deodands

The Wow of physics: the amazing variability of the neutron star teaspoon

Neutron stars are small and very dense stars. So astonishingly, mindbogglingly, dense that they're a rather useful rhetorical device for the people who do public understanding of science. They want to persuade a grudging audience that there is a point to theoretical physics and astronomy. If neutron stars can be dense enough to make you say 'Wow', then they have made some sort of progress. That's why you'll probably have heard of the amazing mass of a teaspoon of neutron star. What this entry is about is how amazing the variability is in the subsequent estimates, and what this teaches us about physics. It uses sarcasm and SI units.

Neutron stars do really exist. Long after the protons and electrons have long given up the struggle to maintain their identity against the force of gravity, all that is left is neutrons, pressed together into one big atomic nucleus a few kilometres across. They're the pulsars that populate the radio sky with signals so regular and apparently the product of a conscious agent that their discoverers worried about how they would talk to their scientific colleagues about the existence of Little Green Men.

There are problems with the Wow approach to physics. Some people, for example, might object to representing theoretical physics as a collection of Wow things, rather than a high minded pursuit of truth. Since I don't think it's very high minded, and there are other more interesting kinds of scientific explanations of the world anyway, I don't mind that. But there is a more practical problem, common to every attempt to explain the mindboggling, which is that the mind gets boggled before it gets to the explanation.

The reasonable solution to the boggle block for the gigantic quantities of cosmology is to shift at least one of the quantities involved to a human scale. This is where the teaspoon appears: as the standard Public Understanding of Science (they really do call it PUS) unit of neutron star volume. A teaspoon is reckoned by cookbooks at 5mls=5x10-6 m-3. The maximum theoretical density of a neutron star is roughly 2x1018 kg m-3 . So a teaspoon of neutron star would have a mass of up to around 1013 kg.

1013 kg is still too boggling to fit in the mind, so the PUSsians still have some work to do. We can lose three orders of magnitude by converting to the still apparently intuitive unit of the tonne. (Apparently intuitive, because we think we know what a tonne is, but there is no conceptually graspable difference between a billlion kilograms and a billion tonnes.) A teaspoon of neutron star might have a mass of 1010 tonnes. Finally we can fit that past the boggle filter by converting it into words. A neutron teaspoon has a mass of 10 billion tonnes. (Actually, some go further and try and convert that figure into the mass of a mountain or - a prevalent recent meme - the mass of all the cars, lorries and trucks on Earth. There's a whole other investigation there, someday).

Wow. 10 billion tonnes. Give these people university departments, superconducting supercolliders, and a cameo on the Simpsons.

And yet. Try a google for 'neutron star teaspoon', a triple which yields almost no false positives. According to NASA, who after all have the most acute PUS mission of any organisation on the planet, a teaspoon of neutron star 'weighs about a billion tonnes'. Well, only a factor of 10 different to my estimate. Perhaps they use those tiny little souvenir teaspoons with an enamelled picture of the Belgian flag on the handle. Another hit, from the University of Tasmania: 500 million tonnes. I suppose it would be quite fiddly to fill the whole teaspoon. Another hit, NASA again, but they seem to have spilt some: 100 million tonnes. From the astrophysics course at Cornell (so not really PUS), a 'teaspoon of neutron star material would weigh about 10 million tonnes'. Ooh. That's a really small teaspoon. In the other direction, the Canadians say 50 billion tonnes , and my world google record so far comes (of course) in California, where the neutron teaspoon has a mass of 1 trillion tonnes . To meta-teaspoon this, you could balance one Californian's neutron teaspoon against one hundred thousand New York teaspoons. When we got married we were given six teaspoons: I would have to get married over 16 thousand times to acquire enough NY ones to balance the Californian, and given UK divorce laws would have to live more than 30000 years.
Wow. Fund that SSC after all.

(It would probably be unfair to mention the student project page at Penn State that assigns the neutron teaspoon a massive 5 tonnes, so I will. Similar sightings welcome).

So where does this teaspoon variability come from? Even the fiddliest Belgian souvenir spoon can't differ by more than a factor of ten in volume from my wedding present ones. Is there difference in the science?
Time for the science bit then, which relies on Shapiro and Tekukolsky, a graduate text from 1983 which starts with the heartening words

An understanding of polytropic stellar models or the equation of an ideal Fermi gas is likely to be useful to the student forever.

Stars are big balls of gases. Their size is determined by the balance between two opposing forces: gravity pulling the gas inwards, and pressure pushing it outwards. Just like the pressure of air in a balloon, pressure reflects the fact that it's hard to push things together. The pressure depends on how many things you're trying to push together (density), but it also depends on how hard they are to push together. At higher temperatures, the air molecules have more energy, so it takes more effort to keep them from bouncing off each other. There's a relation, then, between the amount of matter and the pressure it exerts in a given setting, which is called the equation of state. We have a fairly good idea of this relationship for the interior of stars like our own sun. Eventually, as our sun radiates energy away, the internal pressure will fall and the gravitational force will increase the density until the point at which electrons are forced together (or, more precisely into degenerate states) forming a white dwarf, and for these conditions we also have a fairly good idea of the equation of state. But for more massive stars, the collapse keeps going past this point and is only halted when the remaining neutrons are forced too 'close' together. And at this point we reach some uncertainty (at least according to 1980 era graduate texts) in what the equation of state is. So there is some real scientific uncertainty in the mass of the neutron teaspoon.

Another source of variability is whereabouts in the neutron star we are going to scoop our teaspoon out from. Since it's really, mindbogglingly, dense, remember , there's a lot of gravity about, which accounts for a pretty steep gradient in density from the outside to the inside. Shapiro and Teukolsky give two examples for a neutron star of typical mass, on p251, in which the outer crust can have a density of between 1014 kg m-3 and 1017 kg m-3 depending on the choice of the equation of state, but the variation in the core density is only between 1.28 and 1.93 x 1018 kg m-3 . Oops, soory, too mindboggling...that 's a two-yearly remarriage to keep up for a period between 36 days (outer crust) and 300 years (core). It seems to me the core density is the natural unit to teaspoonize, but I suppose one might also want to use the mean density. Another textbook (Michel, Theory of Neutron Star Magnetospheres, University of Chicago Press, 1991) gives a mean density of 2.67x1017 kg m-3 with an astonishing degree of precision and lack of references.

The next question is how massive is the neutron star we are interested in. The larger the mass, the greater the core density. S&T (Fig 9.6) give a range of observed masses for seven observed neutron stars, of between about one and two times solar mass. But actually Figure 9.2 suggests that within that range, different choices for equations of state account for more change in core density prediction than differences in mass choice, and whatever the mass choice there is a range of core densities of between 3x1017 to 3x1018 kg m-3 . (It's not clear to me how much weight should be given to those different state equation choices, and whether some can be ruled out: this Figure is drawn from a review paper from 1979 so with luck more is known now. ) The upper limit seems to be about the place where the theory of how neutrons pack together gets a bit foggy, and S&T leave open the possibility that exotic, denser phenomena might still be possible.
But it seems clear that if its the core density you are interested in, and neutron (rather than pion) stars is what you are talking about, then theory is confident to within an order of magnitude. The teaspoon of core contains a mass of between 1012 and 1013 kg: between a billion and and ten billion tonnes. If you want to take a mean density, which would take some fancy teaspoon work, perhaps an order of magnitude or so less. If you scoop from the crust, you can push the teaspoon down to maybe 109 kg: a million tonnes, but you probably weren't meaning to do that because at that density you aren't getting the exciting neutron-neutron packing that makes neutron stars do pulsar things and all the other worthwhile theoretical targets you were trying to PUS in the first place. So NASA (1 billion tonnes) gets full marks, Tasmania (500 million tonnes) and Canada (50 billion tonnes) pass with allowance for teaspoon variation, and there are fails for Cornell (10 million tonnes, and they imply they mean the core not the crust), and UC Riverside (10 trillion tonnes).

Why all these errors? None of these pages show their working, which is fine for the two PUS ones (NASA and Canada) who actually get the answer right, but more embarrassing for the academic pages. My guess is that the ten trillion tonnes was meant to be ten trillion kg, but I can't quite imagine where the 10 million came from.

How can these errors persist? Because they don't matter. You still say Wow even if the neutron teaspoon comes from Penn State (at a measly 5 tonnes) because that is already off the scale of how heavy teaspoons can reasonably be. So the effect is achieved in any case. No one does anything with these estimates, which is why they don't get thought about once made. Which slightly makes one wonder about the point of them. Neutron stars are real objects, with a physical presence in our lives from the time a graduate student sat in a field, not far from where I am typing this, and heard the first pulsar. There's an extraordinary human achievement in understanding as much as we do about them: that's the real Wow.

I finish with my favourite estimate for the mass of the neutron teaspoon. It comes via a Yahoo news story, sadly no longer online, but preserved here:

Neutron stars are almost unimaginably dense: a teaspoon of neutron star material weighs a billion tons (1.016 billion tonnes)

(NB I have probably made a mistake or two in the numbers. For the reasons I've suggested it doesn't matter, but let me know anyway. The science is likely to be out of date. Thanks to Andrew Brown for relaying the question that sparked this. I am happily married and have no plans for further teaspoons.)