The better weather has provided more chances of viewing the night sky but my attempts to catch Mercury at the end of April just after sunset were thwarted by hazy clouds on my western horizon. Also, on the 24th April, the weather wasn’t favourable for seeing the pre-dawn planetary alignment.
 
Observing
Okay, it is time to look at the stars. The following charts represent the night sky at 10.00pm BST on the 8th of May and at 9.00pm BST on the 23rd May.

We looked at The Plough last month and we’ll start there this month as it is such a good signpost for finding our way about the night sky. You will find the middle of the handle near your zenith (directly above you) so, facing south, follow the arc of the handle of The Plough downwards to the brightest star in that region, Arcturus, which has the distinction of being the second brightest star visible in the northern hemisphere at a magnitude of about 0.2. It is an orange giant nearing the end of its life and is relatively close at 36 light years. It is the brightest star in the constellation Bootes- The Herdsman but it is difficult to distinguish such a figure whereas the Kite asterism is easy to see and is what most people recognise as Bootes.
Also marked on the chart is the second brightest star in Bootes, Izar, a binary star consisting of an orange giant and a blue star, well worth a look through a telescope if you get the chance.
In the second chart you should immediately recognise the constellation Leo- The Lion which we looked at in March. Between Bootes and Leo lies the constellation Coma Berenices supposedly representing the hair of Queen Berenice of Egypt but it contains no stars brighter than magnitude 4.

Carry on following the arc from Arcturus for about the same distance again until you see another bright star. This is Spica the brightest star in the constellation Virgo- The Maiden, which is one of the zodiacal constellations. Spica is a blue-white star with an average magnitude of about 1 and is 260 light years from Earth. It may be easier to memorise these two star hops using the expression (Arc on to Arcturus and Speed on to Spica).

Something to look out for

As previously mentioned the planets are not favourably positioned for viewing at present so I am repeating the orbital diagram I showed in February. You can see that the planets Venus, Mars, Jupiter and Saturn are visible only in the morning and not visible at all during the evening or nightime. At present, as the Earth rotates on its axis, these planets are positioned such that they rise in the east just before the Sun which proceeds to outshine them when it rises. Now the outer planets orbit more slowly than the Earth and have further to travel during an orbit ( Jupiter takes about thirty Earth years to complete its orbit) so what has to happen is that the faster travelling Earth must overtake them and position itself between them and the Sun and once again they will appear in our night sky in a few months.
            You can still look out for Mercury to the West just after sunset. Otherwise enjoy the Moon and the constellations.
Clear skies.

We passed the spring equinox on the 20th March so we are getting more daylight and the clocks went forward an hour on the 27th March so it will be later before the sky darkens. The good news is that we have had some clear nights and seeing the winter sky has been really good. Like last year, I thought that it would be a good idea to repeat the bit about the celestial sphere and how the sky changes in appearance from night to night and month to month. I hope this will prove useful to any newcomers to the subject and any youngsters who are hopefully embarking on observing the skies as a lifetime’s hobby.
 
The Celestial Sphere
Before we venture outside let us recall some helpful facts. It is useful to think of the sky as a hollow sphere which has the Earth at its centre and to which all the heavenly objects are attached. This sphere is known as the celestial sphere. Just like when you visit a planetarium. The celestial sphere also has north and south poles directly above the corresponding poles on Earth and a celestial equator directly above the Earth’s equator.  Far away objects such as stars and galaxies are in more or less ‘fixed positions’ on the celestial sphere whereas the Sun, Moon and planets continually shift their positions but stay close to a circular path on the sphere’s surface called the ‘ecliptic’ which is tilted to the celestial equator because the Earth’s axis is tilted by 23.5 degrees to the plane of its orbit. In reality of course the Earth revolves round the Sun and the ecliptic is where the plane of the Earth’s orbit cuts the celestial sphere. This makes sense because when we observe the Sun we are looking along the radius of the Earth’s orbit and hence in the plane of its orbit.

The recent equinox marks the point where the path round the ecliptic crosses the celestial equator. This is when the Sun is overhead at the equator and it continues to travel further north until the summer solstice when it is overhead at the Tropic of Cancer. We see from the diagram that the ecliptic is north of the celestial equator during this period of time.
For us in the northern hemisphere we see the stars rotate about the north celestial pole. Don’t worry about some of the additional information on the diagram. The yellow line is the ecliptic and it shows the signs of the zodiac (representing the constellations) and how the Sun appears to pass in front of them as the Earth revolves around the Sun. Remember we are using a model for what we see and this is governed by the movement of the Earth. The Earth spins about its axis from West to East once a day (i.e. 360 degrees in 24 hours or 15 degrees per hour) and that is why we see the Sun move across the sky daily from East to West. It may not be so obvious that the stars are doing the same thing at night and they move across the sky from East to West at 15 degrees per hour as well. Of course, they also do it during the day, but we cannot see them for the glare of the Sun.
 The Earth also revolves about the Sun once a year (i.e. 360 degrees in 365 days or about 1 degree per day or 15 degrees in 15 days) which is why the sky at 10.00pm one day will look like the sky at 9.00pm 15 days later. If you wait till 10.00pm again the celestial sphere has moved on by 15 degrees or 1 hour and all the stars have moved that amount further west.
 
Observing
Okay, it is time to look at the stars. The following charts represent the night sky at 10.00pm BST on the 8th of April and at 9.00pm BST on the 23rd April.

The fact that some stars appear in a group does not indicate that they are close together and their distances can vary by very large amounts. Some groups of stars stand out but may be only part of a constellation and such groupings are called ‘asterisms’. So we will begin with possibly the best known one- The Plough. Start by facing north east and you will readily see The Plough high in the sky standing on its handle. It contains seven stars and the chart shows three of them named. The Plough is part of the constellation – Ursa Major- The Great Bear, but it takes a lot of imagination to see a bear and that region is mostly referred to as The Plough. In North America it is called the Big Dipper and perhaps here in the UK a better name in modern times would be ‘The Pan’. We said in the introduction that the stars rotate about the celestial North Pole and stars close to there never set but are visible all year round when the skies are dark. Stars like this are said to be circumpolar and Ursa Major is a circumpolar constellation. But note The Plough’s orientation carefully because as it continues on its circular journey it will appear upside down in six months’ time as it travels anti-clockwise about the north celestial pole.
            The constellations are used as signposts in the sky and enable us to engage in a fun activity called ‘star hopping’.
The two stars in the Plough, Merak and Dubhe, are called the pointers and a line from Merak to Dubhe continued onwards (shown in yellow on the chart) leads to Polaris- the Pole Star. The distance is about x5 the distance between Merak and Dubhe. Polaris is very close to the celestial north pole and easily found because although not very bright it is the only star visible in that area. Polaris is in the constellation- Ursa Minor- The Little Bear.                       Now consider a line from the star Alioth in the Plough, through Polaris and continued onwards for about the same distance again (also shown in yellow on the chart) until you see a bright star. It will be the central star of a W (or M) formation, an asterism in the constellation Cassiopeia- Queen Cassiopeia in Greek mythology. Most people see the W shape and call it Cassiopeia. The bright star was never given a name in Western or Middle Eastern culture so is referred to as gamma Cas. The convention is to name stars using the letters of the Greek alphabet and an abbreviated form of the constellation. Generally this is done in the order of brightness of the star but it is not a hard and fast rule. However this star has been given the name Navi, allegedly by the American astronaut Virgil (Gus) Ivan Grissom as an anagram of his middle name because it was used for navigation in the early space missions. A fitting tribute to someone who made the ultimate sacrifice for space exploration. The constellation Cassiopeia is also circumpolar and because it is directly opposite the Plough across the North Celestial Pole the two will have exchanged positions in six months so we will see Cassiopeia much better in November. Just imagine the two of them at the ends of a long pole rotating about the North Pole. Do have a look at them from month to month so that you become familiar with their orientation.
 
Something to look out for
As previously mentioned the planets are not favourably positioned for viewing at present but on the 24th of April Saturn, Mars, Venus and Jupiter are all in a line before dawn as the Sun rises in the east. I think I might just get up early to see that!
            However the elusive planet Mercury is making an evening appearance and will be at its highest point in the sky on Thursday 28th April. It will be well placed and shining at magnitude 0.2 but still challenging to see close to the horizon at sunset.
Clear skies.
 

The good news about the James Webb Space Telescope continues because they have obtained satisfactory images of an object from all eighteen segments of the mirror and are now in the process of synchronising them to produce a single composite image. Roll on summertime when they start receiving data.
 
Observing
The following chart represents the night sky at 10.00pm GMT on the 8th of March and at 9.00pm GMT on the 23rd March. To use the chart, face south at the appropriate time with the bottom of the chart towards the southern horizon and you will see the stars in the chart. If you are observing a little earlier in the evening then the view is shifted 15 degrees eastwards for every hour before the specified time.

I promised you an old favourite last month and nobody can be disappointed with the constellation Leo- The Lion. It is well known because it is a large zodiacal constellation and its outline stars do resemble a crouching lion making it readily recognisable. However, if you are not sure, follow a line from Dubhe through Merak, the two pointer stars in The Plough, down towards your southern horizon and you will find Leo. Earlier in the evening it will be towards the East. This constellation contains an asterism known as The Sickle because of its hook shape and outlined in red on the chart. It also looks like a backwards question mark. The brightest star in Leo is Regulus located at the bottom of the Sickle or the dot at the bottom of the question mark. It is a large blue-white star at a distance of eighty light years from Earth and with a magnitude of 1.4 it is the 15th brightest star in the northern hemisphere. The two other named stars, Algieba and Denebola, are not so well known but with magnitudes of 2.1 and 2.2 they help to make Leo stand out in the night sky. Algieba is a double star, consisting of two yellow giants, easily resolved into its components if you have a small telescope.
Don’t forget to enjoy Orion as it continues westwards in the evening sky to the right of Leo.
          Just above Leo lies the small and insignificant constellation Leo Minor- The Little Lion. Its brightest star is around magnitude 4 and even if you could see it there is no resemblance to a lion.
          The next constellation this month is Ursa Major- The Great Bear but better known for its asterism- The Plough, outlined in red. Ursa Major is a circumpolar constellation visible all year round as it rotates about the celestial north pole. The chart is a bit misleading because your zenith (the point directly above you) lies near the front feet of Ursa Major so you need to turn and face towards the north and you will see the plough standing on its handle at this time of year. The pointer stars have already been mentioned in locating Leo but if you go in the opposite direction you find Polaris – The Pole Star. (There will be more about that next month.) The second star along the handle of The Plough, Mizar, is a well known optical double and has been used as a test for how good your eyesight is. However, even a small telescope will reveal that Mizar is part of a true binary system in its own right.
 
Something to look out for
          As mentioned last month the planets are not in favourable positions at present unless you are an early bird and want to view them before dawn.
Remember it is the spring equinox on Sunday 20th March and the following Sunday the clocks move forward one hour as we start British Summer Time (BST).
Clear skies.

1. Introduction

The existence of what we today call Black Holes was first predicted by the Englishman John Mitchel (1724-1793) in a 1783 paper presented to the Royal Society in London. He reasoned that a star’s gravitational pull might be so strong that the escape velocity would exceed the speed of light.

Albert Einstein (1879–1955) published his Special Theory of Relativity (SR), which formalised the concept of space-time, in 1905. Einstein followed this with the General Theory of Relativity (GR) in 1917, which built on SR and provided a new theory of gravity in space-time. GR predicts that gravitational collapse can lead to an object so dense that nothing - not even light - can escape.

Karl Schwarzschild (1873–1916) developed a solution to Einstein’s Field Equations and in that process discovered the radius of the event horizon of a Black Hole. The event horizon is the radius at which if an object were to fall below it, the object would no longer be visible to an external observer. Note that the event horizon is not the same as the Black Hole itself. At the centre of a sphere of radius equal to the Schwarzschild Radius, Rs, as predicted in GR, there is a singularity of r = 0 and of infinite density which is the Black Hole.

Today (2021), It is believed there are at least two different cases of origin for a Black Hole:

  • The final stage of the evolution of large stars (≥ 8M⊙) is gravitational collapse in a Type II supernova event, the final compression leading to the formation of a stellar mass Black Hole.
  • It is also believed that as a result of stellar coalescence, a super-massive Black Hole lies at the centre of most, if not all, galaxies in the universe. The Black Hole at the centre of our Galaxy, the Milky Way, is called Sagittarius A* (pronounced ’A star’).


2 The event horizon of a Black Hole

2.1 The event horizon and popular misconceptions
The event horizon of a Black Hole can be thought of as the boundary within which the black hole’s escape velocity is greater than the speed of light, i.e. Vesc > c. Since nothing can exceed c, this means that any object falling into the event horizon will never be able to escape from it.

A common misconception about Black holes is that they ”hoover up” any matter that approaches them. This is not the case. Black Holes can’t seek out material to consume any more than any other massive object such as a planet or a star.

Another common misconception is that matter can be observed falling into a black hole. This is also impossible. A distant observer will witness the object moving slower and slower, while any light the object emits will be further and further redshifted.

We can detect accretion disks around black holes, where material moves with such speed that friction creates high-energy radiation [4]. Some matter from these accretion disks is forced out at near relativistic velocity along the Black Hole’s spin axis. When this matter collides with the inter-stellar medium, visible jets are created which may be light years in length. An example of this is the active galaxy M87, which is emitting a relativistic jet at least 1,500 parsecs long.

2.2 The Schwarstschild metric
The Schwarzschild metric, sometimes called the Schwarzschild solution, is a solution of Einstein’s field equations in empty space. The solution is valid only outside the gravitating body; for a spherical body of radius R, the solution is valid for r > R. The Schwarzschild solution describes spacetime under the
influence of a massive, non-rotating, spherically symmetric object. Using the Virial Theorem, we may equate the kinetic energy of a small mass, m trying to escape the gravity of a much larger mass M:

The actual Solar radius, R⊙ = 6.96 ∗ 10⁵ km is evidently very much larger than the three km Schwarstschild radius for one solar mass (by way of comparison, the Schwarstschild radius for an object of mass equal to one Earth mass is about 9 millimeters).
 
Equation 4 provides us with a useful calibration tool, in that RS scales as 3kmM⊙−1.

Putting this in the context of the solar system, we know that the mean orbital distance of Mercury is 5.79 ∗ 10⁷km. Hence we can conclude that if the Sun was somehow replaced by the hypothetical one solar mass Black Hole, Mercury’s current orbital parameters would place it well outside the event horizon.

This in turn means that Mercury (or any of the other Solar system planets) would not be in danger of being ”sucked into the Black Hole” as is sometimes popularly suggested.

2.3 Other geometric representations
The Virial Theorem assumes a non-rotating system. Examples of these in astronomy include globular clusters. However, it is known that Black Holes, like stars and spiral galaxies, do rotate.

Whereas the Schwarstschild metric describes a non-rotating Black Hole, the Kerr metric, discovered in 1963 by New Zealand mathematician Roy Kerr (1934-), describes the geometry of empty space-time around a rotating, uncharged, axially-symmetric Black Hole with a quasi-spherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of GR.

Rotating Black Holes have two surfaces with the size and shape of these surfaces depending on the black hole’s mass and angular momentum. The outer surface bounds a region called the ergosphere and has a shape similar to an oblate spheroid. The inner surface marks the event horizon; objects passing into the interior of this horizon can never again communicate with the world outside that horizon. Note neither surface has a physical presence.

In summary, if Q represents the body’s electric charge and J represents its spin angular momentum:

• In the Schwarzschild metric, Q = 0 and J = 0.
• In the Kerr metric, Q = 0 and J ≠ 0.

The Kerr metric can be represented in Cartesian Coordinates as:

Kerr’s mathematical discovery was almost serendipitous. Prior to that time, Black Holes were commonly regarded as interesting theoretical objects which might or might not exist in nature. However, the first quasars (3C 48 and 3C 273) were discovered in the late 1950s, as radio sources in all-sky radio surveys. They appeared mysterious in that the emission lines of quasar spectra did not correspond to any known chemical elements. In 1963, the same year that Kerr published his result, Dutch-born astronomer and Caltech professor Maarten Schmidt (1929-) was studying 3C273 and realized that one emission line was that of hydrogen, albeit very highly redshifted. In turn, that indicated 3C273 was very distant (actually ≈ 3 ∗ 10⁹lyr distant). For the quasar to be so far away and still visible; Schmidt concluded it must be of very high luminosity. (3C273’s apparent V magnitude is 12.9, so within range of amateur telescopes. 3C273 has the highest apparent magnitude of any currently known quasar and is the closest quasar to the Solar system.)

Following these discoveries, Black Holes were no longer a theoretical curiosity and there has been much recent observational work done on quasars. An indication of how technology has advanced in recent years can be discerned from the fact that just one quasar was observable by the European Space Agency Hipparchus mission (launched 1989, ran for 4 years). The follow-on ESA GAIA mission is observing more than 500,000 quasars (Perriman, 2022).

In fact, quasars (which are distant active galaxies, and should not to be confused with pulsars, which are neutron stars) are the intensely powerful centres of distant, active galaxies, powered by an accretion disc of particles surrounding a supermassive Black Hole. 3C273 is now estimated to have luminosity 2∗10¹²L⊙. Yet 3C273 appears to be less than a light-year across (our Galaxy is thought to be                  ≈  100,000lyr in diameter).


3 Sagittarius A*

Sagittarius A* is the designation assigned to the Black Hole at the centre of our Galaxy. It was first discovered in 1951 by Australians Harry Minnett (1917-2003) and Jack Piddington (1910-1997) who used radio observations at 1200MHz and 3000MHz (Winnett and Piddington, 1951) .

This was confirmed by observations in 1994 by a team led by Reinhard Genzel (Genzel1, Hollenbach and Townes, 1994) of the Max Planck Inst. fur Extraterrestrische Physik. Genzel and Andrea Mia Ghez of the University of California, Los Angeles were jointly awarded the Nobel Prize in Physics in 2020 for their discovery that positively identified Sagittarius A* as a supermassive compact object, for which a Black Hole is the only currently known explanation.

Genzel’s team reported ten years of observations in 2002 of the motion of the star S2 orbiting Sagittarius A*. The observations of S2 used near-infrared (NIR) interferometry at λ = 2.2m. The VLBI radio observations of Sagittarius A* were aligned with the NIR images. The rapid motion of S2 (and other stars close to Sagittarius A*) were easily distinguishable from slower-moving stars along the line-of-sight so these could be subtracted from the images. They found that S2 is in a highly eccentric Keplerian orbit (ϵ = 0.8843, one focus of which was at Sagittarius A*. From this, the team deduced the mass of Sagittarius A* to be 4.1∗ 10⁶M⊙, and that the source radius is no more than 17 light-hours (120AU)

Using this result and Equation 4, we may deduce that the Schwarstschild radius of Sagittarius A* is:

i.e ≈ 3 orders of magnitude larger than the Schwarstschild radius of Sagittarius A*.

If you prefer. this can be equivalently expressed as ≈ 5.9 ∗ 10⁻⁴parsecs (recall that M87s jet extends at least 1,500parsecs). Alternatively this is also equivalent to 1.9 ∗ 10⁻³lyr or 120AU.

4 Hawking radiation and Black Hole evaporation

In 1974, Stephen Hawking (1942-2018), determined that a Black Hole should emit radiation with a perfect black body spectrum (Hawking, 1974). Assuming Hawking’s theory is correct, Black Holes are expected to shrink and evaporate over time as they lose mass by the emission of photons and other particles via this phenomenon.

The temperature of this Hawking radiation, the Hawking temperature, is directly proportional to the Black Hole’s surface gravity:
                                                                 TH ∝ gBH                                                        (10)

and in the case of a Schwarzschild Black Hole, it’s gravitational force is inversely proportional to the mass:
                                                              gBH ∝ 1/MBH                                                     (11)

Hence, large black holes emit less radiation than small black holes, and so large Black Holes will take longer to fully evaporate. A stellar black hole of M⊙ has a Hawking temperature of only 6.2 ∗ 10⁻⁸K, whereas the cosmic microwave background radiation has T ≈ 2.7K. Hence, stellar-mass Black holes receive more radiation (and thus, more mass) from the cosmic microwave background than they emit through Hawking radiation and will grow instead of shrinking.

Although the theoretical evidence for Hawking radiation is very strong, the onlyblack holes we have  seen in nature are the size of stars or galaxies, and their Hawking radiation is invisible. To be able to evaporate, a Black Hole must have a Hawking temperature larger than 2.7K and)mass less than the Moon. Using Equation 4, the event horizon of this Black Hole would have a RS < 10⁻⁴m.


5 Conclusions

Black Holes are objects so dense that nothing -not even light can escape their gravitational field. Black Holes are at the centre of a sphere the radius of which is called the event horizon. any object passing through the event horizon can never escape out of it.

The possible existence of Black Holes was first suggested in the eighteenth Century. In the second half of the twentieth Century, the discovery of Quasars prompted renewed interest in Black Holes as the only conceivable source of such vast energy in active galaxies such as M87.

It is now recognised that Black Holes are located at the centre of most galaxies in the Universe. Our own Galaxy, the Milky Way, has a Black hole at the Galactic centre named Sagittarius A*. It is estimated that Sagittarius A* has a mass ≈ 4.15M⊙. This has been confirmed by observations of stars orbiting the Black Hole.


6 References

R Genzel1, R Hollenbach, D and Townes, C (1994). The nucleus of our Galaxy Report on Progress in Physics, Volume 57, Number 5 https://iopscience.iop.org/article/10.1088/0034-4885/57/5/001/pdf Perriman, M. In lecture presented to AGM of Wells and Mendip Astronomers, January 23 2021.

Hawking, S (1974). Black hole explosions?. Nature 248, 30–31 (1974). https://doi.org/10.1038/248030a0

Wang, Q et al (2013). Dissecting X-ray-Emitting Gas Around the Center of Our Galaxy. https://ui.adsabs.harvard.edu/abs/2013Sci...341..981W/abstract.

Winnett, H and Piddington, J (1951). Observations of Galactic radiation at 1200 and 3000 Mc/s https://articles.adsabs.harvard.edu/pdf/1951AuSRA...4..459P

I think I am turning into a ‘grumpy old man’ because last month I was complaining about the lack of clear skies and now that we have had some clear skies I’m complaining because it is too cold! However I was pleasantly surprised early one morning when I drew open my curtains and there was the full Moon about to set in the west- a beautiful sight. The big story in January has to be the successful deployment and positioning of the James Webb Space Telescope at its Lagrange L2 destination. It was quite tense at times as the various stages had to be completed but it all went off flawlessly. It will be exciting times when the data starts coming back.   
 
Observing
The following chart represents the night sky at 10.00pm GMT on the 8th of February and at 9.00pm GMT on the 23rd February. To use the chart, face south at the appropriate time with the bottom of the chart towards the southern horizon and you will see the stars in the chart. If you are observing a little earlier in the evening then the view is shifted 15 degrees eastwards for every hour before the specified time.

With part of Orion, Gemini and the Winter Triangle comprising the stars Betelgeuse, Procyon and Sirius on show there is no problem with navigating the skies but our focus this month is on fainter stars to the east of them. Of course, I’m sure you will still enjoy viewing Orion and the brighter stars that were mentioned last month. The constellation Cancer- The Crab, lies to the southeast of Gemini and being a zodiacal constellation its name is relatively well known although perhaps not its location. It is the faintest of the constellations with its stars typically being of magnitude 4 or dimmer so dark skies are essential. Its brightest star, the one nearest Procyon, is of magnitude 3.5. Cancer does not lie on the Milky Way so is in a dark region of the sky not containing very much apart from an open cluster known as Praesepe- The Manger, or the Beehive Cluster or M44. It is visible to the naked eye, as a small nebulous patch, under good viewing conditions and has been known of since ancient times. Of course, it would be better viewed with a pair of binoculars.
            Below and to the southeast of Cancer is the constellation Hydra- The Water Snake. It is the largest constellation in the night sky with the tail of the snake lying in the southern hemisphere. Unfortunately the chain of stars representing the snake is of an indistinct shape and mostly faint. The ringlet of stars representing the snake’s head is just south of Cancer while the brightest star, Alphard, sits on its own to the southeast of this. It is meant to be the heart of the snake and is an orange giant of magnitude 2. If you find this month’s constellations hard to spot just remember the night sky keeps changing and next month you will be enjoying an old favourite.  
      
Something to look out for
The planets aren’t in good orbital positions for viewing at present. Mars passed through opposition in October and is now a morning object for a short time. Mercury and Venus are close to inferior conjunction while Jupiter and Saturn are close to superior conjunction. This is best seen in the orbital diagram below.

On Wednesday 2nd February there will be a close approach of the Moon and Jupiter just after sunset low above your west-southwest horizon so hopefully we will be able to see Jupiter in a good setting before it reaches conjunction.
            The media seems to get quite excited about full moons these days and this month the full moon is on Wednesday 16th and is popularly known as the ‘Snow Moon’. Theoretically there is an exact time for a full moon but it can be observed at any time of night.       
Clear skies.

We are not being favoured with many clear nights just now but I did manage to catch the close approach of the Moon and Saturn back at the beginning of December. Of course the big event during December was the successful launch of the James Webb telescope on Christmas day using the Ariane 5 rocket from French Guiana. I’m not sure that some members of my family appreciated me being glued to the screen during their visit! The launch was part of the European Space Agency’s contribution to the project and perhaps the Ariane 5 rocket doesn’t get the credit it deserves for being such a reliable launch vehicle over many years.
 
Observing
The following chart represents the night sky at 10.00pm GMT on the 8th of January and at 9.00pm GMT on the 23rd January. To use the chart, face south at the appropriate time with the bottom of the chart towards the southern horizon and you will see the stars in the chart. If you are observing a little earlier in the evening then the view is shifted 15 degrees eastwards for every hour before the specified time.

Last month was mostly about the bright stars which form the Winter Hexagon and the Winter Triangle so this month will be a bit more about constellations. There is no difficulty with navigation because Orion is at its best just now and is the obvious starting point.
To the north east of Orion you can easily locate two prominent stars, Castor and Pollux, the brightest stars in the constellation Gemini- The Twins. It is probably well known because it is one of the zodiacal constellations. The lower star, Pollux, is the brighter of the two stars at magnitude 1.1 and is a single yellow star. On the other hand, Castor is a multiple star system with overall magnitude 1.6 and with the help of a telescope can be resolved into two white stars with a red companion and each of these stars is a double giving a total of six stars in the system! Castor and Pollux represent the heads of the twins while several stars of magnitude3/4 represent their bodies with the magnitude 1.9 star, Alhena, being the foot of Pollux paddling in the Milky Way. You should find it easily on a line between Betelgeuse and Pollux.
To the east of Orion and directly below Pollux you will find the bright star, Procyon, part of the Winter Triangle which we described last month, in the constellation Canis Minor- The Lesser Dog. This constellation is meant to represent one of Orion’s hunting dogs but has little to offer apart from the white star Procyon the 6th brightest star visible from the northern hemisphere at magnitude 0.4.
To the south of a line joining Procyon and Betelgeuse and in line with Orion’s Belt lies the brightest star in the entire sky, Sirius, in the constellation Canis Major- The Greater Dog, said to represent Orion’s other hunting dog.  Sirius, also known as the ‘Dog Star’, shines at a brilliant magnitude -1.4 and is only 8.6 light years distant. There is considerably more to this constellation but being close to the horizon does not yield itself to easy observation at our latitude. However your challenge this month is to attempt to see the open star cluster M41 which is shown on the chart with a red cross just below Sirius. It should be visible to the naked eye as a hazy patch the size of the full moon. You will need clear skies and be free from light pollution.
To the west of Sirius and just below Orion is the constellation Lepus- The Hare. I can’t say I am familiar with this constellation because it is surrounded by other bright stars but its central stars are brighter than magnitude 3 and the distinctive shape should make it easy to pick out albeit close to the horizon.
Finally the constellation, Monoceros- The Unicorn, lies mostly within the Winter Triangle but lacks any bright stars and is outshone by the brightness of its neighbours. Nevertheless knowing its location and distinctive W shape may help you locate it.
      
Something to look out for
The planets continue to provide a spectacle as long as conditions are good. Venus is moving into inferior conjunction (between the Earth and the Sun) so will disappear from the evening sky but reappear as a morning object next month. Mercury is doing the same but a bit behind and on the 4th January it will form a triangle with Saturn and a very young crescent Moon low in the south west just after sunset (4.15pm). Mercury itself will set at 5.45pm. Even the Moon will present a challenge to the unaided eye as it will be only two days old and close to the horizon just before setting.
            Look out for a close approach of the waxing crescent Moon and Jupiter on the 5th and 6th January. There might be a better chance to see Mercury on the 7th January when it is at its greatest eastern elongation (biggest angular separation from the Sun) and sets about ninety minutes after the Sun or on the 12th January when it is at its highest altitude. If you don’t have any success observing Mercury then just enjoy the brilliance of Jupiter. Clear skies.

I hope you managed to see the planets Venus, Saturn and Jupiter all in a line just after sunset on one of our clear nights. On Sunday 28th November I spotted Venus bright in the sky just as the sun was setting from my armchair in my lounge! I did have to get up off my bottom to spot the other two as the skies darkened although Jupiter is so bright that you cannot miss it either.
 
Observing
The following chart represents the night sky at 11.00pm GMT on the 8th of December and at 10.00pm GMT on the 23rd December. To use the chart, face south at the appropriate time with the bottom of the chart towards the southern horizon and you will see the stars in the chart. If you are observing a little earlier in the evening then the view is shifted 15 degrees eastwards for every hour before the specified time.

This month the focus is on 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 Betelgeuse 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.
The rest of this blog is about stars and there will be more to say about the constellations in which they lie next month. Because it is so easily recognisable, Orion is a good starting point for finding your way about the night sky during the winter months and especially for the stars we discussed last month. 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. Continue the line beyond Aldebaran and you find the star cluster- The Pleiades or Seven Sisters.
 Having followed the line to the Pleiades turn ninety degrees to the north and the bright star you see is Capella, in the constellation Auriga- The Charioteer, lying directly above Taurus. It is the 4th brightest star visible in the northern hemisphere and shines at magnitude 0.1.
Now follow a line from Orion’s belt to the south east and you will find the brightest star visible in the night sky, Sirius- the Dog Star. At magnitude -1.4 it is twenty three times more luminous than the Sun and a mere 8.6 light years distant. Sirius is part of the asterism known as the Winter Triangle, formed in conjunction with Betelgeuse and Procyon which lies due east of Betelgeuse. The white star, Procyon, is the 6th brightest star visible from the northern hemisphere so it is little wonder that the Winter Triangle is something to behold. It is outlined in yellow in the chart.
We finish this month with another asterism- the Winter Hexagon. It is roughly centred on Betelgeuse and comprises the stars Procyon, Sirius (both in the Winter Triangle), then continuing anti-clockwise, Rigel, Aldebaran, Capella and Pollux. It is outlined in red on the chart. You can enjoy viewing all these throughout the coming winter months as they make their way westwards in the evening sky.
 
Something to look out for
            A solar eclipse takes place on Saturday 4th December but unfortunately it will not be visible from Europe. Look out for coverage in the media because it is always good to see.
            Having witnessed the planets aligned last month there is a chance to see each one in turn, Venus, Saturn and Jupiter have a close approach with the Moon on the 7th, 8th and 9th of the month respectively.
            Finally the December solstice is on the 21st.
Clear skies.

The clocks have gone back again so there should be more opportunities to enjoy the winter sky in the darker evenings. It has been good watching Jupiter and Saturn in the evening sky, especially the former as it shines so brightly. Unfortunately my attempts to see their close approaches to the Moon have been thwarted by clouds. Still there will be another chance this month.
 
Observing
The following chart represents the night sky at 11.00pm GMT on the 8th of November and at 10.00pm GMT on the 23rd November. To use the chart, face south at the appropriate time with the bottom of the chart towards the southern horizon and you will see the stars in the chart. If you are observing a little earlier in the evening then the view is shifted 15 degrees eastwards for every hour before the specified time.

On the right hand side of the chart there is part of the constellation Andromeda and the two smaller constellations Triangulum and Aries which we mentioned last month. But we will use our old favourite Cassiopeia as our guide because you cannot fail to see it. Remember you don’t have to wait till late in the evening to observe these, make use of the earlier dark skies and look more towards your eastern horizon.
            Look to the south east of Cassiopeia and you find, lying in the Milky Way, the constellation Perseus- named after the hero in Greek mythology. Again it is difficult to recognise a human shape but the lines above the star Mirphak represent his right arm and sword while the bright star Algol represents his left hand holding the head of his victim. That is enough of the gory details. Mirphak is a 1.8 magnitude yellow supergiant while Algol is an eclipsing binary varying in magnitude between 2.1 to 3.4. There are another five stars around magnitude three or brighter making Perseus a prominent northern constellation.
            To the left of Perseus is the constellation Auriga- The Charioteer. It is easily identified because it contains the star Capella, the 4th brightest star in the northern hemisphere, at magnitude 0.1, and ‘only’ 42 light years from Earth. Again it is a binary system composed of two yellow giants. Auriga is in the rough shape of a pentagon although the bottom star Alnath is actually in the constellation Taurus.
            In fact the constellation Taurus- The Bull, is our final constellation this month, lying directly below Auriga. It is one of the oldest constellations having been recognised as early as Babylonian times. It is also one of the constellations of the zodiac. Its brightest star is Aldebaran a red giant of magnitude 1 and readily recognisable because of its colour, supposedly representing the red eye of the angry bull. The two lines with Alnath at the top of the upper one represent the bull’s horns. The ‘V’ shape to the right of Aldebaran represents the face of the bull and is an open star cluster called the Hyades. The more famous open cluster, the Pleiades or Seven Sisters, lies to the north west of Aldebaran in the direction of Algol. The Pleiades is one of my favourite objects to view with the unaided eye and I recall watching Venus pass close to the Pleiades during the spring of 2020.
             
Something to look out for
            Saturn and Jupiter are the gifts that keep on giving as they both make another close approach to the Moon on the 10th and 11th of November respectively. There will also be a partial eclipse of the full Moon on the 19th November but we are not well located for viewing it as the Moon will be close to the horizon and will set partway through the eclipse which takes place between 7.00am and 9.00am.
Clear skies.

Abstract
 
Since the first discovery of an exoplanet system in 1992, it has become apparent that the structure of the solar Solar system as it exists today is by no means typical of planetary systems in general. The early discovery of hot Jupiters orbiting their stars close in, although a result of observational bias in the radial velocity method, has been confirmed by other detection methods and has led to the concept of planetary migration over timescales of millions to billions of years. Studying the likely history and possible future of the solar Solar system reveals a chaotic environment for the inner planets and a semi-stable environment for the giant planets. Long term orbital instability has profound implications in terms of planetary migration, collisions and ejections for both the solar Solar system and for exoplanet systems.

Introduction 

The evolution of planetary systems has been the subject of study for several hundred years, with Kepler (1571-1630), Newton (1643-1727), Laplace(1749-1827), Lagrange (1736-1813), Gauss (1777-1855) and Poincaré (1854-1912) all having made great contributions to the field. Also, as is well known, the precession of Mercury's orbit was first accurately explained by Einstein's, (1879-1955) theory of General Relatoivity.
Approximately 4.5 billion years ago, gravity pulled a cloud of dust and gas together to form the protoplanetary nebula from which the Sun and the rest of Solar system evolved. Subsequently, by processes of both acretionaccretion and condensation in the gas cloud, planetesimals formed and collisions between them led to the formation of planets. The present Solar system structure is like a flat disk, with objects within the disk orbiting in almost the same plane. The objects generally orbit in the same direction and, with the exceptions of Venus and Uranus, rotate in the same direction.
There are several general types of such objects:

• Planets of which there are two types:
  • Terrestrial (Earth-like) planets: Mercury, Venus, Earth, Mars. Mass < ~15 Earth masses
  • Jovian (Jupiter-like) planets: Jupiter, Saturn, Uranus, Neptune. Mass > ~15 Earth masses
    
• Minor planets: Ceres, Pluto
• Debris: Asteroids (of which some are relatively large; the current number of known asteroids is over 1 million), meteors, comets

All these objects are believed to be ~ 4.6 Billion years old.
The Sun contains >99% of the mass of the Solar system, whereas the major planets account for
 ~ 99% of the angular momentum of the Solar system.
Gravitational influence of the giant planets has a major effect on the smallest planets. Jupiter, with the greatest mass of all, perturbs the orbits of the two least massive planets so that Mercury has the most eccentric orbit  and Mars the second most eccentric orbit in the Solar system.

Discussion

Planetary systems 

Using the 305m Arecibo radio telescope (sadly, now defunct),  it was demonstrated that the 6.2ms pulsar PSR1257 +12 is orbited by two or more planet-sized bodies – the first detection of an exoplanet system (Wolszczan & Frail, 1992).  Perhaps more well-known, 51 Pegasi is a Sun-like star located in the constellation of Pegasus at a distance of approximately 15 parsecs from the Solar system. 51 Pegasi was the first main-sequence star (type G2IV) found to have an exoplanet - designated by convention as 51 Pegasi b (Mayor & Queloz, D et al, 1995). Until the early 1990s it was widely assumed most planetary systems would be essentially similar to the Solar system. That this is not the case may be explained by considering the whole time-domain of the evolution of all planetary systems. Each system we detect maybe at any stage in its development, and will therefore most likely have different characteristics to the present-day Solar system.
We begin by considering gravitational effects. Generally, a low mass object orbiting a much more massive body would be expected to be forced gravitationally into an almost circular (i.e. zero-eccentricity) orbit. For example, this is the case with the orbits of the Galilean moons of Jupiter, where each moon's mass is much less than Jupiter’s mass (see  Appendix: resonant and non-resonant orbits).
Considering just two gravitatonally-bound objects, Newton’s law of gravitation can be written as:

Where the respective forces are as shown in Figure 2.

This is mathematically called a 2-body problem, which as may be seen from the equation has a simple algebraic solution.
 
We should also recall Kepler's Laws, where he postulated that all orbits are elliptical. In particular, Kepler's Third Law tells us that

where k is a constant of proportionality given by

It turns out that if P is in Earth years and a is in Astronomical Units, (AU), k=1. The orbits of the Solar system’s major planets at the present time conform exactly to Kepler's Third Law as shown in Figure 3 below:

The preceding evidence gives the impression of a well-ordered and static system. We must however remember that what we are seeing at any instant in time is a snapshot in the history of the Solar system. As we shall see, planetary systems are generally much more complicated.


Complexity of multi-body systems 

Kepler's Laws are regarded as fundamental Laws of planetary systems, so Kepler's Third Law is assumed to apply throughout the life of the Solar system and that of any exoplanet system. However, this does not mean the configuration over the lifetime of the system is static. The orbits of the system will always be elliptical (and hence conform to Kepler’s Laws), but the parameters eccentricity, major and minor axes of the ellipses will change as the planets interact with each other. The only case where the orbit of a planet will acquire an orbital eccentricity e>1 is in the case of a planetary ejection from the system (Laskar, 1994)
The plot in Figure 4 shows the eccentricities of the major Solar system planets. We can see from this plot that whereas the orbits of Venus, Earth, Jupiter, Saturn, Uranus and Neptune have similar eccentricities, the orbits of Mercury and Mars are clearly outliers. Mercury has a particularly  eccentric orbit Ε = 0.2056.

Exactly as one would expect, the Left-to-Right order of the plots is the order outwards from the Sun at which the planets orbit. Of all the major planets in the solar system,. Mars has the second most eccentric orbit with Ε =0.0935.
 
In Figure 5, we plot orbital eccentricity, Ε against planetary mass, M for each major planet.  Here the plots are not in order of distance from the Sun - the rightmost plot is the most massive planet, Jupiter which has therefore the most domiantdominant influence on the rest of the Solar system planets (Hayes, 2010).

Mercury has been described as having the most unstable orbit in the Solar system (Lithwick & Wu, 2014). The root cause of this appears to be that at least the inner Solar system is chaotic, and the outer Solar system is borderline stable (Lithwick & Wu, 2014). What we see here is the situation in the Solar system's time-domain as of today. Beyond a few tens of Myr into the future the motion of the planets based on what we observe today cannot be accurately predicted (Woillez and Bouchet, 2020).
On longer timescales, planetary trajectories can only be studied probabilistically with software using numerical algorithms running on supercomputers (Budrikis, 2020). Simulations using these methods predict that over the Sun’s remaining lifetime around 1% of possible trajectories of the planets show Mercury’s orbit becoming so eccentric it may be involved in a collision with Venus or the Sun (Woillez and Bouchet, 2020). Early studies have discussed the possibility of Mercury being ejected from the Solar system altogether or its collision with Venus (Laskar, 1994). This is also indicated in more recent work which shows the strong influence of the outer planets inducing chaotic variations by the inner planets (Hayes, 2010) . In case these scenaria sound improbable, consider that the most widely accepted theory of the origin of Earth's Moon is that of a proto-Earth colliding with a Mars-size planetesimal.
To better understand this, consider a simple thought-experiment where a planetary system consists of just the Sun, Mercury and Jupiter. The gravitational forces acting on Mercury due to Jupiter and the Sun respectively will be the vector:

The vector components on the right-hand side will be, according to the Inverse Square Law, inversely proportional to the respective vector distances of the Sun and Jupiter respectively, rSUN and rJUPITER:

When Mercury and Jupiter are on the same side of the Sun, these vector components will be at their highest. When Mercury and Jupiter are on opposite sides of the Sun, these vector components will be at their lowest.
As we can see, the two vector sums change continuously as the two planets proceed along their respective orbits, and the calculation of the vectors is relatively tedious..
Now, aside from this thought-experiment, in the real Solar system, we must recall several things, namely:

1. During the time that Jupiter makes one orbit, Mercury makes just over 49 orbits; during each of Mercury’s orbits, the resultants of the vector sum above will be different;
   
2. While this is happening the four terrestrial planets and two minor planets (Ceres and Pluto), plus numerous asteroids and comets, each with differing masses, orbital periods and eccentricity will influence the vectors.
3. The spin axis of each planet is precessing (see September 2020 AstroBlog).
   
4. The orbit of each planet is precessing (Ibid)
5. The largest mass in the system, the Sun, is also orbiting the common centre of mass in the Solar system ( called the barycentre) - the so called "wobble" by which exoplanet systems are detected using the radial velocity method.

 
We can now see that the calculations are of very high complexity, even for a single instant in time, and vastly more complex if we look at the distant past or distant future. This is known as the “N-body problem”, the problem of predicting the individual motions of a group of celestial objects which gravitationally interact. Though analytic solutions have been proven up to N=3 (at least in cases where M1 >> M2) , there is no analytic solution to the N-body problem where N>3 (Heggie, 2005), and instead numerical solutions must be run on computers.
At earlier times in the history of the Solar system, it has been suggested that Jupiter and Saturn were likely in the 3:2 resonance, defined as PSATURN/PJUPITER = 1.5, where PJUPITER and PSATURN are the respective orbital periods of Jupiter and Saturn. Today, this ratio has become 2.49, the resonance having been disrupted by gravitational interactions (Nesvorny, D, 2011)


Planetary migration 

As planets evolve, their mass is subject to changes. Collisions and absorption of dust small objects increase a planet’s mass, and collisions may either increase or decrease mass. This results in an exchange of angular momentum between an evolving planet and the protoplanetary disc, which causes the planet to migrate through the disc. Until the long-awaited discovery of exoplanets, when planetary formation models were based on the Solar system, planetary migrations were usually considered unlikely.
More recently, data from exoplanet systems has provided strong evidence of planetary migration. For example, WASP-107b, a super-Neptune discovered in 2017, is estimated to have a rocky core of ~ 10 Earth masses, and a large gaseous envelope consisting mainly of H and He. This means WASP-107b’s most probably formed several AU from the host star where the protoplanetary disk is rich in gas, ices and dust (Piaulet et al, 2020). However, the current orbital semi-major axis of WASP-107b is only 0.055±0.001 AU (NASA Exoplanet Archive). This leads to the conclusion that WASP-107b has most likely undergone inwards migration (Piaulet et al, 2020).


Decaying planetary orbits 

Have any decaying planetary orbits been observed? Yes, but not that many so far.
The orbital period of exoplanet TrES-1 b, discovered in 2004 (Alonso et al 2004) is getting shorter by around 11 milliseconds per year. This may not sound much, but over an astronomically short time of ~300,000 yr, that's approximately 3 days. The orbital period of TrES-1 b is very close to 3 days according to recognized sources (Exoplanet.eu: P~3.0300722d; NASA Exoplanet Archive: P~3.030070±0.000008d; (see NASA Exoplanet Archive).

Ejection of planets from systems 

As mentioned earlier, ejection of small terrestrial planets is plausible (Laskar, 1994). But what about giant planets? Studies of giant planets’ interaction within the protoplanetary gas disk indicate that planetary migration is usual. Moreover, planets emerging from mergers of planetesimals are expected to be in orbital resonance. Planetary systems formed from protoplanetary disks can become dynamically unstable after the gas disappears, since the gas exerts a stabilising influence. This leads to a phase when planets scatter off of each other. According to this model, Jupiter and Saturn were most likely trapped in the 3:2 resonance (Nesvorny, 2011). We can see from Figure 7 that this is certainly not the case today.

Using 6000 scattering simulations, Nesvorny et al evaluated a historic Solar system with both four and five gas giant planets. With four gas giants (i.e. as in the current epoch) the best results were obtained with disk masses between 35 and 50 Earth masses. The fraction of simulations with an initial four outer planets producing a final system also having four outer planets was only between 10% and 13%, showing an unlikelihood that the Solar system evolved from a four giant planet system. When run with a five outer planet system as the initial configuration, simulations showed it roughly 10 times more likely to obtain a Solar system analog.
The conclusion reached by Nesvorny, postulating a "jumping Jupiter" scenario, is that the fifth giant planet was ejected from the early Solar system about 3Myr ago. It should be noted that to explain the current orbital parameters of Jupiter, there is a dependency on the 3:2 resonance between Jupiter and Saturn mentioned earlier. More recently, investigation into the framework of this jumping-Jupiter model assessed the possibility that the high eccentricity and inclination of Mercury originated during the instability and concluded instability can produce the presently large values of eccentricity and inclination of Mercury. (Roig et al, 2021)
How plausible is it that a planet - in the case of the 2011 study by Nesvorny study, a giant planet - could be ejected from a planetary system, becoming a lone planet? An observational detection of a giant-mass lone planet using gravitational micro-lensing has been made (Mroz et al, 2018). In the case of this study, there are some major uncertainties as to where the lens is located. If it is in the Galactic disk, the lone planet should be of Neptune-mass; if the lens belongs to the Galactic bulge population, the lone planet should be a Saturn-mass. It is interesting to note that today’s micro-lensing surveys are able to detect lone planets as small as a single Earth-mass.


Conclusion 

The configuration of the Solar system as it is today is atypical of planetary systems we have discovered to date. The current configuration would not have existed throughout the history to date of the Solar system, neither will it remain static in the future. The same reasoning apples to exoplanet systems. Since we are examining exoplanet systems as they exist now may explain why most exoplaenet systems discovered to date show very different characteristics and structure to those of the current Solar system structure.
When we are observing exoplanetary systems, we are observing evolving systems at an instant in time. These systems will also have changed over time and will continue to do so in future. The same reasoning applies in the case of the Solar system.  Modeling these changes throughout the expected time-domain is only possible by using numerical methods which require large amounts of computer time.
In planetary systems, events such as collisions, both between orbiting objects with other orbiting objects, and between orbiting objects and the parent star are regarded as normal rather than exceptional. Migration of planets from their current orbits is also not unusual. Planetary ejections also occur, which may be one reason for the existence of some lone planets.

References 

Alonso, R et al (2004). TrES-1: The transiting planet of a bright k0 v star. ApJ, 613:L153–L156, 2004 October 1. https://iopscience.iop.org/article/10.1086/425256/pdf  Accessed September 23, 2021
Budrikis, Z (2020). Nature Reviews Physics. The path to the Solar system’s destabilization.
DOI:https://doi.org/10.1038/s42254-020-0218-0https://doi.org/10.1038/s42254-020-0218-0   Accessed Sep-8-2021
Hayes, W et al 2010). The interplay of chaos between the terrestrial and giant planets. MNRAS 407, 1859–1865 (2010). DOI: 10.1111/j.1365-2966.2010.17027.x Accessed Sep 26 2021.
Heggie, D (2005). The Classical Gravitational N-Body Problem.
https://arxiv.org/pdf/astro-ph/0503600.pdf  Accessed September 14 2021
Kokori, A; Tsiaras, A; Edwards, B et al. (2020) ExoClock project: an open platform for monitoring the ephemerides of Ariel targets with contributions from the public. https://www.exoclock.space/project Accessed September 25, 2021
Laskar, J (1994). Large scale chaos in the solar Solar system. A&A 287 L8-12(1994)
Lithwick, Y; Wu, Y (2014). Secular chaos and its application to Mercury, hot Jupiters, and the organization of planetary systems. Proceedings of the National Academy of Sciences, Volume 111, Issue 35, 2014, pp.12610-12615. https://arxiv.org/pdf/1311.1214.pdf Accessed Sep-8-2021.
Mayor, M.; Queloz, D; et al (1995).  51 Pegasi.   IAU Circ., No. 6251, #1 (1995)
DOI: https://doi.org/10.1038/355145a0   Accessed Oct 15 2021
Morbidelli, A et al (2010) Evidence from the asteroid belt for a violent past evolution of Jupiter’s orbit. ApJ, 140:1391–1401, 2010 November. https://iopscience.iop.org/article/10.1088/0004-6256/140/5/1391/pdf Accessed Sep 25 2021.
Mroz, P (2018). A Neptune-mass Free-floating Planet Candidate Discovered by Microlensing Surveys.  ApJ , 155:121 (6pp), 2018 March https://iopscience.iop.org/article/10.3847/1538-3881/aaaae9/pdf  Accessed Sep 22 2021. Submitted to ApJ
Nesvorny, D (2011). Young Solar System’s Fifth Giant Planet? https://arxiv.org/pdf/1109.2949.pdf Accessed Sep 22 2021.
Piaulet, C et al (2020). WASP-107b’s density is even lower: a case study for the physics of planetary gas envelope accretion and orbital migration. https://arxiv.org/pdf/2011.13444.pdf Accessed Sep-14 2021
Roig, F (2016). Jumping Jupiter can explain Mercury’s orbit. ApJ 820:L30 (5pp), 2016 April 1. https://iopscience.iop.org/article/10.3847/2041-8205/820/2/L30/pdf Accessed Oct-2-2021
Saillenfest, M; Laskar, J; Boué, G (2019) Secular spin-axis dynamics of exoplanets A&A Volume 623, March2019. https://www.aanda.org/articles/aa/pdf/2019/03/aa34344-18.pdf Accessed Sep-8-2021.
Wolszczan, A & Frail, D (1992). A planetary system around the millisecond pulsar PSR1257 + 12.  Nature volume 355, pages 145–147 (1992).  DOI: https://doi.org/10.1038/355145a0   Accessed Oct 15 2021

Acknowledgments

The author gratefully acknowledges two anonymous referees, whose critiques have resulted in improvements to this paper.


Data sources used in this work

Exoplanet.eu
NASA Exoplanet Archive
NASA Index of Planetary Fact Sheets

Appendix: resonant and  non-resonant orbits 

In a system where a low mass satellite is orbiting a much more massive body, we would expect the satellite’s orbit to be forced gravitationally into an almost zero-eccentricity (i.e. very nearly circular) orbit - a process called circularization.
Consider the case of the Galilean moons of Jupiter, where this is indeed the situation. Although there are 79 known Jovian moons (as of September 2021), the Galileans account for 99.007% of the orbiting system’s mass. In this system, each of the four moons’ mass is much less than Jupiter’s mass, and as Table 2 shows, all the respective orbits are of very low eccentricity.
The reason this system’s orbital configuration is not like that of the Solar system is because in the case of the Galilean system, there are no high mass Jovian satellites beyond Callisto, the outermost Galilean moon

A mean-motion orbital resonance occurs when two bodies orbiting a larger primary have periods of revolution that are an in an integer ratio. An example is the resonance between the Galilean moons of Jupiter Io, Europa, and Ganymede, which are in 1:2:4 resonance. That is to say, the furthest orbiting moon, Ganymede makes one orbit, Europa makes two orbits and Io makes four orbits.  The fourth Galilean moon Callisto does not cross the zero point simultaneously with the others and is therefore not in orbital resonance with the other three.

Well the autumn equinox is past now and we should begin to enjoy some dark skies again. Remember the clocks go back on the 31st of this month. The Harvest Moon on the 21st September looked quite spectacular as I caught it rising in the east and just fitting between two nearby houses.
 
Observing
The following chart represents the night sky at 11.00pm BST on the 8th of October and at 10.00pm BST on the 23rd October. To use the chart, face south at the appropriate time with the bottom of the chart towards the southern horizon and you will see the stars in the chart. If you are observing a little earlier in the evening then the view is shifted 15 degrees eastwards for every hour before the specified time.

The chart this month should look familiar because facing south and just above your zenith will be the magnitude 2.3 star ,Caph, a member of five bright stars forming the readily recognisable ‘W’ shaped asterism in the constellation Cassiopeia. Being circumpolar it is visible all year round and six months ago it was low over the northern horizon but now we have a chance to admire it high in the sky.
            Below Cassiopeia and to the right is the Great Square of Pegasus which we looked at last month. However, our focus this month is on the constellation Andromeda- the princess and daughter of the mythological Queen Cassiopeia and King Cepheus. Remember that the brightest of the stars in the Great Square of Pegasus, Alpheratz, is actually in the constellation Andromeda so we shall start from there. The main features of Andromeda are two curved strings of relatively faint stars starting at Alpheratz and lying above it and to the left, below Cassiopeia. The bright stars Algol and Mirphak in the constellation Perseus to the east can also help with navigation. The lower string of stars is fairly easily followed by star hopping from Alpheratz: delta Andromeda, Mirach and Almach have magnitudes of 3.3, 2.1 and 2.2 so no problem. The higher string of stars is fainter and poor viewing conditions and light pollution will make them difficult to see with the unaided eye as they are typically of magnitude 3.5 to 4.5. The constellation Andromeda is home to one of the most famous objects in the night sky- the Andromeda galaxy also known as M31 and shown on the chart by a red cross. 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 as its magnitude is 3.4), start at Alpheratz and by star hopping, jump to the second pair of stars along the curved strings and extend a line from Mirach through the second star and the Andromeda galaxy will be at a distance approximately equal to the distance between the two stars.  Now some mind boggling statistics- the distance to the Andromeda galaxy 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. 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. Perhaps we will be able to do it together when we meet at Oakhill on the 9th October. Let’s hope for clear skies.
            Only two stars in the constellation Pisces- The Fishes, are brighter than magnitude 4 so it offers little to the unaided eye. The constellation, Triangulum- The Triangle, is equally insignificant but because of its compact size, and a shape matching its name, it is relatively easy to spot.
            We’ll finish this month with another zodiacal constellation Aries- The Ram. In Greek mythology it represents the golden ram whose fleece was sought after by Jason and the Argonauts. I can see no resemblance to a ram but its brightest star, Hamal, shines brightly at magnitude 2.0 and can be readily picked out.
                         
Something to look out for
There will be a close approach of the Moon with Saturn and Jupiter on the 14th October and 15th October respectively. Perhaps I should explain that the chart in this blog is selected because it gives the view of the stars shown as they are crossing our meridian and therefore at their highest in the sky and as far away as possible from atmospheric disturbance closer to the horizon. However it is good to take a wider view from time to time and if you look west you can see the Summer Triangle with Altair getting closer to the horizon before it disappears from view later on. Similarly if you look towards your eastern horizon you will catch the beautiful star cluster known as the Pleiades or Seven Sisters. And don’t forget to cross from Cassiopeia through the pole star, Polaris, to The Plough which is currently close to the northern horizon and looking like a plough! Clear skies.