usually when that happens it turns out that the generalization was true under certain circumstances and is still useful, under those circumstances, but that you have to scrap it under others. (At the risk of a spoiler, you’ll see that Newtonian gravity is one of those cases.)

And in so many cases, if we seem to see far, it is because we stand on the shoulders of giants, the men who preceded us, and they stand on the shoulders of the men who preceded them. None of this could happen if we weren’t willing to use information gathered by others and build on it, and in turn that’s a testament to the power of being able to write things down so that knowledge outlives us.