I had to suspend the production of these videos for a month or so while I dealt with examination matters, but after that short hiatus, here is Episode 6 during which I explain just how weak and feeble the force of gravity really is. Combined with the fact that the Universe has such a low density (see Episode 5), the weakness of gravity means that the cosmos evolves extremely slowly.
Follow @telescoper
I’m was mucking out this blog’s blocked comments folder and unsurprisingly found a few from Mr Hine, a regular if sadly deranged correspondent.
One of his blocked comments begins
In the forlorn hope that Mr Hine might some day learn something scientifically correct I thought I’d repost this problem, which is very easy if you have a high school education in physics or applied mathematics but no doubt very difficult if you’re Mr Hine.
Verify that the radius of a circular geostationary orbit around the Earth is about 42,000 km, i.e. find the radius of a circular orbit around the Earth which has a period of 24 hours so that its orbital period matches the Earth’s rotation period, thus ensuring that an object travelling in such an orbit in the same direction as the Earth’s rotation is always above the same point on the Earth’s surface.
(You will need to look up the mass of the Earth.)
Follow @telescoperI’m a bit late passing this on but I think some of my readers might find this interesting, as I did when I came across it a week or so ago. There’s a paper on the arXiv by Nicholas Stone and Nathan Leigh with the title A Statistical Solution to the Chaotic, Non-Hierarchical Three-Body Problem and the following abstract:
The three-body problem is arguably the oldest open question in astrophysics, and has resisted a general analytic solution for centuries. Various implementations of perturbation theory provide solutions in portions of parameter space, but only where hierarchies of masses or separations exist. Numerical integrations show that bound, non-hierarchical triples of Newtonian point particles will almost always disintegrate into a single escaping star and a stable, bound binary, but the chaotic nature of the three-body problem prevents the derivation of tractable analytic formulae deterministically mapping initial conditions to final outcomes. However, chaos also motivates the assumption of ergodicity, suggesting that the distribution of outcomes is uniform across the accessible phase volume. Here, we use the ergodic hypothesis to derive a complete statistical solution to the non-hierarchical three-body problem, one which provides closed-form distributions of outcomes (e.g. binary orbital elements) given the conserved integrals of motion. We compare our outcome distributions to large ensembles of numerical three-body integrations, and find good agreement, so long as we restrict ourselves to “resonant” encounters (the ~50% of scatterings that undergo chaotic evolution). In analyzing our scattering experiments, we identify “scrambles” (periods in time where no pairwise binaries exist) as the key dynamical state that ergodicizes a non-hierarchical triple. The generally super-thermal distributions of survivor binary eccentricity that we predict have notable applications to many astrophysical scenarios. For example, non-hierarchical triples produced dynamically in globular clusters are a primary formation channel for black hole mergers, but the rates and properties of the resulting gravitational waves depend on the distribution of post-disintegration eccentricities.
The full paper can be downloaded here. The abstract is very clear but you might want to read the wikipedia entry for the three-body problem for general background. Here’s a fun figure from the paper:
Let me just add a note of explanation of the word `hierarchical’ as applied here: it means when the mass of one body is very different from the other two, or that two of the bodies have a much smaller separation from each other than they do from the third.
This paper does not present an analytic solution of the unrestricted three-body problem (which is known to be intractable) but does provide some very useful statistical insights into the long-term evolution of three-body systems, for example confirming the generally held opinion that most such systems evolve into a state in which one body is ejected and the other two form a tight binary.
Follow @telescoperWe’ve more-or-less sorted out who will be teaching what next term in the Department of Theoretical Physics at Maynooth University next term (starting a month from now) and I’ll be taking over the Mathematical Physics module MP110, which is basically about Mechanics with a bit of of special relativity thrown in for fun. Being in the first semester of the first year, these is the first module in Theoretical Physics students get to take here at Maynooth so it’s quite a responsibility but I’m very much looking forward to it.
I am planning to start the lectures with some things about units and dimensional analysis. Thinking about this reminded me that I posted a dimensional analysis problem (too hard for first-year students) on here a while ago which seemed to pose a challenge so I thought I would post another here for your amusement.
The period P for an elliptical orbit of semi-major axis a of a moon of mass m around a planet of mass M, depends only on the quantities a, m, M and G (Newton’s Gravitational Constant).
(a). Using dimensional analysis only, determine as completely as possible the relationship between P and these four quantities.
(b). How would the period P compare with the period P′ of a system consisting of a moon of mass 2m orbiting a planet of mass 2M in an ellipse with the same semi-major axis a?
Please submit your efforts through the comments box below.
Follow @telescoper
A paper by Norbert Klein caught my eye as I tried to catch up on my arXiv reading after a couple of days away last week. It’s called Evidence for Modified Newtonian Dynamics from Cavendish-type gravitational constant experiments and the abstract reads:
Recent experimental results for the gravitational constant G from Cavendish-type experiments were analysed in the framework of MOND (Modified Newtonian Dynamics). The basic assumption for the analysis is that MOND corrections apply only to the component of the gravitational field which leads to an accelerated motion of the pendulum body according to Newtons second law. The analysis is based on numerical solutions of the MOND corrected differential equation for a linear pendulum at small acceleration magnitudes of the order of Milgroms fundamental acceleration parameter a0 = 10-10m s-2 for the case of a mixed gravitational and electromagnetic pendulum restoring force. The results from the pendulum simulations were employed to fit experimental data from recent Cavendish-type experiments with reported discrepancies between G values determined by different measurement methods for a similar experimental setup, namely time of swing, angular acceleration feedback, electrostatic servo and static deflection methods. The analysis revealed that the reported discrepancies can be explained by MOND corrections with one single fit parameter. The MOND corrected results were found to be consistent with a value of G = 6.6742 x 10-11 m3 kg-1 s-2 within a standard deviation of 14 ppm.
I have edited the abstract slightly for formatting and added the link to an explanation of MOND. You can find a PDF of the paper here.
I blogged about the discrepancies between different determinations of Newton’s Gravitational Constant G a few years ago here, where you can find this figure:
The claim that Modified Newtonian Dynamics can resolve these `discrepancies’ is very bold and I’m very skeptical of the arguments presented in this paper. It seems to me far more likely that the divergence in experimental measurements is due to systematics. If anyone else has different views, however, please feel free to share them through the comments box.
Follow @telescoperI’ve been sorting through some old problem sets for my course on Astrophysics and Cosmology, and thought I would post this one in the Cute Problems folder for your amusement. The first part is easy, the second part not so much…
Answers to (2) through the comments box please – and don’t forget to explain your working!
Follow @telescoperToo busy today (again) for anything else so I’m going to resort (again) to the Cavendish Problems in Classical Physics. I think I’ll eschew the multiple-choice format for this one, but will say that there is a small hint in the fact that the question is split into two parts:
The Eiffel Tower is 300m high and is situated at a latitude 49° N. What are the magnitude and direction of the deflection caused by the Earth’s rotation to:
Please give your answers, with reasons, through the comments box below. For legal reasons I should make it clear that you are not expected to perform either experiment.
Follow @telescoperWhile I’m catching up on developments over the last week or so I thought I’d post an update on a story I blogged about a few weeks ago. This concerns the the topic of dark matter in the Solar Neighbourhood and in particular a paper on the arXiv by Moni Bidin et al. with the following abstract:
We measured the surface mass density of the Galactic disk at the solar position, up to 4 kpc from the plane, by means of the kinematics of ~400 thick disk stars. The results match the expectations for the visible mass only, and no dark matter is detected in the volume under analysis. The current models of dark matter halo are excluded with a significance higher than 5sigma, unless a highly prolate halo is assumed, very atypical in cold dark matter simulations. The resulting lack of dark matter at the solar position challenges the current models.
In my earlier post I remarked that this study makes a number of questionable assumptions about the shape of the Milky Way halo – they take it to be smooth and spherical – and the distribution of velocities within it is taken to have a very simple form.
Well, only last week a rebuttal paper by Bovy & Tremaine appeared on the arXiv. Here is its abstract:
An analysis of the kinematics of 412 stars at 1-4 kpc from the Galactic mid-plane by Moni Bidin et al. (2012) has claimed to derive a local density of dark matter that is an order of magnitude below standard expectations. We show that this result is incorrect and that it arises from the invalid assumption that the mean azimuthal velocity of the stellar tracers is independent of Galactocentric radius at all heights; the correct assumption—that is, the one supported by data—is that the circular speed is independent of radius in the mid-plane. We demonstrate that the assumption of constant mean azimuthal velocity is physically implausible by showing that it requires the circular velocity to drop more steeply than allowed by any plausible mass model, with or without dark matter, at large heights above the mid-plane. Using the correct approximation that the circular velocity curve is flat in the mid-plane, we find that the data imply a local dark-matter density of 0.008 +/- 0.002 Msun/pc^3= 0.3 +/- 0.1 Gev/cm^3, fully consistent with standard estimates of this quantity. This is the most robust direct measurement of the local dark-matter density to date.
So it seems reports of the dearth were greatly exaggerated..
Having read the paper I think this is a pretty solid refutation, and if you don’t want to take my word for it I’ll also add that Scott Tremaine is one of the undisputed world experts in the field of Galactic Dynamics. It will be interesting to see how Moni Bidin et al. respond.
This little episode raises the question that, if there was a problem with the assumed velocity distribution in the original paper (as many of us suspected), why wasn’t this spotted by the referee?
Of course to a scientist there’s nothing unusual about scientific results being subjected to independent scrutiny and analysis. That’s how science advances. There is a danger in all this, however, with regard to the public perception of science. The original claim – which will probably turn out to be wrong – was accompanied by a fanfare of publicity. The later analysis arrives at a much less spectacular conclusion, so will probably attract much less attention. In the long run, though, it probably isn’t important if this is regarded as a disappointingly boring outcome. I hope what really matters for scientific progress is people doing things properly. Even if it don’t make the headlines, good science will win out in the end. Maybe.
Follow @telescoperI’m a bit late getting onto the topic of dark matter in the Solar Neighbourhood, but it has been generating quite a lot of news, blogposts and other discussion recently so I thought I’d have a bash this morning. The result in question is a paper on the arXiv by Moni Bidin et al. which has the following abstract:
We measured the surface mass density of the Galactic disk at the solar position, up to 4 kpc from the plane, by means of the kinematics of ~400 thick disk stars. The results match the expectations for the visible mass only, and no dark matter is detected in the volume under analysis. The current models of dark matter halo are excluded with a significance higher than 5sigma, unless a highly prolate halo is assumed, very atypical in cold dark matter simulations. The resulting lack of dark matter at the solar position challenges the current models.
As far as I’m aware, Oort (1932, 1960) was the first to perform an analysis of the vertical equilibrium of the stellar distribution in the solar neighbourhood. He argued that there is more mass in the galactic disk than can be accounted for by star counts. A reanalysis of this problem by Bahcall (1984) argued for the presence of a dark “disk” of a scale height of about 700 pc. This was called into question by Bienaymé et al. (1987), and by Kuijken & Gilmore in 1989. In a later analysis based on a sample of stars with HIPPARCOS distances and Coravel radial velocities, within 125 pc of the Sun. Crézé et al. (1998) found that there is no evidence for dark matter in the disk of the Milky Way, claiming that all the matter is accounted for by adding up the contributions of gas, young stars and old stars.
The lack of evidence for dark matter in the Solar Neighbourhood is not therefore a particularly new finding; there’s never been any strong evidence that it is present in significant quantities out in the suburbs of the Milky Way where we reside. Indeed, I remember a big bust-up about this at a Royal Society meeting I attended in 1985 as a fledgling graduate student. Interesting that it’s still so controversial 27 years later.
Of course the result doesn’t mean that the dark matter isn’t there. It just means that its effect is too small compared to that of the luminous matter, i.e. stars, for it to be detected. We know that the luminous matter has to be concentrated more centrally than the dark matter, so it’s possible that the dark component is there, but does not have a significant effect on stellar motions near the Sun.
The latest, and probably most accurate, study has again found no evidence for dark matter in the vicinity of the Sun. If true, this may mean that attempts to detect dark matter particles using experiments on Earth are unlikely to be successful.
The team in question used the MPG/ESO 2.2-metre telescope at ESO’s La Silla Observatory, along with other telescopes, to map the positions and motions of more than 400 stars with distances up to 13000 light-years from the Sun. From these new data they have estimated the mass of material in a volume four times larger than ever considered before but found that everything is well explained by the gravitational effects of stars, dust and gas with no need for a dark matter component.
The reason for postulating the existence of large quantities of dark matter in spiral galaxies like the Milky Way is the motion of material in the outer parts, far from the Solar Neighbourhood (which is a mere 30,000 light years from Galactic Centre). These measurements are clearly inconsistent with the distribution of visible matter if our understanding of gravity is correct. So either there’s some invisible matter that gravitates or we need to reconsider our theories of gravitation. The dark matter explanation also fits with circumstantial evidence from other contexts (e.g. galaxy clusters), so is favoured by most astronomers. In the standard theory the Milky Way is surrounded by am extended halo of dark matter which is much less concentrated than the luminous material by virtue of it not being able to dissipate energy because it consists of particles that only interact weakly and can’t radiate. Luminous matter therefore outweighs dark matter in the cores of galaxies, but the situation is reversed in the outskirts. In between there should be some contribution from dark matter, but since it could be relatively modest it is difficult to estimate.
The study by Moni Bidin et al. makes a number of questionable assumptions about the shape of the Milky Way halo – they take it to be smooth and spherical – and the distribution of velocities within it is taken to have a very simple form. These may well turn out to be untrue. In any case the measurements they needed are extremely difficult to make, so they’ll need to be checked by other teams. It’s quite possible that this controversy won’t be actually resolved until the European Space Agency’s forthcoming GAIA mission.
So my take on this is that it’s a very interesting challenge to the orthodox theory, but the dark matter interpretation is far from dead because it’s not obvious to me that these observations would have uncovered it even if it is there. Moreover, there are alternative analyses (e.g. this one) which find a significant amount of dark matter using an alternative modelling method which seems to be more robust. (I’m grateful to Andrew Pontzen for pointing that out to me.)
Anyway, this all just goes to show that absence of evidence is not necessarily evidence of absence…
Follow @telescoper
In the Dark 