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In my November 2020 Blog we considered colliding galaxies; we saw that the number density of stars in the Galaxy was so small (just one star per 2.63 cubic parsecs) that collisions between stars are very rare events. Let’s look at a much smaller volume of space - the solar system. Here, number densities are much higher – there are eight major planets, thousands of asteroids and an unknown number of comets. Collisions are much more frequent, although less frequent now than in earlier epochs. Anyone who’s observed the Moon through binoculars or a telescope knows that the Moon’s surface has many craters. Craters are the result of impacts between massive bodies in evolving planetary systems. This is believed to be a fundamental process in planetary formation. The Barringer crater in Arizona (Figure 1) is the most perfectly preserved impact structure on Earth. The reasons this crater is so perfectly preserved include the very dry Arizona climate and the fact that the impact event happened very recently in astronomical terms – about 50,000 years ago. The crater is approximately 1.2km wide and 170m deep and was formed by the impact of a nickel-iron meteorite just 50m in diameter.  How could such a small object create a hole so much larger? The answer lies in the enormous kinetic energy of the impact. Kinetic energy scales linearly with mass and exponentially (specifically a square law) with velocity:  Typically, an impacting asteroid will have a velocity between 15 -30 km per second. The kinetic energy of the Barringer impact is estimated to have caused a blast equivalent to the detonation of a 10-12 megaton bomb. The main cause of damage after impact would have been due to the atmospheric shock wave. Two km from the impact site, the shock wave would have arrived approximately 6 seconds after impact. The peak overpressure would have been around 95.1 psi (normal air pressure is 14.7 psi). The maximum wind velocity would have been an astonishing 1360 mph (approximately Mach 1.8) and the sound Intensity 117 dB (i.e. threshold of pain). That’s quite a score sheet. But, as Table 1 shows, the Barringer event was actually a relatively small event in solar system terms. 
Simple and complex impact structures The Barringer crater is an example of a simple impact crater, having a bowl shape with a covering of shattered rock and mineral fragments. On Earth, simple craters are generally less than 4 km in diameter (Ball, Kelley and Peiser, 2007). Larger impactors produce complex impact craters. Large-diameter craters develop not only a central peak, but often one or more peak rings (French, B 1998) and also concentric ring structures. Many examples of this are seen on the Moon, such as the crater Tycho (Figure 2 and Figure 3)  Why are impact craters circular? One might conclude that if the impactor arrived exactly at 90° to the impact site, the crater would be circular. Otherwise it might be more oblate in shape. In fact, nearly all impact craters we observe are more or less circular, as shown by the examples in Figure 4 and Figure 5 below.  The basic mechanism of impact crater formation is an explosion rather than a ‘skid mark’. Earthquake or volcanic events can be quite geographically widespread, and particularly in the case of volcanic activity, take place over relatively long timescales. Impact events are concentrated at a single point on a planetary surface. The release of enormous amounts of kinetic energy takes place in the case of a small crater in a fraction of a second; and even in the case of a larger impactor in just a few minutes over tens or hundreds of kilometres (French, B 1998). Counting impact craters On planetary surfaces, the more craters there are, the older the terrain is believed to be. This is the case of the heavily crated regions of Mercury (Figure 6). However, there are other considerations as well. On Mars, the surface has experienced erosion as well as burial of craters (Figure 7). a surface covered with many small craters on Mars is often one that is more resistant to erosion, and not necessarily older.  Observation of impacts There have been quite a few impacts observed on Earth and elsewhere in the Solar system. A small meteorite impacted Mars’ surface sometime between September 2016 and February 2019 – the uncertainty being because the MRO can’t be everywhere at once. The impactor is estimated to have been about 1.5m in diameter and the resulting crater to be 15 to 16 meters in diameter (Figure 8).  Comet Shoemaker–Levy 9 was a comet that broke apart into 21 main fragments in July 1992 and collided with Jupiter in July 1994. This was the first time a cometary impact with a Solar system planet had been observed. As Jupiter is a gas giant, no crater was formed as such. However, the vast impact scars caused by the explosive entry of the comet were very evident (Figure 9).  The Chelyabinsk meteor was a small asteroid about 17 meters in diameter that struck Earth's atmosphere at an estimated 18km/second over the city of Chelyabinsk, Russia, on Feb. 15, 2013. The incident was captured on dashcam footage and the luminosity of the object was comparable to the solar luminosity. The atmospheric pressure shock wave caused major damage over a very wide area and over 1200 people were injured.
The largest meteorite fall recorded (NB ‘recorded’, not ‘happened’) in the UK occurred in the Leicestershire village of Barwell on the evening of Christmas Eve 1965. Several villagers did what any English person would do: they reported the matter to the Police, who duly took several fragments into custody. Subsequently, many fragments were found around the local area; the largest weighed over 7.7 kg so it was very lucky nobody was hurt. Among those to visit Barwell not long after the event was Patrick Moore (then, plain Mr. Moore, later Sir Patrick). He found a fragment of the meteorite and offered it to the local museum. He later said, “They told me ‘we have plenty of it so you can keep it for display as long as you make sure it comes to us in your will’”. There is a wonderful story about a Barwell resident whose car was damaged in the incident and he tried to claim off his insurance. His insurers helpfully told him it was an Act of God and therefore they were not liable to pay for the damage. So, he went along to the local church and said since it was an Act of God maybe they could pay, but they didn’t do so. References Katz, B (2019). An Ancient Asteroid Crater May Be Hiding Off Scotland’s Coast https://www.smithsonianmag.com/smart-news/ancient-asteroid-crater-may-be-hiding-scotlands-coast-180972393/ Accessed January 6th 2021. Matson, J (2010). Meteorite That Fell in 1969 Still Revealing Secrets of the Early Solar System. https://www.scientificamerican.com/article/murchison-meteorite/ Accessed January 6th 2021. Earth Impact Database (EID) Ball, A; Kelley, S; Peiser, B (2007). Near-Earth objects and the impact hazard. ISBN 978 0 7492 1887 4 French B. M. (1998) Traces of Catastrophe: A Handbook of Shock-Metamorphic Effects in Terrestrial Meteorite Impact Structures. LPI Contribution No. 954, Lunar and Planetary Institute, Houston. 120 pp. Hirata, N; Ohtsuki, K Keiji; Suetsugu, R (2020). A Huge ring-like structure on the surface of Jupiter’s moon Ganymede may have been caused by a violent impact https://www.kobe-u.ac.jp/research_at_kobe_en/NEWS/news/2020_08_05_01.html Accessed January 6th 2021.
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For me personally the ‘great conjunction’ of Jupiter and Saturn in December was a big disappointment because whenever I tried to make an observation the cloud to the SW horizon thwarted me. I hope some of you had better luck. However 2020 wasn’t all bad from an astronomical point of view as witnessed by the reminiscing at our online Christmas party of what had been observed throughout the year so let’s look forward to what this year will bring. Observing We had a broad look at the sky last month so we will focus in more detail on the winter sky facing south this month.The chart below represents the south facing night sky at 10.00pm on the 8th January and at 9.00pm on the 23rd January. No need for navigational help this month because Orion is so obvious but facing south and looking up you will find the bright star Capella just short of your zenith.  With clear skies we are in for a treat because we have seven of the twelve brightest stars visible from the northern hemisphere. You will be familiar with the chart above but I’ll fill in some details for completeness sake. We have already come across four of the constellations- Orion- The Hunter, Taurus- The Bull, Auriga- The Charioteer and Gemini- The Twins. The two new constellations are Canis Major- The Great Dog and Canis Minor- The Little Dog. In mythology they are the dogs of the hunter Orion but from an observational point of view these constellations are small with little to offer apart from their main stars, Sirius (alpha Canis Major, the brightest star visible from the northern hemisphere) and Procyon (alpha Canis Minor, the 6th brightest star). Incidentally they are two of our Sun’s closest neighbours, Sirius being 8.6 light years distant and Procyon 11.4 light years. These two stars along with Betelgeuse in Orion form an asterism known as the Winter Triangle depicted in yellow in the diagram.
But Betelgeuse is roughly in the middle of another asterism- the Winter Hexagon comprising the stars Sirius, Rigel, Aldebaran, Capella, Pollux and Procyon and depicted by the red outline in the diagram. It is obvious with the unaided eye that these stars are different and within that grouping, including Betelgeuse, you will find a yellow giant (binary twin), a red supergiant, a blue supergiant, a red giant, a yellow star and two stars which are part of a binary system with a white dwarf (not visible to the unaided eye). And allowing for variability they all have a magnitude of about 1 or brighter.If that doesn’t make you reflect on what you are looking at in the winter night sky I don’t know what will. Now that we are in lockdown again if you are not sure which star fits into which category why not do a little research to fill in your time of an evening! You may be thinking I’ve said nothing about Castor, the second bright star in Gemini, because it’s not as bright as the others but in fact it is an amazing star in its own right.To the unaided eye, the star Castor appears as a bright pinpoint of light but it’s actually three pairs of binary stars – six stars in all – in a complex dance about a common centre of mass. Even a fairly small telescope will show Castor as two stars and perhaps a glimpse of a much fainter star nearby, also part of the Castor system. Each of these three stars is also double but they cannot be resolved in a telescope and have to be inferred from spectroscopic data. Something to look out for Although the ‘great conjunction’ is now in the past, Jupiter and Saturn continue to be of interest as they have a close approach with the planet Mercury between the 9th and 14th of January, visible above the south-west horizon from around 4.30pm as darkness falls, and you need to be quick as they are visible for only a short time. On the final day they are joined by a crescent Moon. We will be saying ‘Goodbye’ to Saturn and Jupiter as they are lost to view behind the Sun but Mercury continues to its greatest elongation and highest altitude above the horizon on the 27th January. Let’s hope for some clear skies to show off the winter night sky to its best. We have been struggling a bit recently with observations but the close approach of the Moon and Mars produced the goods on the 25th November. It was good to see. Let’s hope ‘the great conjunction’ lives up to expectations but more on that later. Observing December is the last month in the year and the winter solstice takes place on the 21st so I thought it was time to take a look at the bigger picture rather than focussing on the detail and where better to start than the winter night sky.The charts below represent the whole night sky at 10.00pm on the 8th December and at 9.00pm on the 23rd December. The first chart is the night sky facing north.  Most people believe that it is a good idea that we all know some basic first aid. How to deal with cuts, bruises, stings and nose bleeds. More serious problems can be left to the professionals. By the same token I think everyone should know something about observing the night sky and I’ve come up with my list of five objects in the winter night sky which I think everyone should recognise. (Yes, I did have a job limiting it to five!). Obviously this is my personal list and some of you may come up with your own different selection. These are the things which I think you should be pointing out to your children and grandchildren on a crisp clear winter’s evening. It might just get them hooked on astronomy.
 This second chart is the night sky facing south. 3. Orion As I said last month, Orion has everything and I would defy anyone not to have it on their list of things to see in the winter’s evening sky. It is a brilliant object in the sky and attracts everyone’s attention. Its bright stars and distinctive belt are likely to produce questions so you might want to be able to name Betelgeuse and Rigel if asked. Like The Plough, Orion also leads you on to other objects like my next choice. 4. The Pleiades ( or Seven Sisters) The Pleiades doesn’t form a large object in the sky but it does catch the eye when looking at the sky hence on the list. Found by extending a line from Orion’s belt through the red star Aldebaran. Use a pair of binoculars to enhance the view. 5. Sirius Simply a star but it does happen to be the brightest star in the night sky when it is present so it should be up there with the tallest mountain and longest river as an item of general knowledge. It is also of historical interest because its appearance was used by the early Egyptians to tell that the river Nile was about to flood. Again easily located to the bottom left hand side of Orion. I’m sure many of you will have your own favourites and might be wondering why I have not included them but I wanted to restrict the list to five. Remember this is just a starting point. Something to look out for Mars continues to be easily observed during the evening but this month it will have to take second place to ‘the great conjunction’. First though, there is a close approach of a three day old Moon with Jupiter and Saturn on the 17th December. Visible above the south-west horizon from around 4.30pm as darkness falls, this will provide an opportunity to get your bearings right for the ‘great conjunction’ four days later. On the evening of the winter solstice, 21st December, the planets Jupiter and Saturn will be within about a tenth of a degree of each other. Remember that the Moon has an angular width of about half a degree so these two planets will be separated by about one fifth of the Moon’s diameter. Jupiter is much brighter than Saturn so they might be difficult to resolve. Of course these planets are widely separated in space and this optical effect comes about because Jupiter revolves around the Sun in just under 12 years whereas the orbital period of Saturn is over 29 years and so Jupiter laps Saturn every twenty years. This is the closest they have appeared since 1623 and won’t appear as close again until 2083 so definitely a once in a lifetime event.  I’ve included the chart just to show that Jupiter and Saturn will be close to the horizon and will set at 6.30pm. The Sun will set at 4.03pm so you will need to be outside during twilight to be ready to observe either side of 5.30pm. and you will need an unobstructed view to the south-west/west horizon. Let’s hope the skies are clear. The planets will still be within a degree of each other the weeks either side of the winter solstice so try to catch them whenever you can. The diagram below shows their separation over a two week period.  As 2020 draws to a close we can look forward to a once in 20 year astronomical event, the conjunction of Jupiter and Saturn, on the 21st of December. What is a Conjunction? Occasionally two or more objects meet up with each other in our sky. Astronomers use the word conjunction to describe these meetings. Technically speaking, objects are said to be in conjunction in that instant when they have the same right ascension on our sky’s dome. Practically speaking, objects in conjunction will likely be visible near each other for some days. The word conjunction comes from Latin, meaning to join together. In language, conjunctions relate to clauses brought together in sentences with words like and. In astronomy, conjunctions relate to two or more objects brought together in the sky as we see them. An inferior conjunction is when an object passes between us and the sun. Any object in space that orbits the sun closer than Earth’s orbit might pass through inferior conjunction from time to time, assuming its orbit lies more or less close to the ecliptic. Usually, though, when you hear the words inferior conjunction, astronomers are speaking of the planets Venus and Mercury, which orbit the sun inside Earth’s orbit. Astronomers sometimes refer to Venus and Mercury as inferior planets. When they’re at or near inferior conjunction, we can’t see them. They’re hidden in the sun’s glare. Occasionally, though, Venus or Mercury can be seen to transit across the sun’s disk at inferior conjunction. Consider also the moon. It passes between the Earth and sun at new moon once each month. Therefore you could say that the Moon is at inferior conjunction when it’s a New Moon. A superior conjunction is when an object passes behind the sun from our point of view. Think of Venus or Mercury again. Half of their conjunctions with the sun – when they are brought together with the sun on our sky’s dome – are inferior conjunctions, and half are superior conjunctions. It’s kind of fun to imagine them on an endless cycle of passing in front of the sun, as seen from Earth, then behind it, and back again, like watching squirrels running around a tree. Meanwhile, the superior planets – or planets farther from the sun than Earth such as Mars, Jupiter, Saturn, Uranus and Neptune – can never be at inferior conjunction.  Diagram showing the poistion of planets relative to each other during an inferior conjunction, superior conjunction and at opposition. However there are many more conjunctions that occur which fall into neither of these categories. They occur when planets, or a planet and the moon, appear close together in the sky. The conjunction between Jupiter and Saturn is one of these and is often called a Great Conjunction. How often do they happen? As a result of their long orbits, Jupiter and Saturn meet in the sky only once every 20 years. In this period of time, Saturn completes two-thirds of its 30-year orbit (since 20 is two-thirds of 30). In the same period, Jupiter completes one 12-year orbit, plus, in the remaining 8 years, two-thirds of its next orbit (since 8 is two-thirds of 12). In other words, 20 years is the time it takes Jupiter to catch up and pass Saturn again as they circle the Sun. The last conjunction of Jupiter and Saturn occurred on 28 May 2000, but was almost impossible to view because it occurred while the two planets were just 14.9° west of the Sun from our point of view. So the spectacle was largely lost in the Sun’s glare. After 2020, the next great conjunctions will occur on November 2, 2040 and April 7, 2060. On both these occasions, the minimum separation of Jupiter and Saturn will be 1.1 degrees—which means they will appear 11 times further apart from each other than on December 21, 2020. In fact, the 2020 great conjunction of Jupiter and Saturn is exceptionally close. Over a period of one thousand years, from 1600 to 2599, there are only six great conjunctions where the minimum separation between Jupiter and Saturn is less than 0.2 degrees: 1623, 1683, 2020, 2080, 2417, and 2477. Though each successive ‘great conjunction’ takes place around 117º apart in the sky, the planets’ orbital resonance is such that each conjunction returns to the same part of the sky roughly every 800 years. How and when to see it The conjunction actually occurs at 13:30 GMT – that’s when the right ascension of the two planets will be the same, but the Sun will still be high in the sky at that point. When the Sun sets at 16:05 (according to Heavens Above website) the two planets will only be 14 degrees above the south-western horizon and set less than 2hrs 30mins later.  Sky chart from Heavens Above for 17:00GMT on December 21st. Location set to Wells. But don’t just rely on clear skies on the night. The two planets are already nice and close together as we look at them so just in case the great British weather lets us down on the 21st make sure you take a look in the evenings leading up to the big night.  Image from Sky at Night magazine. Saturn and Jupiter appear to close in on each other as the Great Conjunction 2020 approaches on 21 December (apparent distance given in degrees and arcminutes). Credit: Pete Lawrence. Regular readers of my Blog will know that I come more from the data analysis side of astronomy than the pretty pictures side. So this month just for a change, we’ll have some wonderful images as well as data. So, let’s start off with … a pretty picture!  In fact, Figure 1 is far from being ‘just a pretty picture’ – this stunning image shows two galaxies, NGC2788 and NGC2789 which are in collision. How do we know these galaxies are interacting and that what we are seeing is not just a trick of perspective? If we look at the NASA Extragalactic database, we see the data in Table 1:  The first thing we can see is that both galaxies should be visible in modest sized amateur telescopes. The two galaxies have similar magnitudes, as shown above, although remember that each order of magnitude decreases brightness by approximately 2.5. The fact that NGC2798 and NGC2799 are close together is revealed by two parameters: their redshifts are similar within two parts in 10,000. And their Hubble distances with respect to the Cosmic Microwave Background (CMB) are approximately only 2.78% different. Will stars collide? The NASA/ESA press release that accompanied this picture states “While one might think the merger of two galaxies would be catastrophic for the stellar systems within, the sheer amount of space between stars means that stellar collisions are unlikely”. How can that be established? As a first approximation, we can model a spiral galaxy such as our Milky Way (MW) as a disk where the radius of the disk, is about 50,000 light-years or 15.33 kpc and the thickness of the disk is about 0.5kpc. The number of stars in the MW is estimated to be N ~ 10¹¹ We are now interested in parameter called the number density of stars - simply, the number of stars divided by the volume of the disk. However, we can’t work out the volume of the disk simply by calculating the volume of an equivalent cylinder. In reality, the disk of a spiral galaxy is not homogenous – it has been known since the’60s that the spiral arms are density waves (Lin & Shu, 1964). Let's assume 70% of stars are in the spiral arms; there are no stars in the voids between the arms; and that the arms make up 50% of disk. These are of course gross assumptions, good only as a first approximation. The results (calculations are available in Excel® if you are interested) are shown in Table 2 below:  In the ‘Single galaxy’ column the number density of stars is less than one star per cubic parsec; expressed differently: in one cubic parsec of space, there is on average less than one star. In fact, on average we’d need to search 2.63 cubic parsecs of space to find a single star. In the ‘Two similar colliding galaxies’ column, we imagine two spiral galaxies colliding head on. Here, the number density is doubled as the spiral arms collide. Correspondingly, the volume of space on average we expect to encounter a star is still 1.31 one cubic parsecs. Let’s just remind ourselves how large a volume of space a cubic parsec is. Imagine a cube of space one parsec on each side. That’s 3.26 light-years on each side or, if you prefer, 3.1*1013 km on each side.  Even a very large star such as the red supergiant Betelgeuse is a very small object in all that space, so even in a galactic collision, as the NASA press release says, the chances of two stars colliding is very small. How common are galaxy collisions? The answer is – not uncommon. The next two images show some well-known examples.  The image in Figure 3 shows the distorted disk of NGC4631. This is caused by NGC4631’s interaction with two much smaller galaxies, NGC 4627 and NGC 4656.  The plot in Figure 4 shows the results of radio emissions of NGC4361 reported in Neininger & Dumke (1999). The small dark object above the disk of NGC4631 is the dwarf elliptical galaxy NGC 4627 with which NGC4631 is interacting.  Another source of data about colliding galaxies is Galaxy Zoo, a project that enables citizen scientists to categorise and analyse images of galaxies taken by professional observatories. One of Galaxy Zoo’s initiatives is the Galaxy Merger project (Holinchech 2016). The data collection phase is now complete and comprises 62 colliding galaxy pairs (I have this data as an Excel table for anyone interested). A good example from the Galaxy Zoo merger table Is ARP148 in Ursa Major (Galaxy Zoo mergers table ID49). This is also called Mayall’s Object, named after American astronomer Nicholas U. Mayall (not John Mayall), who discovered the object in 1940.  Can it happen here? There is a short answer: ‘Yes’. For example, the Sagittarius Dwarf galaxy has been involved in multiple collisions with our Milky Way galaxy and this is the probable reason for the warped nature of the Milky Way’s disk (Law & Majewski, 2010). A recent determination of the radial velocity of the Andromeda galaxy, M31 with respect to the Milky Way indicates a velocity of −109.3 ± 4.4 kms per second, the negative sign indicating M31 is moving towards the Milky Way. The same research indicates a low transverse velocity of 17 km per second, indicating the probability of a head-on collision between the two galaxies (van der Marel, 2012)  The end result of this collision will be a very massive elliptical galaxy.
Acknowledgements This Blog is not a scientific paper, although it lists various scientific papers as sources in the References section. I am indebted to Hugh Allen for drawing my attention to the paper by Neininger & Dumke (1999). Data used in Table 1 was obtained from the NASA Extragalactic Database. The NASA/IPAC Extragalactic Database (NED) is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. References Holincheck et al (2016). Galaxy Zoo: Mergers – Dynamical models of interacting galaxies. MNRAS459,720–745 (2016). doi:10.1093/mnras/stw649 Law, D; Majewski, S (2010). The Sagittarius dwarf galaxy: a model for evolution in a triaxial Milky Way halo. 2010 ApJ 714 1. DOI: 10.1088/0004-637X/714/1/229 Lin, C & Shu, F (1964). On the Spiral Structure of Disk Galaxies. ApJ, vol. 140, p.646 . DOI: 10.1086/147955 Neininger, N; Dumke, M (1999). Intergalactic cold dust in the NGC 4631 group. DOI: 10.1073/pnas.96.10.5360 van der Marel, R et al (2012). The M31 Velocity Vector.II. Radial Orbit Towards the Milky Way and Implied Local Group Mass. ApJ, 753:8 (14pp), 2012 July 1. DOI: 10.1088/0004-637X/753/1/8 https://iopscience.iop.org/article/10.1088/0004-637X/753/1/8/pdf We haven’t been favoured with particularly good observational weather recently so let’s hope that with darker evenings following the change from BST to GMT we also get clearer skies. Observing The chart below represents the night sky at 10.00pm on the 8th November and at 9.00pm on the 23rd November. Best viewing of what is discussed will be towards the end of the month and going into December.  If you face south, as usual, and look directly overhead you will easily find the ‘W’ shape of the constellation Cassiopeia which we continue to enjoy on its journey westwards in the evening sky. Look to the west and you should see the bright star Deneb, the tail of the swan in the constellation Cygnus, as it flies to the western horizon. So, as Altair slips below the horizon and out of sight, it’s time to say goodbye to the Summer Triangle, but this month we are looking to the east because as one constellation sets in the western sky another one appears in the east.
This month we will have the arrival of the constellation Orion- The Hunter. It’s my favourite constellation because of its distinctive shape and because it appears to have everything. It doesn’t take much to visualise a hunter from the stars in Orion and what stars they are! Orion’s right shoulder is represented by the star Betelgeuse, a variable red supergiant, varying in magnitude from about 0.3 to 1.2 and the 7th brightest star in the northern hemisphere. If Betegeuse were to replace our sun it would reach out all the way to the orbit of Jupiter. It also has the potential of going supernova but of course we do not know exactly when. Then, representing his left foot, is the blue supergiant Rigel the 5th brightest star in the northern hemisphere with a magnitude of 0.2. Between these stars is a line of three stars going from south east to north west and they represent Orion’s belt and at magnitudes of around 2 they are unmistakable. Less bright but still visible to the unaided eye is Orion’s sword hanging from his belt. The bottom star of the sword should be visible in good conditions and above this is a misty fuzzy patch which is the Orion nebula (aka M42) where star formation takes place. Try to observe it through binoculars or a telescope if you get the chance. Because it is so easily recognisable, Orion is a good starting point for finding your way about the night sky during the winter months. Follow a line from Orion’s belt to the upper right, underneath the star Bellatrix representing his left shoulder, and you will find the star Aldebaran, a giant red star of magnitude 1 and the 9th brightest star in the northern hemisphere. It is in the constellation Taurus- The Bull, and is said to represent the angry eye of the bull. The ‘V’ shape of stars outlining the bull’s face is an open star cluster called the Hyades. Continue the line beyond Aldebaran and you find the better known star cluster- The Pleiades or Seven Sisters. Remember back in springtime we watched Venus pass close to the Pleiades which catches the unaided eye but much more is revealed if you use a pair of binoculars. Bright stars are like the proverbial bus, you wait ages to see one then four come along at once. Our fourth star this month is Capella in the constellation Auriga- The Charioteer, lying directly above Taurus. Having followed the line to the Pleiades turn ninety degrees to the north and the bright star you see is Capella. It is the 4th brightest star visible in the northern hemisphere and shines at magnitude 0.1. Auriga is in the shape of a pentagon although the most southerly star is in Taurus. As mentioned earlier the constellations and stars described above are presently in the east in the evening and will be better viewed later on but are highlighted so that you can enjoy them throughout the winter months. Something to look out for Mars continues to be an attraction and it will have a close approach with the Moon on the 25th November and this will visible throughout the evening. We often say that the planets in the solar system ‘orbit the Sun’; or that the Moon ‘orbits the Earth’. But what exactly is an ‘orbit’? Geocentric theories For many centuries it was believed that the Earth was at the centre of the universe and that all the stars and planets revolved in circular motion around the Earth. It is a curiosity that this theory, usually attributed to Ptolemy, persisted so long. It could only explain the observed pattern of movements of the planets by explaining that they moved in epicycles – effectively circles within circles, but how this movement came about was unexplained. Copernicus and heliocentric models The Prussian Nicolaus Copernicus proposed a theory of a heliocentric system, with the Sun at the centre of the universe, in which Earth was one of the six planets known at the time. Copernicus was not the first to espouse this theory, but was the first to extensively document it in his book published shortly before his death in 1543. As in the Ptolemaic system, Copernicus’ orbits were circular, so epicycles still had to be invoked to explain some of the observed motions of the planets. In fact, more epicycles were needed in the Copernican system than the Ptolemaic system, and yet there were still unexplained gaps in explaining astronomical observations. Kepler’s breakthrough Johannes Kepler, a prolific German astronomer and mathematician made the key breakthrough in the study of orbital motion by a combination of observation and deductive reasoning empirically deriving three laws which stand in good stead to this day. Kepler’s key discovery was that planetary orbits are NOT circular, but elliptical. An ellipse is an example of a conic section. In case you’re not familiar with conics, and with terms such as ‘semi-major axis’, ‘eccentricity’ and ‘focus’, the next section contains a quick primer on the subject. If you are familiar with conics, you could skip this and go to the following section.    Newton’s Laws of Gravitation: formalizing Kepler’s laws Due to the operation of gravity, two massive objects will move in space-time around their common centre of mass, known as the barycentre. This was established by Sir Isaac Newton who defined his laws of gravitation, our first formal description of gravity. This provides our first definition of what an orbit is. Newton’s Laws added formal mathematical reasoning to Kepler’s empirically-derived Laws. In his words: ”I deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centres about which they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the Earth; and found them answer pretty nearly.” Kepler’s First Law Planets orbit on elliptical paths, with the Sun at one focus of the ellipse. As we’ve seen, an ellipse is an example of a conic section. More precisely, Newton defined the focus as being at the barycentre of the Sun–planet pair. So, using Newton’s Laws, how can we determine where the barycentre is? Let’s consider a very theoretical solar system consisting of just the Sun and Jupiter. With only two objects in a system, it's simple to locate the barycentre of the system around which the two bodies orbit. The Sun’s mass, M⊙=1.99*1030kg. Jupiter’s mass, MJ=1.898*1027kg. Using Newton’s Laws, we have:  Which means the ratio of Jupiter’s mass to the Sun’s mass is about 1/1,000. In turn that means that the distance from Sun to the system's barycentre approximately 1/1,000 times the distance from Jupiter’s distance to the barycentre. The mean distance from the Sun to Jupiter is 7.785*1011m; we shall see how to work that out in a moment, using Kepler’s Third Law; please accept it for now. One thousandth of that distance is 7.785*108m. The radius of the Sun is slightly less: 6.96*108m. This means the barycentre of our very simplified two-body solar system is just above the Sun’s surface. In the real solar system, consisting of the eight major planets, several known minor planets, asteroids etc., the position of the barycentre of the whole system changes all the time. Jupiter is by far the most massive planet and so has the greatest influence on the position of the barycentre. However, the alignment of the other solar system bodies is constantly changing as the planets orbit at different speeds (see Kepler’s Second Law below). So the relative influence of their respective masses changes the position of the barycentre constantly. Technically, this is known as an “n-body problem”, the mathematics of which is very complicated, so we’ll conveniently declare outside the scope of this Blog. Kepler’s Second Law Planets sweep out equal areas in equal times.  What this means is that the angular momentum is conserved throughout the course of the orbit.  If the orbit was circular, then the distance of the orbiting body from the primary, r, would be constant. However, since the orbit is elliptical, then r changes. The mass, m is constant, so the velocity, v must change to keep the angular momentum constant. In practice,
Kepler’s Third Law The orbital period squared is directly proportional to the size of the semi-major axis cubed.  Where k is a constant. In his gravitational theory, Newton formalized this constant as:  Where M1 and M2 are the masses of the two bodies. As we have seen, in the solar system even the largest planet, Jupiter, has a mass of only about 1/1,000 solar masses, and this simplifies the equation to become:  However, this can be simplified even further if we pick our units carefully. If the orbital period, P is in Earth years (yr); and the semi-major axis, a is in AU, then by definition k=1. So we don’t even need to know the mass of either the star or the planet to calculate the furthest orbital distance between them.  Let’s again take the case of Jupiter. From observations, we know that the orbital period of Jupiter, PJ=11.86 yr. So:  So the mean distance of Jupiter from the Sun is 5.20AU is approximately 7.78*1011m, just as we assumed when describing Kepler’s First Law above. Putting it all together
 This plot clearly demonstrates the huge effect Jupiter has on the solar system. Jupiter’s mass is greater than the sum of the masses of all the other major planets. Notice that Mercury has by far the greatest orbital eccentricity, despite being closest major planet to the Sun. The huge mass ratio between the Sun and Mercury would mean a near circular orbit were it not for the influence of other large bodies – most notably Jupiter.
The high orbital eccentricity of Mars is due to the high mass ratio between Jupiter and Mars, and Mars mean orbital distance being relatively close to the mean orbital distance of Jupiter. Saturn, despite being further from the Sun, has a higher orbital eccentricity than Jupiter, due to the more massive planet’s gravitational influence. Uranus and Neptune, each orbiting further from the Sun than the mean distance at which Jupiter orbits the Sun, are less gravitationally influenced by Jupiter, and so have lower orbital eccentricities. Less easy to explain is the very low orbital eccentricity of Venus – the lowest of any planetary body in the solar system. This may have something to do with Venus’ retrograde rotation affecting its angular momentum. I’d be very interested in comments on this (NOT a trick question: I don’t have the answer!)
Newton’s Laws were unable to account for the orbital precession of Mercury, yielding values well below the observed movements. GR predicts this to very close accuracy (see my August Blog).
September wasn’t a particularly good month weatherwise but on the 5th, Mars and the Moon were good to see between the rolling clouds. Mars has continued its retrograde motion and is now below the lower arm in Pisces (more about that later) while Jupiter and Saturn have continued westward in the evening sky. Poor weather prevented the viewing of the setting Sun on the equinox so will have to wait till the spring equinox in March to fix due West. Observing I was rather dismissive of two of the watery zodiacal constellations, Aquarius and Capricornus, in last month’s blog and as a ‘fishy’ constellation features this month it is probably time to say something about stellar magnitudes. It is OK looking at a stick presentation of a constellation in a diagram but they don’t look like that in the sky! Originally the brightness of a star was classified on a scale of 1 to 6, 1 being the brightest and 6 being just visible to the unaided eye. (Note the the bigger the number the dimmer the star. In the modern scientific era measurements have shown that a difference in magnitude of 1 means the brightness differs by a factor of 2.5, ie a magnitude 2 star is two and a half times as bright as a magnitude 3 star and a magnitude 1 star is one hundred times brighter than a magnitude 6 star. Really bright objects have a negative magnitude). But that was over two thousand years ago in the Middle East with clear skies and no light pollution. What can we expect to see today at a latitude of about 50 degrees North with the associated weather that brings and the light pollution from modern towns and cities. If we face South and look above us just past our zenith we see again the reassuring sight of the ‘W’ shape of Cassiopeia. The three brightest stars Caph, Schedar and Navi are close to magnitude 2 and clearly visible while epsilon Cas on the extreme left is magnitude 3.4 and considerably dimmer but easily visible if conditions are reasonable. (See diagram below). If we drop down to the horizon towards the right we locate as we did last month the Great Square of Pegasus with the star Alpheratz at magnitude 2.1 and Algenib about half as bright with the other two stars of the square in between. The four stars stand out because they are in a fairly empty part of the sky. Now for the tricky part. Again as we did last month, starting from Alpheratz we look for the two curved strings of stars which make up Andromeda. The lower string isn’t too bad because from Alpheratz; delta Andromeda, Mirach and Almach have magnitudes 3.3, 2.1 and 2.2 respectively so no problem. However the higher curved string of stars from Alpheratz; pi Andromeda, mu Andromeda and 51 Andromeda have magnitudes of 4.3, 3.9 and 3.6 respectively. We are now in a situation where poor atmospherics and light pollution become critical if the stars are to be visible to the unaided eye. For comparison, if you are trying to locate the Andromeda galaxy, M31, it has a magnitude of 3.4. If you are struggling to see the fainter stars even in a clear sky you need to leave the comforts of your home and find a more rural dark sky site. Sorry! That’s all my excuses made now so we can return to sky gazing. Below Andromeda and to the south east of the Great Square of Pegasus lies the constellation Pisces- The Fish. Supposedly two fish, one the Circlet and the other the group of stars to the East of Alpheratz, tied together with a ribbon. I use the word ‘lies’ advisedly because unfortunately only two stars in Pisces are brighter that magnitude 4 and even then, only just, so it is unlikely that you will see anything if you are in your back garden! If any readers follow their ‘stars’ in the newspapers or were born under the star sign Pisces perhaps now is the time to consign astrology to the rubbish bin. Why did I bother to mention Pisces? At present the planet Mars is in Pisces and at magnitude -2.3 it is more than a hundred times brighter than a magnitude 3 star and outshines anything nearby. It will have a close approach with the Moon on the 3rd October just three days after the full Moon. It will be at its closest to the Earth on the 6th October and at opposition (on the far side of the Earth from the Sun) on the 14th October so visible all night. Now that is something to look forward to. The diagram below has more named stars than usual not because they are bright but because I used them in the text to explain the variation we see in stellar magnitudes and again I have omitted some minor star groupings to help with clarity.  With the idea of stellar magnitudes firmly in mind let us look at three further constellations. To the southeast of the lower string of Andromeda and due East of the Great Square of Pegasus is another zodiacal constellation, Aries- The Ram. (Remember you probably cannot see anything in Pisces apart from Mars). Aries contains two brightish stars, Hamal at magnitude 2.0 and Sheratan at magnitude 2.7 which are readily seen but there is not much more. How you make the shape of a ram from that I do not know. However Aries has a claim to fame in that it was the location of the spring equinox about two thousand years ago and that event is still called the ‘first point in Aries’ even though it is now in Pisces.
Between Aries and Andromeda is the constellation Triangulum- The Triangle. It has the great redeeming feature that it is what it says on the tin- a triangle! However it doesn’t have any stars brighter than magnitude 3 but because of its compact nature it is readily recognisable if seen. Finally to the northeast of Aries and Andromeda and southeast of Cassiopeia is the fairly prominent constellation Perseus- another hero from Greek mythology. It contains the stars Mirfak and Algol both around magnitude 2 and another five stars around magnitude 3 or brighter. Something to look out for As mentioned above Mars is going to be the major attraction in the night sky this month so don’t miss it and see if you can follow its retrograde motion to the first week in November. ( I used my binoculars to locate eta Pisces and epsilon Pisces). If you want to see a ‘falling star’ your best chance will be on the 21st October when the Orionid meteor shower is at its peak. At the end of the month there are two lunar close approaches to look out for. The Moon and Jupiter on the 22nd and the Moon and Mars on the 29th. Clear skies and happy viewing. My attempts to see the Perseid meteor shower were thwarted by cloud cover but the close approach of the planets Jupiter and Saturn with the Moon at the beginning of the month was good to see. At the time of writing there has been a lot of cloud cover so I’m not very optimistic about seeing the close approach at the end of August. Observing I hope you are keeping your eye on Cassiopeia on its journey west because it is approaching its optimal viewing position and with the Plough low in the northern sky, Cassiopeia is better placed to help us find our way among the stars. Also it is just lovely to look at! The diagram below seems to contain a lot this month but some of it you are already familiar with and I have omitted some minor star groupings to help with clarity. As usual we start facing south and look up to the zenith and just short of it and to the right hand side we see the bright star, Deneb, the tail of Cygnus the swan and along with Vega and Altair we quickly pick out the Summer Triangle. From the line joining Deneb and Altair turn left by about 45 degrees to look east of south and you will spot the asterism, the Great Square of Pegasus, which stands out not because of the brightness of its stars but because it is away from the Milky Way and there are few stars visible in this area. This asterism is part of the constellation Pegasus (the winged horse in Greek mythology). Again it is difficult to imagine a horse and no obvious signs of wings. Pegasus is quite a large constellation but its other stars do not stand out as much as the square. The square isn’t actually a square and to add insult to injury the star, Alpheratz, at the top of the square isn’t part of Pegasus! However on a positive note, the Great Square of Pegasus is easily picked out and is another good signpost to help us find our way around the skies.  Courtesy In-in-the-Sky.org edited by Brian Davidson Alpheratz, is part of the constellation Andromeda (the princess, daughter of the mythological Queen Cassiopeia and King Cepheus) and being the brightest star is also referred to as a And (alpha Andromedae). The main features of Andromeda are two curved strings of relatively faint stars meeting at Alpheratz and it is readily found because of its association with Pegasus. The constellation Andromeda is home to one of the most famous objects in the sky- the Andromeda galaxy also known as M31. It is marked on the diagram with a red cross and labelled M31. The Andromeda galaxy is the nearest large galaxy to Earth and is similar in many ways to our Milky Way galaxy and is the only one visible to the unaided eye in the northern hemisphere. To locate it for observing, (your eyesight needs to be pretty good), start at Alpheratz and by star hopping jump to the second pair of stars along the curved strings and the Andromeda galaxy will be to the right hand side at a distance approximately equal to the distance between the two stars. There is no rush as Andromeda will be in a good position right through till November. Now some mind boggling statistics- the distance to Andromeda is about two and a half million light years which means that the light entering your eyes from Andromeda set out two and a half million years ago, about the time the first members of the genus Homo appeared on Earth using stone tools and long before Homo sapiens arrived on the scene! Andromeda is the most distant object you can see with the unaided eye but you will need a dark site with no light pollution and clear skies. Don’t expect to see something like the images shown in the gallery of the WMA website, you will have to settle for something which might be described as a smudge or fuzzy star but that doesn’t detract from the sense of achievement. Good luck! If we drop down from the Great Square of Pegasus along the diagonal from Alpheratz to the horizon we find another zodiacal constellation, Aquarius (the Water Carrier). Unfortunately to the unaided eye Aquarius has no bright stars and is of an indistinct pattern. Since antiquity it has been seen as a figure pouring water from a jug but I am obviously lacking in imagination. However I remember a time in the late sixties when any radio channel you switched to was likely to be filling the air with the song ‘Aquarius’ from the musical ‘Hair’ or from a version by an American pop group. It was a song to cheer you up and had the memorable line, ’This is the dawning of the age of Aquarius’. This might lead us in to discussing ‘the precession of the equinoxes’ but luckily this has already been done in a recent blog by Gordon Dennis (Dennis, July 2020). Now follow the line from Vega through Altair down to the horizon and you will find the right hand edge of another zodiacal constellation, Capricornus (the Sea Goat, associated with many myths from ancient times) just to the south east of Aquarius. This constellation is relatively small and the second faintest zodiac constellation so doesn’t have much to offer the casual observer and I can’t recall a pop song called Capricornus. However, precession is a slow process so there might be one by the year 4750 give or take a few hundred years! Let’s have a grand tour. Find the Great Square of Pegasus and follow the left hand side of the square upwards past Andromeda on your left till you see Cassiopeia. From the star Navi go across the top of the constellation Cepheus to the pole star, Polaris, in Ursa Minor, and continue in a straight line to the star Alioth in the handle of the Plough, part of Ursa Major. Just as before, follow the arc of the handle down till you find the bright star Arcturus in the constellation Bootes then follow round eastwards to spot the Keystone of Hercules not missing out the small but attractive constellation, Corona Borealis, on the way. One more step eastwards and you are back at the Summer Triangle comprising Vega in Lyra, Deneb in Cygnus and Altair in Aquila from where we found the Great Square of Pegasus at the start. Now give yourself a pat on the back because you have gone round half the visible night sky and identified twelve constellations. Erratum: In last month’s blog, ’Looking to the Skies August 2020’, the line pointing to the variable star delta Cep should have gone past the first star to the top right hand one of the three stars in the bottom left hand corner of Cepheus. Sorry for the ambiguity. Something to look out for This is the month of the autumn equinox when as the Earth revolves around the Sun, the Sun’s apparent path round the ecliptic crosses the celestial equator and the Sun is overhead at the equator and everywhere on earth has almost equal amounts of daylight and darkness. It will happen on the 22nd of September but of course it happens at a specific time which is about a quarter of a day later each year until it is corrected for with the extra day of a leap year and so the date of the autumn equinox can vary by a day or two but it is usually on the 22nd or 23rd. Another feature of an equinox is that the sun rises due east and sets due west so note some landmark on the horizon in line with the sun at sunrise and sunset and you have your east and west directions. As far as astronomers are concerned it means we are into Autumn with more starry evenings. There are exciting times ahead because the planet Mars is going to be a major attraction in the night sky in the coming months. Try to catch it early in the month because it rises in the east with the Moon on the 5th September at 9.30pm BST. I say catch it early in the month because there is a little project you might enjoy doing- observing the retrograde motion of Mars. By the 10th of September it stops its apparent eastward motion against the background stars of Pisces and then appears to move westwards until November. Use the stars nPsc, mPsc and ePsc on the lower branch of Pisces as your guide. It should pass between the first two of these stars about the 30th September and it will be getting brighter throughout the month. I hope the diagram clarifies the situation. The orange line is the path which Mars will follow during its retrograde motion from 1st September to the first week in November.  Courtesy In-in-the-Sky.org Finally there will be a conjunction, I use the term loosely to mean a coming together rather than the more technical definition, of the Moon, Jupiter and Saturn on the 25th September, best viewed to the south just after 8.00pm BST.
Hertzsprung-Russell diagrams In July’s WMA webinar, we looked at Hertzsprung-Russell diagrams. H-R diagrams are one of the most important tools available in stellar physics, since they indicate a great deal about the characteristic of stars. The H-R diagram below, known as a theoretical H-R diagram, is a scatter plot of stars photosphere temperature vs. luminosity. Brian Davidson’s last couple of Blogs have mentioned the Summer Triangle asterism, consisting of Vega, Deneb, and Altair. Taking a sample of ‘nearby stars’ as shown below we have marked the Summer Triangle stars on the H-R diagram, showing how different these three stars are:  The spectral classes of the stars are indicated by the familiar ‘OBAFGKM’ legend. Before looking at the three stars, look at the track running from Alnitak at the top left to Proxima Centuri at the bottom right. This track is the Main Sequence, where in the stars interior hydrogen is converted into helium by nuclear fusion. Stars leave the main sequence once about 11% of the hydrogen-mass has fused to helium and the core of the star becomes unstable. About 90% of stars are on the main sequence. It may not appear like this from the diagram, but that’s because of our sample, which is ‘nearby stars’. Altair is a main sequence star, larger and hotter than the Sun. It will therefore have a shorter main sequence life than the Sun. Altair’s luminosity is 10.6 L⊙. Recall that luminosity is a measure of the power radiated by a star; the unit of luminosity is the Watt. Vega has a photospheric temperature similar to that of Altair. But Vega’s hydrogen burning phase is now over. The star has entered the main sequence turnoff, on the way to becoming a red giant, before shedding its outer layers to form a planetary nebula. At that time, what will remain of Vega is a hot white dwarf star at the bottom left of the H-R diagram. This sequence of events will also be what happens to the Sun when main sequence turnoff occurs. Deneb is a super-giant star. Although Deneb’s Photospheric temperature is comparable to both Vega and Altair, its luminosity is very much greater. Deneb has a luminosity of more than 104 L⊙, a mass of 19M⊙ and is destined to end its life in a Type II supernova event. This will leave a supernova remnant perhaps like the Crab Nebula, M1 and a neutron star which is way off scale at the bottom of the H-R diagram. Open clusters and globular clusters For this month’s Blog, we’ll consider a question asked at the July WMA webinar, “Can we make H-R diagrams of star clusters to help determine their characteristics?” The answer is yes. Astronomers are familiar with both open clusters and globular clusters. Their main characteristics are shown in the table below.  How can we conclude that open clusters consist of young stars and globular clusters consist of old stars? The amount of hydrogen that a star has available for fusion is directly proportional to the star’s mass. In simple terms, the greater the mass of hydrogen packed in, the faster the reaction rate, and the higher the luminosity. The star’s luminosity determines how quickly the star will fuse the hydrogen into helium, and hence how long the star lives on the main sequence according to the relation:  Since from the mass-luminosity relation we know that:  Substituting  The diagram below summarises how stars of different spectral classes leave the main sequence - the “main sequence turnoff” - as they evolve:  Plugging the numbers into the equations, this means that a star of 10M⊙ will have a lifetime of only about 13 million years. Bear in mind that we know that about 80% of stars are red dwarfs, smaller than the Sun. A low mass star of about 0.6M⊙ has a life of ~34 billion years. That time is much greater than the age of the universe which means that no low mass star has yet completed its main sequence lifetime. H-R diagram for open cluster M45 So, let’s plot an observational H-R diagram, (also known as a “colour-magnitude diagram”) for open cluster M45, known of course as the Pleiades:  The observational H-R diagram above is a plot of absolute magnitude (VMag) vs. colour index (BMag-VMag). The scale ranges are (x: colour index 0.2 à 1.45; y: absolute magnitude 8 à -2) . We can see at the top left of the H-R diagram, only a few of the larger, more luminous (which of course implies more massive) stars in the Pleiades have begun their main sequence turnoff. These are the dots at the top left which are turning upward and to the right. The majority of the (less massive) stars in the plot remain very much on the main sequence. They are so young that hydrogen burning has a while to progress. It is generally thought that open clusters disperse after a short time (in cosmological terms) before the stars in them have commenced main sequence turnoff. It is also thought that then Sun formed in an open cluster which subsequently dispersed and that this accounts for the fact that the Sun is isolated and not part of a multiple star system, although that is far less certain. H-R diagram for globular cluster M14 Now, let’s look at the H-R diagram for globular cluster M14. This H-R diagram is markedly different to that of M45. The plot is a subset of the total data of over 1,000 stars and is plotted with the same scale ranges as the M45 plot.  In the M14 H-R diagram, just about no main sequence stars are evident. The reason is that most stars in M14 are very old, and have completed hydrogen burning and moved off the main sequence. Low mass stars are either ascending the red giant branch or have already become red giants. Like Altair, Vega, and the Sun, they will end their lives as white dwarfs. A few at the top right of the H-R diagram are supergiants and, like Deneb, will finish their lives in Type II supernova events.
The fact that the majority of stars in this globular cluster are grouped at the red end of the colour index confirms the generally red appearance of the globular cluster. References Data sources Contreras Pena, C et al (2013). The globular cluster NGC6402 (M14). A new BV color-magnitude diagram. ApJ, September 2013. DOI: 0.1088/0004-6256/146/3/57 Accessed August 10th 2020 Australia Telescope National Facility |
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