| Modulo | Universe

Cosmology, Astronomy and Abstract Mathematics


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Solar Day at Abriachan

We were blessed with a lovely sunny day on Saturday for our day of Solar learning up at Abriachan.  We were fully prepared for indoor activities as forecasts were looking pretty grey.  But as the weekend swung around skies cleared and we ended up seeing plenty of Sun all day.

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A day of fun solar learning

Since conditions were so good we moved everything outside, including the talk I’d prepared which was originally put together on powerpoint.  I demonstrated basic shadow time keeping and direction finding, and how solar eclipses take place using a scale model of the Moon and Earth (with the moon’s orbit inclined at 5 degrees).

Based on our model the Earth and Moon were around 3 meters apart with the former about the size of a large orange.  At this scale the Sun would be 10 meters in diameter and over a mile away!  At this scale the relative rarity of total solar eclipses becomes clear (on average one every 18 months).

During the talk we also touched upon:

  • Sun gods and how our ancestors perceived the Sun as a perfect orb with no imperfections
  • The human fear of eclipses
  • The discovery of Sun spots and how they revealed that the Sun is spinning
  • How spectroscopy showed that our Sun was in fact a star (in very close proximity)
  • Why the Sun is loosing mass – over 4 billion tons of hydrogen per second
  • The ultimate fate of our Sun – how it will eventually and briefly flare up as a red giant star before cooling and shrinking down to a white dwarf

After the talk Clelland took over for some fun outdoor activities including a scale walk of the solar system, DIY spectroscopes and solar lasers using big magnifiers.  We also did a fun experiment simulating the colour of the sky and sunsets using milk in water bottles.

In terms of solar viewing, I setup the 200mm with a full objective white light filter, and we also had a Sunspotter, kindly on load from Glasgow Science centre.  Both setups produced clear views of the Sun’s photosphere, but unfortunately there were no sunspots to see.  This isn’t entirely surprising given we’re currently bang in the middle of the 11 year solar cycle minimum, although large ones can appear suddenly at any time.  We hope to one day invest in a good quality hydrogen alpha filter for these events, as these reveal many more interesting features, like edge prominences and coronal loops.

Overall a fun day of learning with great interaction and questions from the adults and little ones alike.


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Next Abriachan Astronomy Dates

I’m excited to be hosting two more astronomy events alongside the Abriachan forest team in March and April 2018.  Details and ticket links below.

Star Cluster Special – March 10th (moved from Feb 10th) 7pm-9pm

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The Hyades and Pleiades Star clusters

Explore the great winter open clusters under moonless dark skies with campfire stories to follow. Outdoor binocular guiding under clear skies. Indoor talk, astronomy activities and virtual guiding in the classroom in the event of poor weather. Refreshments provided.

Ticket link: https://www.eventbrite.co.uk/e/dark-sky-observingwith-a-sta…

Solar Special and the Life of Stars – April 14th 2pm – 4pm

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A typical G-type main sequence star – locals have dubbed this one ‘The Sun’

A Sun special exploring our nearest star and the life of giant stars. Outdoor sun projections and activities, with illustrated talk and refreshments. Suzann even has plans for a solar pizza oven!

Ticket link: https://www.eventbrite.co.uk/e/some-sunny-science-and-the-l…

All stargazing events organised in collaboration with the Abriachan team, astronomer Stephen Mackintosh and learning coordinator Suzann Barr. Campfire tales delivered by forest ranger Clelland.

For group bookings please email: abriachanforest@gmail.com


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Exoplanet Hunting and other Habitable Worlds

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Artists impression of a giant exoplanet occulting part of its parent star.

Recent advances in telescope and sensor technology have finally allowed us to start answering one of the great unknowns of the 21st century – how many habitable worlds are there in our Milky Way galaxy?

Until recently it was simply assumed that a fairly high fraction of stars ‘probably’ contained planets, and that some further fraction of those would be in the so-called habitable zone.  These assumptions were based from simple extrapolation from our own solar system (along with some inconclusive inferential data gathered in the 1990s).  But as has been painfuly demonstrated with the hunt for life in our solar system, assumptions like this need careful confirmation, and extrapolating from a sample size of one is seldom convincing.  Often, what we want to believe is, regrettably, not compatible with reality.  The video link from the late Carl Sagan at the end of the piece will give you a flavour for what people ‘guessed’ in the 1980s.

The good news is we now have concrete evidence that an abundance of other planets are out there orbiting other suns – in fact roughly 70% of all stars are accompanied by other worlds.  How do we know this?  There are two simple detection methods which I’d like to explain in more detail, which both ‘indirectly’ detect the presence of other planets.  These are:

  1. Transit Photometry
  2. Radial velocity

Transit Photometry

The first method involves closely observing a star over a period of time with a photometer, looking for any subtle changes in its brightness.  Any dip in brightness which is periodic and cannot be explained by general stellar dynamics is then assumed to be a large body passing between us and the star.

Incredibly, even amateurs with 12 inch backyard telescopes have been able to detect these illumination changes for very large planets, producing crude light curve plots that have later been verified by professional astronomers.

transit

As the exoplanet passes in front of the host star the number of photons reaching the objective lens of the telescope drops in a characteristic way.

This method of detection works out to distances of several thousand light years, and allows an approximate calculation of the planet’s size, or radius.  Additionally, for certain nearby stars (if clear spectra can be obtained), drops in light from specific elements can be detected, telling us about the possible composition of the planet’s atmosphere.  For example, if the planet contained a Nitrogen rich atmosphere, we might detect a dip in the intensity of the spectrum corresponding to the wavelength of Nitrogen – this process is called ‘absorption spectrometry’.

The main disadvantage of the transit method is that it relies on a nearly perfect edge-on view of the planet-sun ecliptic from our earth bound position.   Otherwise we simply would not detect any occultation.  Thankfully, we can calculate roughly how often this orientation occurs and account for it statistically.  There’s no shortage of candidate stars out there with systems well aligned for detection.

Radial Velocity

The second method is similar, but instead focuses on the relative motion of the star.  Despite the huge difference in mass between a star and its satellites, as a planet orbits it will impart enough of a gravitational tug to make the star  rotate about a small local axis – tiny but detectable.  The larger the planet the bigger this wobble will be.

Orbit3

An exaggerated illustration of how a large star will rotate about a local axis due to the gravitational influence of a planet.

To detect this regular movement we can look at the light emanating from the star over time and try and detect if its spectra is being shifted by tiny amounts.  By the doppler effect, if the wavelengths of the star’s spectra appear shifted towards the red it must be moving away from us, and the opposite if the spectra is shifted towards the blue.  In this way the sun’s speed and local orbital radius can be calculated, and from that a determination made of the mass of the orbiting planet.

As a quick example, the Sun moves by about 13 m/s due to the influence of Jupiter, but only about 12 cm/s due to Earth.  Incredibly, velocity variations down to 1 m/s or even less can be detected with modern spectrometers, such as the HARPS.  The major limitation of this method is distance.  At the moment it’s generally only useful for star systems up to 200 light years away.

Bringing both methods together, however, lets us form a picture of an exoplanet’s size and mass, and therefore it’s overall density.  From that,  inferences can even be made about the internal structure of the planet.  All this information without even observing the planet!

The Complete Picture

So what do these methods tell us, so far, about the likely number of habitable or earth like planets in our Milky Way galaxy?  The answer is absolutely staggering.

Based on Kepler mission data, as many as 40 billion earth like planets in the habitable zone could be orbiting around red dwarf and sun-like stars.   Taking away the red dwarfs leaves an upper calculation of 11 billion around Sun like stars.  Just think about those numbers for a moment.

That’s as many as one earth-like planet in the habitable zone for every 10 stars in our local galaxy!  A stupendously high number of candidate worlds from which life may have originated.

But here again, we must be cautious.  11 billion is 11×10^9.  But what if the probability of life forming on rocky planets within the habitable zone was actually as low as low as 11 x 10^-9,  or even 11 x 10^-99?  Then there might only be one or no candidate planets containing life.  A depressing possibility, but one we should never allow our natural bias to discount.  The lesson here is that any large number can quickly be diminished in stature by an equally small probability.

At the moment we simply don’t know what this probability of emergent life is.  Some biologists are more optimistic and consider it relatively high for simple single cell life, but other figures, for more complex multi cellular organisms, are much more pessimistic.  But when we do know this figure, calculating the number of planets on which life has arisen will be comparatively simple, and Frank Drake’s famous equation for estimating the number of ‘technical civilisations’ will be one step closer to a final solution.

 

How did we view the question of ‘other planets’ in the context of life outside Earth in the 1980s.  Watch the late Carl Sagan to find out.


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Hyperspace and La Grande Hypercube

Hyperspace.  What does it mean?  Science fiction has largely hijacked the word but its real foundations are embedded in the mathematics of higher or n-dimensional space pioneered by Bernhard Riemann in the late 19th century.  Riemann lay the foundations for extra dimensional spaces and so-called non-euclidean geometries that were self consistent and logical.  His Riemann tensor generalised geometry to curved surfaces, breaking free from over two millennia of Euclid’s dominance over the subject.  In this new geometry of the curved surface, parallel lines can intersect and the angles of a triangle do not necessarily add to 180 degrees.

Bernhard Riemann broke past the established doctrine of 'flat' Euclidean geometry.

Bernhard Riemann broke past the established doctrine of ‘flat’ Euclidean geometry.

‘Therefore, either the reality on which our space is based must form a discrete manifold or else the reason for the metric relationships must be sought for, externally, in the binding forces acting on it’ – Bernhard Riemann

Riemann’s curvature tensor was part of a developmental framework encapsulating the idea that forces could be considered as alterations in the fabric or ‘geometry’ of space rather than an instantaneous phenomena acting at a distance.  The reason a three dimensional creature felt the effects of a force could then be explained by a warpage in the fabric of the four dimensional geometry it moved through.  The analogy of a 2D ant residing on an undulating 3D surface is a good way to understand this thought process.  As the ant moves around it cannot see the curvature of the larger 3D space it inhabits yet it feels the effects as a force as it moves across the world.

Hyperspace in Science Fiction

Han-Solo and Chewbacca are seasoned hyperspace users in the film Star Wars.

Han-Solo and Chewbacca are seasoned hyperspace users in the film Star Wars.

‘Traveling through hyperspace ain’t like dusting crops, boy!’ – Han Solo

Hyperspace has been popularised in science fiction since the late 1920s and early 1930s, coincident with Albert Einstein’s special and general theories of relativity which became widely accepted around this time, and involved the idea of time as an additional dimension.  By directly applying the Riemann tensor to his work on universal gravity Einstein was able to powerfully simplify his approach and construct his general theory of relativity.  It was this application of the multi-dimensional to our understanding of gravity, light and space-time that propelled hyperspace into fiction and art. In a science fiction context the word represents a vague notion of transcending into higher dimensions to facilitate quick interstellar travel (Star Wars), or in the case of Howard Phillips Lovecraft’s ‘Dreams in the Witch House’, a gateway into a shadow realm of cosmic horrors.  It also spawned the notion of the worm hole – a higher dimensional bridge permitting a traveler to move between normally disconnected and independent spaces (or possibly times).

Lovecraft used concepts from hyperspace and non-euclidean geometry in his cosmic horror tale 'Dreams in The Witch House'.

Lovecraft used concepts from hyperspace and non-euclidean geometry in his cosmic horror tale ‘Dreams in The Witch House’.

‘One afternoon there was a discussion of possible freakish curvatures in space, and of theoretical points of approach or even contact between our part of the cosmos and various other regions as distant as the farthest stars or the transgalactic gulfs themselves – or even as fabulously remote as the tentatively conceivable cosmic units beyond the whole Einsteinian space-time continuum.’ – The Dreams in the Witch House by H P Lovecraft.

So how far away is the real mathematics of hyperspace from this popularised notion?  And can mathematics let us glimpse into these mysterious spaces?

Mathematical Hyperspace – a journey into the 4th Dimension

Mathematics can represent higher dimensions with relative ease and in doing so can often reduce formerly complex problems to more elegant and simpler forms. But what can mathematics reveal to us visually or conceptually about higher dimensions?   It’s a vast topic but let’s try exploring one geometrical dimension up – the fourth dimension – by constructing one of the simplest 4D objects to conceptualise, the hypercube.  Follow me on an imaginative journey into the fourth dimension!

Journey into the fourth dimension

  • To kickstart our journey into hyperspace lets begin by imagining a single point in space – a singularity.  A singularity has no dimensions so resides in zero dimensional space – the sort of place the universe may have originated from or a black hole will take you.
  • Now take this point and draw an imaginary straight line from it to a second point.  We’re now in 1D space and have 2 points with a line segment connecting them.
  • Now take each point of our line and drag it up or down to make a square with 4 points.  Here we are in two dimensional space.  How many line segments do we have now?
  • Next  grab  those four points and pull them up or down at a right angle to make a cube.   We’re now in familiar 3-dimensional space with 8 points and 12 connecting line segments.

Now for an imaginative leap into hyperspace.  Let’s pull those 8 points out of our reality and into the fourth dimension!  What do you see?  Or let’s put it another way – what should we expect to see? Although we can’t directly visualise it we can infer some of the properties of this four dimensional cube from mathematical extrapolation.  For starters it should have 16 points. How about edges?  Well each point will connect to 4 edges so that’s 4 x 16.  But each vertex will share all its edges with another vertex so we half this value.  Therefore it must have 4 x 16 /2 = 32 edges.

Dimension Descriptor Points Edges
0 Singularity 1 0
1 Line 2 1
2 Square 4 4
3 Cube 8 12
4 Hypercube 16 32

This 4D analogue of a cube is also known as a Tesseract.  Now that we’ve inferred some of its properties can we say anything about what it might look like?  Well in the same way that a sphere only reveals a shadow of itself on a flat plane, we can only visualise a 4D hypercube from its shadow projected into our 3D world.  As it happens such a shadowy denizen from another dimension actually exists on earth.

La Grande Arche a la Hypercube

Most Parisians walking in La Defense are probably oblivious to the fact that there’s a manifestation of the fourth dimension nearby. La Grande Arche was designed by Danish architect Johann Otto von Spreckelsen.  It’s a mystery whether or not this grand design was ever consciously inspired by mathematics but it’s nonetheless a beautiful and radical piece of architecture.

A real earth based projection of a four dimensional cube - The famous Paris arch

A real earth based projection of a four dimensional cube – The famous Paris arch

Despite being only a shadow of a 4D cube this structure contains most of the defining properties of a 4D cube – crucially it has 16 points connecting 32 edges.  One of the properties it lacks is uniformity of distance along its edges.  In the same way a square in two dimensions has equal sides, a real hypercube in 4D space would have all its connecting edges exactly the same length.  But this projection clearly doesn’t (note the lengths of the inner edges are shorter than the outer edges).  That’s because any projection from a higher to a lower dimension doesn’t necessarily preserve the distance between points, or indeed the angles between edges and faces.  Looking at the shadow of a cube on a piece of paper is a good example of this; no orientation can preserve the three dimensional distances between points.  In other words information is lost whenever we try to represent a higher dimensional object in a lower dimensional setting.

Projection of a hypercube  rotation about a single plane bisecting its middle.

Projection of a hypercube rotating about a single plane bisecting its center.

Mathematics can even rotate the hypercube and we can observe the weird and impossible dance made by its three dimensional shadow (above).  The object is seen to move through itself in a way that defies the rational behavior of solids in our 3D world.  The same effect can be seen by rotating a 3D cube in front of a lamp and watching the behavior of its two dimensional projection.  You would really see these  sort of visual oddities if four dimensional objects or beings entered our 3D world.

n-cubes

Of course mathematically cubes don’t stop in 4 dimensions. You can move into five, six or even more dimensions and build n-cubes in all these worlds. For example, a hypercube in n dimensions will have  2^n vertices. From each of these vertices there will be n edges emerging, each of which are counted twice, so the N-dimensional cube has n.2^{n-1} edges.  I’ll leave the task of imagining such n-cubes to the reader!

Dimension Descriptor Points Edges
n n-cube  2^n  n.2^{n-1}