I was outside watching the planets and then much later for any sign of the aurora.  I didn't see the northern lights but I did see some noctilucent clouds.

I took this picture near my home in Red Row overlooking the cemetary at about midnight last night.  It wasn't the brightest display and the camera certainly captures the colour and structure more easily than the eye.  But it's sign that NLC season, which lasts from about now until early august, has arrived again.

As twilight fell the planets came out to shine.  Just a few minute walk from home, past St John's Church in Red Row I took this picture looking towards the northwestern horizon.  The brightest spot in the picture is Venus.  Just above and to the right is Mercury (about to be eclipsed by a bird).  Jupiter is at the top left of the picture.

I wrote about this particular dance of the planets in an earlier post.  The nearest planet to Earth in this picture is Mercury, sitting just on the farside of the Sun 1.18AU (177 million km) away.  Beyond Mercury is our evil twin Venus - 1.65AU (247 million km) away.  Jupiter is way beyond both of those planets - about 6.06AU (765 million km).  And they all look so cosy together as we see them along nearly the same line of sight. 

In 1904 Swedish mathematician Helge von Koch published a description of a strange and beautiful mathematical shape.  This is what it looks like:

This shape is not described by a simple function of the form $y=f(x)$.  It has a recursive definition and it can be constructed in stages by iteration.

• Start with an equilateral triangle.
• Remove the middle third of each side.
• Add two line segments to make new equilateral triangles on each side of the previous triangle.
  

The shapes generated by these rules evolve like this:

The Koch snowflake is the shape we get as the number of iterations increases to infinity.  Koch snowflakes have lots of interesting mathematical properties.  In fact, this was one of the first examples of a mathematical shape called a fractal to be studied.  The snowflake is notable for the following properties:

• It is continuous everywhere (there are no breaks or discontinuities) but it's differentiable nowhere.  That means you can't calculate the slope at any point on it.  Think about it!  Try to imagine drawing a tangent line to determine the slope from first principles....you just can't.  The line wiggly and bumpy no matter how much you magnify the shape.  It's just not smooth anywhere along it.
• The total area of the snowflake is finite but the perimeter is infinite.  The mathematical proof of this is given below.  But the fact it fits on your computer screen or smartphone shows it has a finite area.  You can see that the perimeter increases at each step by looking at the pictures.  But not all increasing sequences have infinite sums.  The series $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots$ also increase but doesn't get bigger than 2.  The proof that the perimeter of the snowflake is infinite is also given below.

This week I'm teaching integration by parts - a method for integrating functions in product form.  At school this is the method that made me want to quit my A-level!  The xkcd cartoon on the left pretty much sums up what it felt like at the time.
For students new to calculus I think it's a pretty daunting method for several reasons:

1. It uses integration and differentiation in the same problem
   
2. Might require chain rule / reverse chain rule in the same problem
   
3. Needs a good choice at the start: which function to integrate or differentiate?
   

And as for the second integral $\int u dv$ looking easier...well that's sometimes subjective and the obviousness only comes with experience.  In exams that I've marked students will often pursue the solution of that second integral after the initial bad choice of $u$ and $dv$.

My best advice for getting used to this method is to start with easy examples to see how exactly they work.  Also, being good at the chain rule (for differentiation) or reverse chain rule (for integration) is a must.  If you can do those slightly easier methods with confidence then the "by parts" method shouldn't be so filled with problems.
Some nice examples of integration by parts, with solutions, can be found here.

It's been a great year for comets so far.  First there was 2012 K5 LINEAR and then the superb apparition of 2011 L4 PANSTARRs.  The UK gets to see a third comet over the next few weeks as 2012 F6 Lemmon makes an appearance in the morning sky before dawn.

Comet Lemmon was discovered in 2012 by A.R.Gibbs at the Mount Lemmon Observatory, Arizona, USA.
In early 2013 it was naked eye object from the southern hemisphere and lots of memorable photos showed it chasing Comet PANSTARRs across the southern constellations.
The comet reached a peak brightness of magnitude +4 in mid-March but it wasn't visible from the UK.  It's now just emerging from the morning twilight but it has faded to 7th magnitude - too faint to see without optical aid but should be easy enough for small telescopes - and cameras attached to them!

Comet PANSTARRs emerged in the northern sky in mid-March.  Now it is joined by Comet Lemmon - following a similar path through the constellations Pegasus and Andromeda.  The finderchart below shows the path of the comet from the end of May to mid-June.  

You'll notice in the chart that Comet Lemmon will drift past the Andromeda Galaxy (M31) in the second week of June - just like PANSTARRs did at the start of April.  It'll be very hard to get a picture of the two because the sky just doesn't get so dark in Northumberland as we approach the solstice.  However, I'm determined to get a picture of Comet Lemmon before it fades and strong moonlight wipes out the dark background sky. 

Comet Lemmon is tracking rapidly north in the sky and will soon be circumpolar from the UK - it just won't set.  The best time to observe is after midnight when the comet is rising in the northeast part of the sky.  There's only a short window of opportunity to view it against anything like a dark sky.  The encroaching dawn light will end the observing session by around 3.30am.

Hope I get to see it but failing that....Comet Lemmon will probably be back in our skies in about 8 thousand years.

An interesting gathering of planets begins to take shape in the last ten days May.  Jupiter is slipping closer to the setting Sun but before it leaves the evening sky it will be joined by Mercury and Venus for a rare triple conjunction.

The above picture shows the scene on the evening of May 26th when the three planets form the smallest group.  Mercury, Venus and Jupiter are contained in a circle less than $3^{\circ}$ (about 5 moon diameters) in size.  The days leading up to this evening and those after it are also worth seeing.  
The animation below shows the view towards the northwest horizon each evening at 10pm from May 20th until the end of the month.   Conjunctions are a great chance to see the clockwork motion of the planets - the shifting aspect of the planets over each evening is very obvious.

In the UK the best time to go out and look for these planets will be between 10pm and 10.30pm.  This is not long after sunset and the evening twilight will be bright - particularly from further north in the UK.  Binoculars might help with seeing the planets initially but they should be obvious from about 10.30pm until they set.

Venus is the brightest planet (magnitude -3.9) and about six times brighter than Jupiter (magnitude -1.9).  Mercury is the dimmest of the three planets.  It races around the Sun faster than the other two and the portion of it illuminated by the Sun (the phase) changes more dramatically.  On May 20th Mercury is magnitude -1.3 with 90% phase.  By May 31st it has faded to magnitude -0.3 with phase 63%.  You can see from the animation that Mercury climbs higher into the evening sky so that fall in intrinsic brightness is counteracted by it being visible against a slightly darker sky (and less atmospheric extinction).

Viewed from above the solar system you can see the immense distances between the planets - and exactly why look close together in the sky.  We're seeing them all in the same direction - more or less in a straight line beginning at Earth and ending at Jupiter - just over 900 million km away. 

Mercury and Venus are in the foreground - just 171 million km and 246 million km respectively.  Beyond them - way beyond - is Jupiter nearly a billion km from us.

Conjunctions between bright planets are not rare - we usually get at least a few per year.  But conjunctions involving three or more planets are a bit special.  There was a gathering involving Mercury, Venus, Mars and Jupiter in May 2011 but it happened in the wrong part of the sky at the wrong time of the year for UK observers to see it.

The picture here was taken in the evening sky in March 2011.  Mercury and Jupiter were strikingly obvious once I'd picked them up in the twilight sky.  This scene will soon be repeated with Venus making an appearance too.

On the next clear evening take look at that familiar pattern of Cygnus. You might notice it has an extra star!

Usually, for a couple of months out of every 14 months the variable star $\chi$ (Chi) Cygni is bright enough to rival the brighter stars in Cygnus.  I was in the garden the other night and it caught my eye - for a moment I thought it was a satellite changing the appearance of the constellation and then realisation dawned - chi Cygni was back!

At this time of the year Cygnus can be seen in the eastern sky during the late evening.  The picture shows the cross-shaped pattern of its brightest stars along with the location of chi Cygni (and my garage).

$\chi$ Cygni is a Mira type variable star.  Mira variables are pulsating red giant stars with periods of 100 days or more and displaying large changes in brightness.  They are named after the star Mira (meaning "wonderful") in the constellation Cetus.  Stars become red giants during the later stages of their lives - after the stable period of hydrogen fusion has ended.  Eventually they'll blow off their outer layers to form a planetary nebula and in the middle of the beautiful wreckage will the exposed core of the old star - a hot white dwarf.  But before this stage is reached the red giant star may go through a period of pulsations as a Mira variable.
Mira stars undergo changes in size and temperature - which affect the overall brightness of the star.  $\chi$ Cygni is a dramatic example.   It is sometimes as bright as 3rd magnitude - see the picture of Cygnus above to get an idea of what that looks like.  But it will soon fade by a factor of 20 thousand!  The star will drop to 14th magnitude - as faint as Pluto - and only visible through very big telescopes.  Despite the big brightness changes the overall energy output doesn't change that much.  When the star is near minimum brightness visually it is still shining fairly brightly in infrared. 

$\chi$ Cygni was discovered by the Astronomer Royal  of Germany Gottfried Kirch in 1686.  The periods between successive maxima (or minima) is around 405 days.  There's a bit of variation in this figure and a bit of variation in the brightness of each peak.  The light curve of the star over the last 10 years looks like this:

Variable star observing is something that I've been interested in for a long time but never given it as much time as it deserves.  It takes practice to accurately estimate the brightness of stars visually.  And it's an area of astronomy where amateurs can contribute real data to the professionals.  As the astronomer and writer David H Levy notes in his book "Observing Variable Stars":

It's interesting to read a few paragraphs earlier in that book that $\chi$ Cygni was the first variablesLevy tried to get an estimate of brightness for ... although he was trying in vain to locate it among the rich star fields of the Milky Way while it was near minimum.

But you won't have that problem for the next few weeks!  If it's clear go and look for the wonderful star of Cygnus.

All night twilight has arrived in Northumberland and the waxing moon is growing brighter each evening.  All of that as PANSTARRs continues to recede from both Earth and Sun.  It means that this is probably one of the last pictures I'll take of this icy visitor in our sky.

Although the comet is fading and activity dying down it remains an interesting sight.  There are two tails emerging from the nucleus - a curved dust tail to the north and a narrow antitail to the southeast.

At the time of the photo Comet 2011 L4 PANSTARRs was 1.67AU (250 million km) from Earth and 1.57AU (234 million km) from the Sun.  It was shining at about magnitude +8.4 in the constellation Cepheus.

Some of the hardest problems in mathematics are the easiest to state.  Two of the more famous examples concern prime numbers - the building blocks of the integers.  Prime numbers are numbers with no factors other than themselves and 1.  The list of primes begins 2,3,5,7,11,13,17,23,29,31,37,41,43,47,53,... and goes on forever.
 The most celebrated prime number theorem is probably Fermat's Last Theorem which states there are no positive integer values $x$, $y$ and $z$ which satisfy $x^n+y^n=z^n$ for $n\ge 3$.

For $n=1$ it is easy to find three numbers, for example $3^1+5^1=8^1$ (which is just the same as $3+5=8$).
For $n=2$ we can find an infinite number of Pythagorean triples: $3^2+4^2=5^2$ (meaning $9+16=25$) being the most familiar.
In 1637 French mathematician Pierre de Fermat conjectured that no such integer values could be found for larger values of $n$.  In fact he proved it for the case $x^4+y^4=z^4$.  Fermat claimed to have a general proof of his conjecture but none was never found - even following a careful search of his papers after his death.  Fermat's last theorem remained unproved until 1995 when British mathematician Sir Andrew Wiles published a proof using mathematics unknown to Fermat.

The twin prime conjecture is just as easy to state.  Look at the list of prime numbers from earlier.  In the list are some so-called twin primes - primes separated by an a single (even) number.  The first cases are (3,5) then (5,7).  They grow more rare as the numbers get bigger:  (11,13) is followed by (17,19), (29,31) and (41,43).  Mathematicians speculate that there are an infinite number of pairs $p$ and $p+2$ but a proof has so far eluded everyone.  Nevertheless mathematicians have been attacking the problem for years - chipping away at the foundations and some interesting mathematics has developed as a result.
The prestigious journal Nature is carrying a story today connected to the twin prime conjecture.  Mathematicians have shown that infinite pairs of primes exist.  The difference is that they've proved that an infinite number pairs exist spaced by not more than 70 million.  This is somewhat larger than prime pairs spaced by a single even number.  But it represents significant progress on one of the hardest, open problems in mathematics.

Why should we care whether or not twin primes are infinite in number or not?

Prime numbers are at the heart of mathematics.  You can think of them in the way that atoms were once regarded - building blocks out of which everything is built.  Every integer can be written as a product of prime numbers.  Learning more about prime numbers pays us back with mathematical techniques that help solve other mathematical problems.  Prime numbers are not just abstract mathematical entities.  They keep online transactions secure.   Biologists observe that evolution by natural selection has favoured some creatures with hibernation patterns based on prime numbers!  They are a part of our universe - naturally and embedded in our technology.

This month promises to be an interesting one for amateur astronomers.  A gathering of planets in the evening sky is shaping up for the last week of the month.  Jupiter has been waiting for months and now Venus is rounding the far side of the Sun to join it in the sky.  Mercury will follow shortly.

Astronomers past have dubbed Venus the Earth's twin because of it's slightly smaller size and mass.  Venus may once have possessed oceans like the Earth but a runaway greenhouse effect has heated the planet enough to strip the water, bring plate tectonics to a grinding halt.  The upshot: explosive volcanism periodically resurfaces the entire planet.  The plate tectonics on Earth act like a valve - releasing the heat of the interior through more gentle volcanic events and Earthquakes.  On Venus the next big one really is a Big One.  The temperature is now a hellish $450^{\circ}$C at the bottom of an atmosphere which would crush humans in the same way that being a couple of miles under an Earth ocean would.  And they named this place for the Godess of Love!