###### or why Vera Rubin and Albert Bosma deserve a Nobel Prize

Natural Law: a concise statement describing some aspect of Nature.

In the sciences, we teach about Natural Law all the time. We take them for granted. But we rarely stop and think what we mean by the term.

Usually Natural Laws are items of textbook knowledge. A shorthand all in a particular field known and agree to. This also brings with it an air of ancient authority, which has a flip side. The implicit operating assumption is that there are no Natural Laws left to be discovered, which implies that it is dodgy to even discuss such a thing.

The definition offered above is adopted, in paraphrase, from a report of the National Academy which I can no longer track down. Links come and go. The one I have in mind focussed on biological evolution. To me, as a physical scientist, it seems a rather soft definition. One would like it to be quantitative, no?

Lets consider a known example: Kepler’s Laws of planetary motion. Everyone who teaches introductory astronomy teaches these, and in most cases refers to them as Laws of Nature without a further thought. Which is to say, virtually everyone agrees that Kepler’s Laws are valid examples of Natural Law in a physical science. Indeed, this sells them rather short given their importance in the Scientific Revolution.

Kepler’s Three Laws of Planetary Motion:

1. Planetary orbits are elliptical in shape with the sun at one focus.
2. A line connecting a planet with the sun sweeps out equal areas in equal times.
3. P2 = a3

In the third law, P is the sidereal period of a planet’s orbit measured in years, and a is the semi-major axis of the ellipse measured in Astronomical Units. This is a natural system of units for an observer living on Earth. One does not need to know the precise dimensions of the solar system: the earth-sun separation provides the ruler.

To me, the third law is the most profound, leading as it does to Newton’s Universal Law of Gravity. At the time, however, the first law was the most profound. The philosophical prejudice/theoretical presumption (still embedded in the work of Copernicus) was that the heavens should be perfect. The circle was a perfect shape, ergo the motions of the heavenly bodies should be circular. Note the should be. We often get in trouble when we tell Nature how things Should Be.

By abandoning purely circular motion, Kepler was repudiating thousands of years of astronomical thought, tradition, and presumption. To imagine heavenly bodies following elliptical orbits that are almost but not quite circular must have seemed to sully the heavens themselves. In retrospect, we would say the opposite. The circle is merely a special case of a more general set of possibilities. From the aesthetics of modern physics, this is more beautiful than insisting that everything be perfectly round.

It is interesting what Kepler himself said about Tycho Brahe’s observations of the position of Mars that led him to his First Law. Mars was simply not in the right place for a circular orbit. It was close, which is why the accuracy of Tycho’s work was required to notice it. Even then, it was such a small effect that it must have been tempting to ignore.

If I had believed that we could ignore these eight minutes [of arc], I would have patched up my hypothesis accordingly. But, since it is not permissible to ignore, those eight minutes pointed the road to a complete reformation in astronomy.

This sort of thing happens all the time in astronomy, right up to and including the present day. Which are the important observations? What details can be ignored? Which are misleading and should be ignored? The latter can and does happen, and it is an important part of professional training to learn to judge which is which. (I mention this because this skill is palpably fading in the era of limited access to telescopes but easy access to archival data, accelerated by the influx of carpetbaggers who lack appropriate training entirely.)

Previous to Tycho’s work, the available data were reputedly not accurate enough to confidently distinguish positions to 8 arcminutes. But Tycho’s data were good to about ±1 arcminute. Hence it was “not permissible to ignore” – a remarkable standard of intellectual honesty that many modern theorists do not meet.

I also wonder about counting the Laws, which is a psychological issue. We like things in threes. The first Law could count as two: (i) the shape of the orbit, and (ii) the location of the sun with respect to that orbit. Obviously those are linked, so it seems fair to phrase it as 3 Laws instead of 4. But when I pose this as a problem on an exam, it is worth 4 points: students must know both (i) and (ii), and often leave out (ii).

The second Law sounds odd to modern ears. This is Kepler trying to come to grips with the conservation of angular momentum – a Conservation Law that wasn’t yet formally appreciated. Nowadays one might write J = VR = constant and be done with it.

The way the first two laws are phrased is qualitative. They satisfy the definition given at the outset. But this phrasing conceals a quantitative basis. One can write the equation for an ellipse, and must for any practical application of the first law. One could write the second law dA/dt = constant or rephrase it in terms of angular momentum. So these do meet the higher standard expected in physical science of being quantitative.

The third law is straight-up quantitative. Even the written version is just a long-winded way of saying the equation. So Kepler’s Laws are not just a qualitative description inherited in an awkward form from ancient times. They do in fact quantify an important aspect of Nature.

What about modern examples? Are there Laws of Nature still to be discovered?

I have worked on rotation curves for over two decades now. For most of that time, it never occurred to me to ask this question. But I did have the experience of asking for telescope time to pursue how far out rotation curves remained flat. This was, I thought, an exciting topic, especially for low surface brightness galaxies, which seemed to extend much further out into their dark matter halos than bright spirals. Perhaps we’d see evidence for the edge of the halo, which must presumably come sometime.

TACs (Telescope Allocation Committees) did not share my enthusiasm. Already by the mid-90s it was so well established that rotation curves were flat that it was deemed pointless to pursue further. We had never seen any credible hint of a downturn in V(R), no matter how far out we chased it, so why look still harder? As one reviewer put it, “Is this project just going to produce another boring rotation curve?”

Implicit in this statement is that we had established a new law of nature:

The rotation curves of disk galaxies become approximately flat at large radii, a condition that persists indefinitely.

This is quantitative: V(R) ≈ constant for R → ∞. Two caveats: (1) I do mean approximately – the slope dV/dR of the outer parts of rotation curves is not exactly zero point zero zero zero. (2) We of course do not trace rotation curves to infinity, which is why I say indefinitely. (Does anyone know a mathematical symbol for that?)

Note that it is not adequate to simply say that the rotation curves of galaxies are non-Keplerian (V ∼ 1/√R). They really do stay pretty nearly flat for a very long ways. In SPARC we see that the outer rotation velocity remains constant to within 5% in almost all cases.

Never mind whether we interpret flat rotation curves to mean that there is dark matter or modified gravity or whatever other hypothesis we care to imagine. It had become conventional to refer to the asymptotic rotation velocity as V well before I entered the field. So, as a matter of practice, we have already agreed that this is a natural law. We just haven’t paused to recognize it as such – largely because we no longer think in those terms.

Flat rotation curves have many parents. Mort Roberts was one of the first to point them out. People weren’t ready to hear it – or at least, to appreciate their importance. Vera Rubin was also early, and importantly, persistent. Flat rotation curves are widely known in large part to her efforts. Also important to the establishment of flat rotation curves was the radio work of Albert Bosma. He showed that flatness persisted indefinitely, which was essential to overcoming objections to optical-only data not clearly showing a discrepancy (see the comments of Kalnajs at IAU 100 (1983) and how they were received.)

And that, my friends, is why Vera Rubin and Albert Bosma deserve a Nobel prize. It isn’t that they “just” discovered dark matter. They identified a new Law of Nature.

## 21 thoughts on “Natural Law”

1. EuroSpin says:

This sounds like a pretty good argument. But how many discoveries of laws of nature does it take for the ball to fall on “galactic studies”, at the Nobel roulette?

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2. If the standard is “galactic studies,” then never – there is no Nobel prize for Astronomy.
If the standard is the discovery of a new law of nature that has a huge impact on physics, then this is a slam dunk.

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3. EuroSpin says:

Yes, and there is no Nobel prize for ethology either, but that didn’t prevent ethologists like Nikko Timbergen to get the Nobel under the “Physiology or Medicine” label (presumably because it was recognized that what he had found on innate behavior had an impact on both of these fields). So Nobel committees can be open-minded !

Maybe there’s also something to say about the way the discovery was made ? I suppose that if e.g. Planet 9 was observed tomorrow, there would be no doubt about who would get calls from Stockholm. What makes this more Nobel-worthy than the asymptotic flat galaxy curves?

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4. Actually, I doubt that Planet 9 would be considered Nobel worthy. If so, why not Eris? or any of the other Pluto sized objects in the Kuiper belt. What would make planet 9 more worthy? Size? How big need it be? Is it more important because we’re calling it a planet instead of a dwarf planet? Does @plutokiller get to eat planet 9 and discover it too?
A discovery that will win the Nobel prize at the first opportunity is the LIGO detection of gravitational waves. We were already pretty sure these had to exist (something carries energy away from the orbit of the binary pulsar of Hulse & Taylor – one of the few astronomical observations deemed to be Nobel worthy) so arguably LIGO is no more surprising than a planet 9. I personally find LIGO more compelling, and it is certainly more physics-y.
To come back to rotation curves, I think a better comparison is the Nobel Prize awarded for SN Ia. Widely portrayed as the discovery of dark energy, what the SN Ia data really show is that the cosmic metric differs from what GR predicts for a formerly sensible universe – one without a cosmological constant. I think that is Nobel worthy: GR cosmology fails unless we admit this mysterious anti-gravity. But rotation curves are at least as worthy, and should have been recognized decades ago.
I suspect SN would not have been recognized either if one of the awardees (Perlmutter) were not known as a physicist to the physics community. It was a purely astronomical experiment, after all. If it had been done solely by astronomers (as it mostly was), then it would not have been considered a physics result.

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5. EuroSpin says:

For the general public, and even for scientists in other disciplines, I would guess that high energy physics, condensed matter physics, astronomy and cosmology all constitute different fields in physics. I for one was completely unaware of the chasm between physics and astronomy, before hearing about it from you. Sad ! (forgive the trumpism : ). Dark matter is precisely the topic that should bridge across the divide. Effects observed in galaxies, yet particles to be discovered in the lab (if any..). But I suppose your point is precisely that astronomy should have its Nobels without reference to particle physics, hard to disagree with that.

You say: “Never mind whether we interpret flat rotation curves to mean that there is dark matter or modified gravity or whatever other hypothesis we care to imagine.”
Could you please elaborate on whether and how e.g. MOND would explain this natural law, or point to a paper?

Planet 9, if it’s here, is a big chunk of our solar system that has probably been cast away in its youth. It would be much larger than Pluto, and could have influenced the sun in important ways (i.e. solar obliquity, https://arxiv.org/abs/1607.03963). Who knows what other features of our solar system it may have helped shaped. I bet it would be considered well worth a Nobel.

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1. Ron Smith says:

I’m sure Prof. McGaugh can, and will, explain it much better than I can, but my understanding is that the a -> sqrt(aN*a0) part of Mond results in ‘flat’ rotation curves out to infinity, or at least as far as the galaxies stars and gas extend.
This leads me to my own question. Is there any theory, Mond based or otherwise, as to why galaxies have an end? In light of the flat rotation curve law there does not seem to be any internal reason for the galaxies to have an edge, they could just go on indefinitely. The only restraint I can see would be the result of finite amounts of gas/dust available when the galaxy formed.

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1. Where galaxies end isn’t always clear. The light profiles of disk galaxies fall off exponentially, so the number of stars becomes arbitrarily small at large radii (the integrated number is finite albeit large). The harder we look the more we tend to see – my colleague Chris Mihos here at Case and his collaborators have done extensive work on this. Some galaxies do seem to truncate while others just keep getting ever thinner and thinner. So thin that it hardly adds to the total mass, but there can be a wealth of structure at very faint surface brightness.

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2. MOND was motivated by the observation of flat rotation curves, and constructed to produce them (Milgrom 1983 ApJ, 270, 365). So it explains this natural law by design. In contrast, dark matter is inferred from the flat rotation exceeding the Keplerian expectation. There is nothing about dark matter that requires that rotation curves become flat. There was much discussion of this apparently fine-tuning (sometimes called the disk-halo conspiracy) in the early literature. This petered out as people got use to expecting flat rotation curves. They achieved the status of de facto natural law, so it no longer seemed an unusual conspiracy but rather part of the natural order.

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1. EuroSpin says:

I see, thanks.
On recent news, what do you think about the LUX non-result ??

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6. Context.
Mike’s comment is a good example of using the same words to mean different things. I have seen a lot of grief come of this, usually because people don’t realize they mean different things by the same words, so get really upset when the other person doesn’t agree to their oh-so-obvious use of them.
Natural Law in science has the meaning I discuss here. It has a history of being used in this context for many centuries. This is a science blog, so its use in the scientific meaning is, well, natural.
Philosophers do not have a unique claim on this phrase. How they use this term is utterly irrelevant here.

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7. Dear Stacy. I am Amarashiki, the author of the Spectrum of Riemannium blog. I have already read all your posts on flat rotation curves. They are really AMAZING. I also recalled for the data to the radial acceleration profile. I want to play with those data. If they are public, of course. Could you give a link if positive? My other concern is a proposal. I want you to summarize in format of “numbered laws” the Galactic Motion Rules, but I wonder if you could including both, Spirals AND Elliptical galaxies, and more…Would you like to sketch in for a short blog post on my page? What do you think? What I would look for is a post like the statements of the Kepler laws but referred to Galactic motion (including if possible, a naming, e.g., Vera’s law -a tribute would be welcome-, Renzo rule, or whatever, since you are the expert on all this). You can email me. You can also use LaTeX, since my blog have QuickLaTeX and so it can render LaTeX formulae with no problems.
BTW, I loved your words for Vera Rubin. I really really think her Nobel was deserved and it has been a missed opportunity to do justice. Anyway, I believe she was happy even without it…I offered recently a book as a gift to one of my High School FEMALE students, plus a book on Einstein life; also another book to one male student to correct his way (let’s see if he turns back to light or falls to the Dark Side).
I can not change the world alone but I do want to seed it to BE ABLE to change through future generations, so I am doing little contributions due to my “condition”. Would you help me? A more specialized question, to answer here, and the reason why I puzzled and was surprised when reading you first on the arxiv and now here is your formula for radial acceleration: Have you tried more functional forms? Is yours the best and simpler fit you could find? Let me add…One activity I would like to explore with my students (it entangles dynamics and computers) is to play with data and Newton laws. Galactic motion is so different to planetary motion that it shocks the minds when you see those flat rotation curves…Best regards,…JF

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