# Amateur Astronomy

#1241

Given the braking distance of a train at 50 km/h and the range over which an average human can throw a tennis ball, I’d say that understanding how a gravitational slingshot works is the least of one’s worries under those circumstances.

#1242

Yes, the sensible experimenter will stand on the platform, not on the tracks. In that case, multiply the ball’s velocity by the cosine of the angle between the ball’s trajectory and a head-on trajectory. You should be able to make the angle as small as makes no difference to all but the most tedious quibbler.

While I’m at it, gravity assists may be accompanied by an engine burn, which maximises fuel efficiency by using the potential energy of the fuel as well. See Oberth effect. I’m afraid there’s no useful analogy we can make with trains for this one.

#1243

Excellent stuff. Thanks.

#1244

So I mentioned that the virial theorem gives us a way to measure the dynamical mass of a galaxy. I also mentioned that there were other ways of estimating a galaxy’s mass. One of those to look at the brightness of the galaxy as a proxy for the mass of all the stars. There is a major pitfall with this approach, though. Compare these two galaxies:

The visually stunning arms of spiral galaxies like the one on the left are sites of active star formation. We get young O and B type stars which are both the brightest and the most massive. A star which is three times the mass of our Sun will be a hundred times as bright. A star which is fifty times the mass of our Sun will be ten thousand times as bright. This means it’s also burning fuel at ten thousand times the solar rate. Such stars are always young, as they don’t live very long. Even with fifty solar masses an O type star will burn through it in less time than primates have been on earth, about fifty million years. Our own galaxy, if we could see it from the outside, looks like the one on the left. Yet only one in three million stars in the solar neighbourhood is of the most massive O type. You can do the sums. O and B stars dominate the light of a spiral galaxy but only contain a small fraction of its mass.

The galaxy on the right is a more evolved* elliptical. It doesn’t have the dazzling jewels of the spiral, and is definitely not as bright, but it’s probably more massive. It’s composed of more numerous older, fainter stars. They are not as flashy as their spiral cousins, but ellipticals are more massive on the whole as they have formed from galactic mergers. In large clusters where there are lots of mergers we find some truly massive galaxies. Whereas our Milky Way weighs in at around half a trillion solar masses, the aptly named El Gordo cluster contains 3,000 trillion solar masses, including many mind-bogglingly huge elliptical galaxies. Indeed, it is composed of two sub-clusters which are still hurling themselves at each other at several million kilometres per second.

*** Galaxy classifications were invented at a time when it was believed that ellipticals evolved into spirals, when in fact the opposite is true. The terminology was never fixed so ellipticals are called “early type” and spirals are “late type”. Just another addition to the 3,000 years of accumulated cruft in astronomical terminology.
**

So, it should be obvious that estimating the mass of a galaxy from its starlight is not entirely straightforward. We need to know the distribution of star sizes so as not to be misled by the comparatively rare bright ones. We use something called an initial mass function (IMF) which is a model of the distribution of star sizes in a galaxy. As a general rule, there are more and more stars the further you go down the size scale. Obviously this progression cannot continue indefinitely: there is a lower limit to the size of a star based on the mass needed to ignite nuclear fusion. Though even there the waters are muddied by the existence of brown dwarfs, which could be considered failed stars or super-planets depending on your point of view. The problem is that even in our own Milky Way, let alone distant galaxies, the smallest stars are too faint for us to be able to take a census. Consider that a decent size telescope was required to discover our very nearest stellar neighbour, Proxima Centauri (about a century ago in South Africa, using a scope built by Grubb in Rathmines whom we mentioned up thread). Proxima is 20,000 times fainter than the Sun, but is by no means the smallest that stars can get.

Anyway, to cut a very long story short, there is considerable uncertainty over the masses of galaxies as measured by their starlight, due to uncertainties in the IMF especially concerning the lower end of the stellar mass scale which probably dominates galaxy masses. There is various theoretical modelling, with a bit of voodoo thrown in, and some empirical observations such as that the IMF can be different for populations of stars with different metallicities. (It is known that the quantity of heavier elements in gas clouds from which stars form affects the ease with which smaller stars are made).

But one thing is for certain. When astronomers started comparing dynamical masses of galaxies (computed from the velocities of stars) with estimates based on starlight, they were very different even allowing for uncertainties in the latter. This so-called dynamical-mass-to-light-ratio was strong evidence that the masses of galaxies were not dominated by stars at all, but by some unseen component which came to be known as dark matter.

Since the 1970s it has become conventional wisdom that all galaxies contain an abundance of dark matter. Indeed, the standard model of how galaxies first formed is now explained by postulating dark matter density fluctuations in the early universe, which became the gravitational seeds within which “normal” matter congregated to make the first stars.

That’s why a couple of recent discoveries have unsettled astronomers … but I’ve run out of time and will need yet another post.

#1245

An interesting chat about the virial theorem in action in M70. The thermodynamics jargon will sound a bit strange, but he’s just talking about kinetic and potential energy being exchanged between the inner and outer cluster members. It’s a bit like the way many minor bodies were flung into very distant Kuiper Belt orbits as our solar system’s larger bodies formed and sank toward the centre.

#1246

I nearly forgot the final piece of the post … we already looked at dynamical and luminosity-based estimates of galaxy masses, and why dark matter was invoked to try to explain the differences between them.

But a year ago, a group of astronomers found a galaxy in which the dynamical mass-to-light ratio was in balance. There didn’t seem to be any need for dark matter. This is contrary to the conventional wisdom about how all galaxies are thought to form. All sort of criticisms were leveled at the measurements that led to this discovery, such as too much (or too little) emphasis being placed on globular clusters in the galaxy instead of “normal” stars. But last month the same group published another paper about a second galaxy that seems to be missing its dark matter component. Speculation is rife about whether it’s possible for a galaxy to lose its dark matter by some sort of tidal stripping, though this seems a tad unlikely.

The new discovery paper is here. A summary discussion is here. For now these dark matter free galaxies are in the “unexplained” category.

#1247

I make a point of avoiding pseudoscience and conspiracy theories on this thread. I’m making an exception for the flat-earther below, because he asks such a damn good science question. He just doesn’t realise the question makes perfect sense and science has the answer. (I subsequently talked to him about it – and realised no explanation from me would be able to penetrate his very bizarre belief system, but for those a bit more accepting of science, read on ).

So he asks how Polaris, the North Star, can stay in a fixed position above the pole if Earth is wobbling on its axis while orbiting the Sun, while the Sun itself careens through space. And of course the answer is that it can’t … and it doesn’t!

Let’s ignore the question of the Earth’s wobble initially. What we’d expect to see is Polaris tracing a circle in the sky over the course of a year, due to Earth’s orbit around the Sun. And that circle should itself be moving across the sky due to the Sun’s motion through space. That’s exactly what we see! But we have to make exquisitely sensitive measurements because the movements are tiny. And the movements are tiny because Polaris is far away (even though quite close in astronomical terms).

Let’s cut to the chase and answer most of his question with just one Wikipedia lookup. We need a little background knowledge to be able to decipher the answer. The parallax of a star is just precisely the radius of the little annual circle it makes on the sky against the more distant background stars due to the Earth’s orbit.

Actually, if you think about it, only stars more directly “above” the plane of the solar system (like Polaris) will make circles; stars off to one side will make flattened circles or ellipses, but it doesn’t affect our calculations. And so we read off the parallax of Polaris from its Wikipedia page and get 8 mas. What sort of unit is a mas? We divide a circle into 360 degrees, a degree into 60 minutes, a minute into 60 seconds, and a second into a thousand milliarcseconds, or mas. So there are 3.6 million mas in just one degree! We are talking about a very, very small angle. The annual parallax circle that Polaris makes on the sky is more than a hundred thousand times smaller than a full moon. No wonder the average person can’t see it – neither can the biggest telescopes in the world, a point I’ll come back to.

First let’s look at another motion – Polaris’s movement in a straight line across the sky due to it’s motion through the galaxy relative to the Sun. We call this the proper motion. In fact it’s only one part of the overall three-dimensional space motion which would include radial motion toward or away from us as well. The two-dimensional proper motion has two components – declination in the direction of the celestial equator (equivalent to latitude on earth) and right ascension or rotation around the pole (equivalent to longitude on earth). Because of Polaris’s proximity to the pole the R.A. has a negligible effect¹ so we just read off its proper motion in declination from the Wikipedia page. This tells us it’s moving at 15 mas per year, about the same width as the parallax circle, or less than a hundred thousandth of a moon diameter.

How can we measure such tiny movements. Well, it’s damn hard! The effects of the earth’s atmosphere is to distort the image of a star so that instead of a needle sharp point it appears as a little blurred circle. Even with the best adaptive optics it has been difficult to achieve angular resolutions better than 10 mas, though this limit has been pushed in recent years. So for most stars, the dancing, blurry star image is bigger than the annual movement we are trying to detect. This has a direct effect on the distances we can measure by stellar parallax, as the distance in parsec is just the inverse of the parallax. That is, a parallax of 10 mas gives a distance of 1000/10 parsec, or about 320 light years. Polaris is just beyond the limits of this so its best earth-based measurement is imprecise. This is where space-based astrometry enters the scene.

Above the blurring effect of the atmosphere, spacecraft can achieve very high angular resolution. Also they can use techniques like drift scanning where the camera is very slowly panned as a star’s proper motion is measured. This gives a streak across the camera’s CCD pixels so that interpolation can be used to yield sub-pixel resolution. It’s all very ingenious!

The Gaia mission currently under way will provide the highest resolution astrometrical measurements to date. Until the results are in, our next best is the catalogue from the 1990s Hipparcos mission. You can look at all this data yourself at sci.esa.int/hipparcos/. Here is the astrometry for Polaris, or HIP 11767 in the catalogue:

This data is simply magnificent. Here, in the form of the green cycloid, you can see approximately² what the actual movement of Polaris in the sky would look like if you zoomed in really close and stood there for three and a half years. Unfortunately I can’t show you the animated version that the ESA app provides, which shows Polaris as the yellow dot (lower right of cycloid) orbiting the moving orange dot which slides along the purple barycentric line (the path of the solar system’s centre of gravity relative to Polaris). If you want to play around in detail, a link to the app and a description of the data is here.

Finally there is the question of the Earth’s wobble. I’ve ignored it until now that we’ve covered the Hipparcos data. The reason is obvious – Hipparcos is in space so the Earth’s axial precession doesn’t affect it. (Actually, the Hipparcos spacecraft had its own axial precession which was well known and corrected for). But yes, for an earth-based telescope there would be an additional movement which affects the whole frame of the sky and not just any individual star. The Earth’s spin axis which points very close to Polaris at present is precessing along a wide circle which takes 26,000 years to complete:

This motion, though still small, is on the order of a thousand times greater than the proper motion of Polaris. We’ve known about it for over two thousand years. (It’s discoverer was Hipparchus, after whom the Hipparcos satellite was partly named). But because it affects the whole sky it doesn’t affect Polaris’s proper motion relative to the background stars.

Coming back to our flat-earther’s incredulity about Polaris being apparently static above the north pole, it’s all a question of how long you watch and wait. Polaris moves exactly as expected, but those movements are tiny. The thing our flat-earther hasn’t appreciated is the distance involved. On the scale of his scribbled diagram, where the Earth is about an inch from the Sun, Polaris is over 700 miles away!

If you’ve ever watched the International Space Station pass overhead, it’s about a third of that distance. I’ve watched the ISS while on the phone to another observer a hundred miles away so that we could measure the parallax shift. But now imagine how much the space station would seem to move against the background stars if you took a step of just two thirds of an inch to the left! That’s the same amount that Polaris moves by when the Earth moves 300 million kilometres to the other side of the Sun in six months.

1. To be scrupulous, the proper motion calculation with δ =Dec and α=RA is: https://latex.codecogs.com/gif.latex?\theta%3D\sqrt{\delta^2+\left(%20\alpha\cos\delta_*%20\right%20)^2}

2. This is assuming you rotate with the Earth, and allowing for the fact that the R.A. scale would be compressed because of Polaris’s proximity to the pole. In reality you’d have to squeeze this diagram horizontally by a factor of a hundred or so for the true apparent ratio of R.A. to Dec. Note the Hipparcos value of proper motion in declination (pmd) of -11.7 compared to the Wikipedia value of -15 mas/yr.

#1248

Thanks. I remember seeing an article about the earth’s magnetic north changing dramatically over the last few months/ years. Is the changing magnetic field likely to have any impact on the earth’s orbit. I don’t mean anything major but measurable effects

#1249

The Earth’s magnetic field is generated by the rotation of its fluid metallic outer core, and may be affected by plumes of material that rise in the deep mantle. Those cause the orientation of the field to change over time, so that the magnetic pole wanders around, essentially randomly, with respect to the axis of rotation. It’s true that the declination of true north has been changing more rapidly in recent years. The magnetic field is also weakening in the Canadian Arctic and strengthening in Siberia.

There’s no particular reason that I know of to expect any relationship between Earth’s orbit and its magnetic field. There are two ways that I can think of that it could be related to Earth’s rotational moment of inertia. The earthquake that caused the Japanese tsunami was supposed to have been so large that it altered the Earth’s axis of rotation. We also know that crustal movements from isostatic rebound are changing the rotational rate. Those are changes to the mass distribution caused by the ice disappearing at the end of the ice age. So mantle plumes that cause magnetic field changes could do the same. The second way is more subtle and is to do with the way the planet interacts with the solar wind. At times of high solar activity the atmosphere heats up and expands. This also changes the moment of inertia and can result in a measurable alteration to the rotation rate and the length of the day (due to conservation of angular momentum).

The thing is, such changes to the Earth’s moment of inertia wouldn’t necessarily have any effect on its orbit. Gravity doesn’t care (much) about the rotation of rigid bodies. I say “much” because there is tidal coupling between rotating bodies, but it’s pretty small between the Sun and the Earth (even compared to the effect of the Moon) because of the large distance. There’s also an effect in General Relativity called frame dragging, but again it’s not something we have to consider on solar system scales.

#1250

If anyone read about Comet Iwamoto’s flyby this week, I’m not holding my breath for a sighting. Last night’s closest approach was clouded out, and tonight the chances are marginal at best. In theory the comet is visible with binoculars as a fuzzy green blob. Tonight it’s in the constellation of Leo, about 60 degrees altitude at culmination due south about midnight. You’ll definitely need a finder map to locate it. And probably a cloud-penetrating radar. Not to worry, it’ll be back in 1,400 years

#1251

It’s a coincidence that this came up quite recently on the thread. To recap, it’s because stars are so small (by angular width) that distortions in the atmosphere when they are low to the horizon can refract their light wholly or partly away from our line of sight. The refraction is wavelength-dependent, hence why colours are affected unevenly.

When you go to places like Iberia, it’s noticeable how much more steady and less twinkly Sirius is. That’s because it’s both higher in the sky due to the more southerly latitude, and because the atmosphere is more stable in arid places.

The American southwest is another similar location and, as it happens, on Monday night it’s in the path of an occultation of Sirius by an asteroid. Occultations of bright stars are incredibly rare. You might think there’s a lot of stuff flying around up there, but it’s easy to forget how big the sky is compared to these small objects. There’s a lot of uncertainty in the angular width of Sirius and of asteroid 4388 Jürgenstock which will eclipse it, but they are both on the order of two millionths of a degree wide. To see one superimposed on the other requires a very high precision of alignment.

Anyone lucky enough to be at exactly the centre of the narrow eclipse path across the southwest and midwest will see Sirius dim or disappear for just a few tenths of a second. It’s not visible from Ireland, but might easily be mistaken for a twinkle if it were! Amateur observers who record the phenomenon will be able to help scientists get a better idea of the shape and size of the asteroid.

#1252

I mentioned in passing that:

Here’s a nice discussion of recent investigations into the evolution of barred spirals:

Along the way it includes Hubble’s mistaken “tuning fork” galaxy classification, which explains where the terminology came from:

#1253

I shouldn’a switched on the telly. Oh gawd. Not another moon with a stupid name! This month it’s the super snow moon. Last month it was the super blood wolf moon. Last year we had super blue moons. IT’S JUST THE FRICKIN’ MOON. It’s full every month! The BBC were oohing and aahing about this one on the news yesterday. The presenters are as clueless as the public they’re misleading.

The average person can’t tell the difference between a super moon and any other full moon. But they’ll see it near the horizon and think they’ve seen the “super moon”, not realising it’s an optical illusion. Or more likely they won’t even venture outdoors but will see a moon the size of a mountain in a photo like the one above. Of course the moon is the size of a mountain … a very far away mountain photographed with a very big zoom lens. Let’s face it – if you line your shot up right the moon is actually the size of a continent.

#1254

Here’s the non flat earth description of the creation of earth magnetic field . The rotating mantle acts as a dynamo to sustain magma circulation using the hall effect. The circulating magma creates the magnetic field.

hal.archives-ouvertes.fr/hal-00 … F5Gq7-9Jhl

#1255

Sounds interesting TF, but the link seems to be broken … do you have a better one?

#1256

apologies, I’m not very dextrous when manipulating links with my phone

link to pdf on the RHS
hal.archives-ouvertes.fr/hal-00302039

#1257

Ta, found it now. Think you mean the rotating core acts as the dynamo. The novelty in this theory is how currents in the core couple to those in the mantle, which produce the field. As peer-reviewed publications go, this one doesn’t look to have gained much traction. It’s got a total of six citations in other works, which includes three from the author himself and one which purports to be a refutation. The refuter claims that any Hall effect in the dense mantle will be negligible, and insufficient to explain the observed effect. I wouldn’t claim enough expertise myself to be able to choose between them … though nobody seems to be challenging the basic idea that the Earth’s magnetic field is somehow generated by the rotating core and circulating mantle material.

#1258

Ok, I had a little rant last week about the portrayal of the so-called “supermoon” in the media. But now even an astrophysics PhD is doing the same thing:

Have a look at the segment from 1:40 to 2:30. For a start her numbers are dodgy, saying the moon looks about 15% brighter and 7% bigger. That’s about right if you compare it to the average moon. But why not compare it to the smallest possible “micromoon”, in which case the numbers are doubled: 30% and 14%. That’s the comparison she actually shows in the picture of the moon at perigee and apogee. But then she commits the cardinal sin of showing one of those zoomed-in pictures of the moon on the horizon, and actually claims that this is a realistic portrayal of the moon “compared to other objects nearby” … but only “when it’s quite low on the horizon”. This is the same combination of confusion about the moon illusion and downright trickery using zoom lenses that I was complaining about before. Ironically, Dr. Becky then goes on to complain about the media hype herself.

By the way, here’s another relative comparison of the supermoon and micromoon:

Obviously you can tell the difference, right? That’s because they’re side by side. But when you see the moon in isolation in the sky, there isn’t a chance in hell you can estimate the size.

Don’t believe me? Well, here’s a trickier question. Look at that those half moon images on your screen again. No cheating now … how do think they compare to the actual size of the moon in the sky if you were to go outside and look at it now. A bit bigger or smaller? Most people are likely to say smaller.

What are we really asking though? There’s clearly no comparison between the 6.5 cm supermoon above and the 3,500 km diameter of the real moon. What we want to know is the angular width of the image versus that of the real moon. And that depends on how far away you are from your screen. So to rephrase: how far away should you hold the screen from your eye in order for the angular width of the moon in the photograph to be same as the real moon? Answer below (and by the way it doesn’t depend on whether the moon’s near the horizon – it doesn’t change).

If you’re viewing this on a typical laptop screen, you’d have to stand back more than seven metres to see the moons in the image above at their “real” size. From a more typical laptop viewing distance, the picture is about ten times too large. It should be the size of a pea. If you don’t believe me, get an actual pea, go outside and hold it up to the moon at arm’s length. They’re about the same size.

#1259

How can you see something that’s too small to see? No cheating – I mean with the naked eye.

This is one where we have to dissect the question. What does it mean for something to be “too small to see”? It means we can’t form an image of it as a distinct object. Equivalently, we can’t distinguish it from another adjacent object. In technical terms we say that we can’t resolve it. But why should there be a limit to what we can see in the first place? Is it a defect of our eye – an eyeball not transparent enough, a lens too imperfect, a retina not sensitive enough? While some of those things might be true, the theoretical limits to optical resolution stem from the nature of light itself.

Think of rays of light arriving from a source like splashes in a pond. Each point in a potential image is a splash:

Like the rebounding water droplet, each splash has a central peak or maximum, surrounded by concentric alternating minima and maxima. This wave nature of light sets the limit on resolution. If two splashes are so close that the central peak of one coincides with the first trough of another, we cannot form an image of the two as distinct splashes. We call that limit the Rayleigh criterion.

Actually, our water splash picture is a little misleading. In reality the waves result from light passing through an aperture such as the lens of our eye, or of a telescope or microscope. Any wave passing through a gap – such as the water waves in the ripple tank below passing through a gap in a screen – will spread out by diffraction. Peaks and troughs will arrive at slightly different times on a subsequent screen, which is what we see as the splash.

The smaller the gap, the more spreading occurs. The wavelength of the light itself is also important – shorter waves make smaller splashes. And so we expect the Rayleigh criterion to be improved for larger apertures and shorter wavelengths. That’s exactly what we find – the following is the Rayleigh criterion for a circular aperture:

https://latex.codecogs.com/gif.latex?\theta_R%3D1.22\frac{\lambda}{d}

This is saying that the minimum resolution angle in radians is proportional to the wavelength of the light (λ) and inversely proportional to the aperture size (d).

So now we can do some sums. Put the typical human pupil size of 5 millimetres and the wavelength of red light at 700 nanometres into the above equation. We get a result of one hundredth of a degree as the maximum resolving power of the eye. In practice it’s a bit worse – about a sixtieth of a degree or one arcminute. That’s the size of a printed full stop in a newspaper viewed from a metre away.

Optical microscopes sometimes use near-ultraviolet light which has less than half the wavelength of red light to increase their resolving power. Electron microscopes don’t use light at all. One of the discoveries of quantum physics was that matter particles can travel as waves too, and electron waves are much tinier than visible light.

In the world of astronomical telescopes we have less choice about which wavelengths to use. It’s not us that’s illuminating the source, but the intrinsic light of the source which reaches us. And very often we want to look in the near infrared which is even longer wavelength than red light. That’s because infrared light manages to penetrate through dust clouds which would otherwise obscure an awful lot of the celestial objects we are interested in. If we can’t shorten the wavelength then we have to increase the aperture: from the one-inch aperture of Galileo’s first astronomical telescope, to the 10+ metre apertures of today’s largest.

Before we go further, let’s return to the original question. How can we see something that’s too small to see? Hopefully it’s now obvious that just because we can’t resolve an object doesn’t mean the light from it doesn’t reach us. If you look out from Dublin bay to the Kish lighthouse eleven kilometres away, the giant Fresnel lens of the lamp is only half the size your eye can resolve at that distance. Yet the light can be seen from 40 km! You can see aircraft landing lights from more than 50 km away, although the lamps are ten times smaller than the lighthouse lamp, and a hundred times below your eye’s resolving power. The planet Venus is a hundred million times bigger than the aircraft lamp, but a million times further away. It’s just below the limit of naked eye resolution. There’s speculation that ancient astronomers might just have been able to see “horns” on Venus when it was a crescent. Had they done so definitively we would probably have had earlier and wider acceptance of the heliocentric theory, as it would have been obvious that the planets were lit at different angles by the Sun. As it happened, Galileo saw phases of Venus almost as soon as he turned his paltry little telescope on the skies in 1610.

We can see stars, but the vast majority of them are below the limits of optical resolution even by the biggest telescopes under perfect conditions. A handful of the giant nearby ones have been resolved into discs with surface features (something that would have been unthinkable just a few decades ago). But our very nearest neighbour, Proxima Centauri, is a dwarf star that would need a telescope with a 150 metre aperture to resolve.

The implication of something being unresolvable is that we can glean no spatial information about its features. Yet, miracle of miracles, when certain types of movement occur on a star the light signal alters in a way that can be used to extract good information, in spite of the star being far below the resolution limit. And it’s not just stars – some of the most distant and powerful objects in the universe are being probed with the same techniques. I’ll talk about it next time.

#1260