I came across the following question in a first-year physics examination from Cambridge (Part 1A Natural Sciences) and, since I have posted anything in the Cute Problems folder for a while I thought I would share it here:
Answers through the comments box please! And please show your working!
P.S. The preamble does not say whether you can also assume irrelevant formulae without proof…
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I’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 @telescoperYesterday I saw a paper by George Efstathiou and Steve Gratton on the arXiv with the title The Evidence for a Spatially Flat Universe. The abstract reads:
We revisit the observational constraints on spatial curvature following recent claims that the Planck data favour a closed Universe. We use a new and statistically powerful Planck likelihood to show that the Planck temperature and polarization spectra are consistent with a spatially flat Universe, though because of a geometrical degeneracy cosmic microwave background spectra on their own do not lead to tight constraints on the curvature density parameter ΩK. When combined with other astrophysical data, particularly geometrical measurements of baryon acoustic oscillations, the Universe is constrained to be spatially flat to extremely high precision, with ΩK = 0.0004 ±0.0018 in agreement with the 2018 results of the Planck team. In the context of inflationary cosmology, the observations offer strong support for models of inflation with a large number of e-foldings and disfavour models of incomplete inflation.
You can download a PDF of the paper here. Here is the crucial figure:
This paper is in part a response to a paper I blogged about here and some other related work with the same general thrust. I thought I’d mention the paper here, however, because it contains some interesting comments about the appropriate choice of priors in the problem of inference in reference to cosmological parameters. I feel quite strongly that insufficient thought is given generally about how this should be done, often with nonsensical consequences. It’s quite frustrating to see researchers embracing the conceptual framework of Bayesian inference but then choosing an inappropriate prior. The prior is not an optional extra – it’s one of the key ingredients. This isn’t a problem limited to the inflationary scenarios discussed in the above paper, by the way, it arises in a much wider set of cosmological models. The real cosmological flatness problem is that too many cosmologists use flat priors everywhere!
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Following on from Sunday’s post about the trials and tribulations caused by Storm Dennis, here is a clip of a plane (an Airbus 380) landing at Heathrow airport on Saturday.
There are other clips of this same event on Youtube and some of them describe this landing as `dangerous’. Although it undoubtedly involved skill and concentration by the pilot it’s not actually dangerous. Aircrew are trained to land in windy weather like this, and it’s fairly routine. My plane to Dublin (an Airbus 320) landed like this on Saturday evening and, although the pilot got a well-deserved round of applause on landing, nobody was ever really at risk.
As it happens, this week I start teaching vector algebra to my first-year Engineering students, so the weekend’s weather events have given me a good illustration of vector addition. The plane has to have a velocity vector relative to the air such that the sum of it and the wind vector adds to a resultant vector directed along the runway. Lots of people seem to think this is just guesswork but it isn’t. It’s applied mathematics.
This is in principle simple as long as the crosswind is steady, but obviously the pilot needs to be alert to gusting and make adjustments along the way. When the plane has slowed down enough to land in normal conditions, the wind over the wings is still causing a bit of extra lift. You can see that in the last moments before touchdown this aircraft is gliding because of this effect. I’m told that because of this, in windy conditions planes usually descend at a steeper angle than usual.
The interesting bit for me is that the plane touches down in such a way that its body is at an angle to the runway. As soon as it has landed it has to correct this and point along the runway. I think this is done with the rudder rather than the undercarriage, but I don’t know. Perhaps any experienced pilots that happen to be reading this could give more details through the comments box?
P.S. The title of this post is a reference to the film Airplane!
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I thought I would give an update with some bibliometric information about the 12 papers published by the Open Journal of Astrophysics in 2019. The NASA/ADS system has been struggling to tally the citations to a couple of our papers but this issue has now been resolved. According to this source the total number of citations for these papers is 532 (as of today). This number is dominated by one particular paper which has 443 citations according to NASA/ADS. Excluding this paper gives an average number of citations for the remaining 11 of 7.4.
I’ll take this opportunity to re-iterate some comments about the Journal Impact Factor. When asked about this my usual response is (a) to repeat the arguments why the impact factor is daft and (b) point out that we have to have been running continuously for at least two years to have an official impact factor anyway.
For those of you who can’t be bothered to look up the definition of an impact factor , for a given year it is basically the sum of the citations for all papers published in the journal over the previous two-year period divided by the total number of papers published in that journal over the same period. It’s therefore the average citations per paper published in a two-year window. The impact factor for 2019 would be defined using data from 2017 and 2018, etc.
The impact factor is prone to the same issue as the simple average I quoted above in that citation statistics are generally heavily skewed and the average can therefore be dragged upwards by a small number of papers with lots of citations (in our case just one).
I stress again we don’t have an Impact Factor as such for the Open Journal. However, for reference (but obviously not comparison) the latest actual impact factors (2018, i.e. based on 2016 and 2017 numbers) for some leading astronomy journals are: Monthly Notices of the Royal Astronomical Society 5.23; Astrophysical Journal 5.58; and Astronomy and Astrophysics 6.21.
My main point, though, is that with so much bibliometric information available at the article level there is no reason whatsoever to pay any attention to crudely aggregated statistics at the journal level. Judge the contents, not the packaging.
This post is based on an article at the OJA blog.
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I’ve had a busy day doing various things that took me to various places on the Maynooth University campus, culminating in my first visit to the Department of Mathematics and Statistics (which is based in Logic House) to give a colloquium. On my travels I passed this sign, which I hadn’t noticed before:
Since I’ve been too busy to write a post today I thought I’d rehash an old one about the person named in the plaque, which is above Callan Hall (on the South Campus)
Father Nicholas Callan (or, more formally, The Reverend Professor Nicholas Joseph Callan) was born in County Louth in 1799 went to the seminary of St Patrick’s College, Maynooth, in 1816 to train as a priest. During his time as a seminarian Callan studied ‘Natural Philosophy’ and became interested in experiments involving electricity. In 1823 Callan was ordained as a priest, and went to Rome in 1826 to obtain his doctorate in Divinity. At the time Italy was a centre for research into electricity and here Callan became familiar with the work of the Italian physicist Alessandro Volta who had developed the world’s first battery. Callan returned to Maynooth where he was made chair of Natural Philosophy, a post he would hold until his death in 1864.
Callan is most famous for inventing the induction coil (in 1836). By connecting two copper wire coils to a battery and electromagnet and then interrupting the current he was able to generate much larger voltages than could be obtained from batteries alone. His 1837 version that used a clock mechanism to interrupt the current 20 times a second is estimated to have produced 60,000 volts – the largest artificially generated charge at that time. It is said that his induction coil could produce sparks 15″ long, which must have been fun to watch.
Callan’s biggest induction coil, unfinished at the time of his death, can be found in the National Science Museum of Ireland (which is in Maynooth). This was one of the largest in the world at the time. The iron core is 109 cm long. The secondary windings are 53 cm in diameter and consist of about 50 km of iron wire insulated with beeswax. They were made in three separate rings separated by air gaps, so wires carrying large voltage differences would not lie adjacent to each other, reducing the risk of the insulation breaking down. At the left end is a vibrating mercury ‘contact breaker’ in the primary circuit, actuated by the magnetic field in the primary, which interrupted the primary current to generate potentials of over 200,000 volts.
Sadly Callan’s work was forgotten for quite a period after his death – experimental electromagnetism was not a priority for St Patrick’s College at this time – for which reason the invention of the induction coil has often been attributed to Heinrich Ruhmkorff who made his first device (independently) about 15 years after Callan. More recently, however, Callan’s achievements have been more widely recognized and in 2000 the Irish government issued a stamp in his honour.
The Callan Building
Nicholas Callan was laid to rest in the College Cemetery at Maynooth in 1864. The Callan Building (above) on the North Campus of the present-day Maynooth University is named in his honour, as of course is Callan Hall which I mentioned at the start.
Follow @telescoperTime for a quick plug for a paper by Nadathur et al. that appeared on the arXiv recently with the title Testing low-redshift cosmic acceleration with large-scale structure. Here is the abstract:
You can make it bigger by clicking on the image. You can download a PDF of the entire paper here.
The particularly interesting thing about this result is that it gives strong evidence for models with a cosmological constant (or perhaps some other form of dark energy), in a manner that is independent of the other main cosmological constraints (i.e. the Cosmic Microwave Background or Type 1a Supernovae). This constraint is based on combining properties of void regions (underdensities) with Baryon Acoustic Oscillations (BAOs) to produce constraints that are stronger than those obtained using BAOs on their own. The data used derives largely from the BOSS survey.
As well as this there’s another intriguing result, or rather two results. First is that the the BAO+voids data from redshifts z<2 gives H0 = 72.3 ± 1.9, while, on the other hand adding, BAO information from the Lyman-alpha forest for from z>2 gives a value H0 = 69 \pm 1.2, favouring Planck over Riess. Once again, the `tension’ over the value of the Hubble constant appears to be related to using nearby rather than distant sources.
Follow @telescoperThere are some wonderful images and movies going around from the Daniel K Inouye Solar Telescope which has produced the highest resolution images of the solar surface ever seen.
Here’s a snapshot:
And here’s a movie:
In the above image you can see the granular structure of the Sun’s photosphere. The cells you can see are a manifestation of the large-scale convective motions that transport energy from the Sun’s inner regions to the surface. This energy is created by nuclear reactions in the solar core and it sets up convective motions in the outer layers rather like those in a pan of boiling water set up by heating from below (or perhaps the gentler motions that appearin a lava lamp).
The surface structure looks surprisingly regular but the highly turbulent magnetized plasma is responsible to an extraordinary range of activity, from sunspots, flares and prominences, to the heating of the solar corona and the generation of the solar wind.
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Now that we have started a new year, and a new volume of the Open Journal of Astrophysics , I thought I would give an update with some bibliometric information about the 12 papers we published in 2019.
It is still early days for aggregating citations for 2019 but, using a combination of the NASA/ADS system and the Inspire-HEP, I have been able to place a firm lower limit on the total number of citations so far for those papers of 408, giving an average citation rate per paper of 34.
These numbers are dominated by one particular paper which has 327 citations according to Inspire (see above). Excluding this paper gives an average number of citations for the remaining 11 of 7.4.
I’ll take this opportunity to re-iterate some comments about the Journal Impact Factor. When asked about this my usual response is (a) to repeat the arguments why the impact factor is daft and (b) point out that we have to have been running continuously for at least two years to have an official impact factor anyway.
For those of you who can’t be bothered to look up the definition of an impact factor , for a given year it is basically the sum of the citations for all papers published in the journal over the previous two-year period divided by the total number of papers published in that journal over the same period. It’s therefore the average citations per paper published in a two-year window. The impact factor for 2019 would be defined using data from 2017 and 2018, etc.
The impact factor is prone to the same issue as the simple average I quoted above in that citation statistics are generally heavily skewed and the average can therefore be dragged upwards by a small number of papers with lots of citations (in our case just one).
I stress again we don’t have an Impact Factor for the Open Journal. However, for reference (but obviously not direct comparison) the latest actual impact factors (2018, i.e. based on 2016 and 2017 numbers) for some leading astronomy journals are: Monthly Notices of the Royal Astronomical Society 5.23; Astrophysical Journal 5.58; and Astronomy and Astrophysics 6.21.
My main point, though, is that with so much bibliometric information available at the article level there is no reason whatsoever to pay any attention to crudely aggregated statistics at the journal level. Judge the contents, not the packaging.
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In the Dark 