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Assignment 6: Model Answer. Not quite so lost in space…. Model Answer. This model answer, once again, is designed to show you how a professional astronomer might have approached this puzzle. I would not expect any of you to have come up with this (though a few of you came pretty damn close…).
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Assignment 6: Model Answer Not quite so lost in space…
Model Answer • This model answer, once again, is designed to show you how a professional astronomer might have approached this puzzle. • I would not expect any of you to have come up with this (though a few of you came pretty damn close…). • There are alternative interpretations of the data that could also get excellent marks.
New Facts • Most of the new facts backed up what you’d learned from the first data set. • In particular, the geometry of space was measured to be not significantly different from that it our own universe (in statistics it is normal to require something to be at least two standard deviations away from your expectation to be considered significant). • This does not prove that space is flat, but that it must be quite close. In the absence of compelling evidence to the contrary, we’ll assume it is flat for the moment, and see where that leads us.
New Objects • By following up the radio pulses, we’ve found a lot of new “stars”. • These new “stars” are in some respects quite like those we saw earlier, but have some striking differences: • They typically have much longer wavelengths and pulse periods. • They are typically much fainter. • Their parallaxes are much smaller (typically too small to measure) indicating that they are much further away.
Two Hypotheses • This leads to the first puzzle. Are these new radio emitting “stars” basically identical to the normal stars, or are they some new class of fundamentally different object? • If they are the same type of thing, then their faint fluxes, long wavelengths and long periods would have to be explained by some cosmological effect, such as the expansion of space. • It could be that every “star” emits occasional radio pulses. • If they are different, all bets are off.
Testing the Hypothesis. • Let’s test the hypothesis that these radio emitting “stars” are basically the same as the nearby “stars”. • If so, any radio sources that happen to lie close by should have identical properties to the “stars” we studied in the first part of this assignment. • There are a couple which do have large parallaxes. These two do seem to be very much like all the other nearby “stars”. • Their pulse periods, fluxes, pulse amplitudes, wavelengths etc seem to fit perfectly onto all the correlations we’d previously identified. • So thus far, everything seems to fit this hypothesis. So let’s run with it and see where we end up.
If they are the same? • The curse of this data-set is that we cannot use parallaxes to measure distances, for all but a couple of nearby “stars”. • But if these radio emitters are really the same as nearby “stars”, we have a handle on how distant they are. • First - a crude distance measure. All the nearby “stars” had roughly the same luminosity (albeit with a huge scatter). Using their parallax-derived distances, their average luminosity was 1.28x1016W. • If we assume that all these new, far away “stars” have the same average luminosity, we can thus use the inverse square law to compute distances.
Crude • This is a crude measure - after all not all nearby stars have the same luminosity. But it gives us somewhere to start. • And what do we see? • These new objects are typically tens of thousands of AU away - about 1000 times more distant than the nearby “stars”. • The further away they are, the fainter their radio flux, the longer their pulse period, the longer both their optical and their radio wavelengths.
Consistent • So far, so good - all these correlations are consistent with the idea that these “distant” stars are much like the nearby ones, only further away and redshifted. • These is a lot of scatter, but considering the crudity of how we are measuring the distance, it’s not too bad. • We can even try to deduce Hubble’s Constant from the plot on the previous slide. • Nearby objects have optical wavelengths of 400 - 800nm, while objects lying around 1015m away have optical wavelengths in the range 1500 - 3000nm.
First Guess Hubble’s Constant • So crudely speaking, objects at this distance have had their light stretched (if you believe our hypothesis) by about a factor of 1500/400 = 3000/800 = 3.75. So the universe has expanded by 275% in only the last 3.3 million seconds (the time it takes light to travel 1015m - ie. 38 days!). • If true, this universe must be expanding a bloody sight faster than ours… • But is our hypothesis (all “stars” are the same) really true?
Accuracy • In the absence of anything better to do, a good rule of thumb in science is to improve the accuracy of your deductions. • We should be able to measure distances and redshifts a whole lot better than in the previous plot. • For a start, we deduced in the first part of this assignment that the luminosity of a “star” correlates with its pulse period, pulse amplitude and wavelength. • Can we use this correlation to improve the accuracy of our distance measurement?
Better Distances • This is an almost exact analogy of what Brian Schmidt does with his supernovae. He observed a correlation between how rapidly their brightness decays and their peak luminosity. He used this correlation to improve the calculation of their true luminosities and hence their distances. But nobody understands where this very useful correlation comes from - it’s empirically observed in nearby supernovae but that’s it. • So let’s do the same thing - use all the correlations we found in the first part of this assignment to get better distances.
Pulse Amplitude • Redshift will alter wavelength and pulse period, but it should not change pulse amplitude. So let’s look for a correlation between pulse amplitude and luminosity. NB - luminosity is how bright some object really is, as compared to flux, which is how much energy we receive from an object at our location, per unit area. • So what does this correlation look like in the nearby sample data?
So there is a correlation • So there is a correlation, though not a stunning one. • You can fit this in different ways. The log plot looks tidier than the linear one, which implies that the best curve to fit is a power-law. • Roughly speaking, these plots show you that whenever you double the pulse period, you halve the luminosity. So let’s use that as our fit. As a constant, note that nearby “stars” with 2% pulse amplitudes have luminosities of around 1x1015W. • There is a lot of scatter in this correlation, but considering luminosities are derived by multiplying flux by distance squared, and distance depends on rather poorly known parallaxes, this is to be expected.
So we have… • So we have: • where L is the luminosity measured in Watts, and a is the pulse amplitude (in percent).
Redshift • We can do something similar for redshift. We use the excellent correlation between pulse amplitude and wavelength of the nearby “stars” to calculate the true wavelength of any given star, and hence (by comparing with the observed wavelength) the redshift of these more distant objects.
New Numbers Triumph! • Using these new, more accurate numbers, we get an incredibly tight correlation. • The tightness of this correlation really tells us that we are on to something here. • It seems that our assumptions really do work. • We get similarly tight correlations between redshift and the radio flux, and the radio wavelength. • Divide all the observed radio wavelengths by these new redshifts, and it turns out that all the radio bursts had an identical wavelength: 20cm. • Suddenly we are finding brilliant correlations everywhere.
Why so tight? • How can these correlations be so tight, when we used such a crap correlation (between distance and luminosity), and the somewhat less crap but still imperfect correlation between wavelength and pulse amplitude to derive them? • It must be that the scatter in the correlation between luminosity and parallax measured distance of the nearby “stars” was due to the crap parallaxes. The correlation must really be a perfect one.
What can we learn? • So - the tightness of these correlations tells us that we’ve done something right (or the universe is conspiring against us). • It seems that space is expanding, just as in our universe. Distances and redshifts correlate together perfectly. • The correlation isn’t a straight line. This curvature is actually because the inverse square law (which we used to derive distances) doesn’t work perfectly when you factor in the expansion of space. • As space expands, each photon is stretched, and hence looses energy. Also, the gaps between photons stretch. Both effects cause the observed flux to drop faster than predicted. • Take out this effect (which I wasn’t expecting anyone to get) and the correlation becomes a straight line.
Hubble’s Constant • We can thus deduce Hubble’s Constant again, with greater accuracy. • Use a nearby “star” so that all the relativistic corrections to the normal Doppler Effect equation aren’t too big. • There is one, for example, at a redshift of 0.546 and a distance of 9.71x1013m. This redshift corresponds roughly to a velocity of 54.6% that of light, ie. 163,800 km/s. The distance is 647 Astronomical Units. • So Hubble’s Constant in this universe is roughly 250 km/s/AU. Vastly greater than in our universe (71 km/s/Mpc - note different units).
A Puzzle • From this value, we can estimate the age of this universe. • An object 647 AU away travelling at 250 km/s would reach us in only 7 days. • So if the universe was expanding at a uniform rate, it would only be 7 days old! • But this cannot be true, not only because we’ve been in this universe longer than 7 days, but because we see “stars” at distances of over 100 light days away. • The light from them would have had to have set out before the “big bang” in this universe!
Steady State. • This combined with the lack of a microwave background (which by itself could have been due to some other cause) suggests that the USS Drongo has landed in a steady state universe. • A Steady State Universe with a rightly flat geometry and a whopping great expansion rate, filled with rather strange objects that look like stars but aren’t.
The Master Plan • That was my master plan when faking this universe. It is a universe containing no matter (other than the USS Drongo). • It’s only contents are the bizarre pulsing things, which are some form of wormhole not found in our universe. • The wormholes are not all identical, but only have one variable parameter (wavelength). Everything correlates or anticorrelates perfectly with this parameter: luminosity, pulse period, pulse amplitude. • The radio bursts are the mini “big Bangs” created when any of these wormholes form. They all have identical fluxes and wavelengths. • You never had any chance of understanding the wormholes (“stars”) as I just made up a set of correlations for them.
The Master Plan • This universe is expanding like crazy, but new wormholes are constantly popping into existence to keep the average density of these things constant. • These correlations were, however, regular enough to allow you to deduce the cosmology of this universe. • This was designed to mirror real cosmology - there are lots of crucial things we do not understand (eg dark matter, dark energy, Type 1a supernovae, primordial fluctuations) but that need not stop us from using the observed properties of all these things to deduce the cosmology of our universe.
An admission • One embarrassing admission. • I did make one mistake in faking this universe. Given the colossal expansion rate of this universe, you should have been able to see an increase in the redshift and a decrease in the parallaxes of the nearby “stars” even over a few days or weeks. • I’ve fixed this ready for the next bunch of students on whom this exercise is to be inflicted.
Comments Please • These types of exercise are experimental, and I welcome your comments about what you did and did not like about them. • They are designed to give you a feel for real research in all its messy glory, and to teach you practical skills on how to approach astronomical puzzles and play with data. • Please let me know whether you feel they achieved these goals, and suggestions for how they might be improved.
