The supermoon-bloodmoon-harvestmoon phenomenon, and how scientists knew when to be looking

I only heard about the lunar event on the BBC evening news the night before the stunning lunar eclipse occurred. Having been deeply disappointed by the solar eclipse earlier this year, standing out on the school playing field during my maths lesson on a chilly Friday morning in March with my ridiculously geeky solar glasses, gazing up at the hopelessly cloudy sky for a good half hour before resigning myself to gawping at the Faroe Islands' live stream, it seemed a hardly decent replacement service by the heavens - lunar eclipses are less highly regarded, since on average a total one can be seen from any given location on the Earth every 2.5 years [1]; however it really was an amazing sight, seeing the Earth's largest satellite in beautiful mars-like hues, close enough to be viewed by the naked eye of an unequipped enthusiast.

Supermoon 

The reason this particular lunar eclipse, the type which produces the colloquially-dubbed "blood moon", was so special is that it converged with yet another astronomical phenomenon, a perigee of the moon (or "supermoon"). Our moon goes through regular perigees and apogees due to its slightly non-circular orbit, which causes its distance from the Earth to vary slightly over its 28-day cycle (27.322 days more precisely [2]). According to mathematics, the eccentricity of an ellipse is given by e = ^c ⁄ _a = ( ^{distance from centre to focus} ⁄ _{distance from focus to a vertex on the major axis} ) = ( ^{distance from centre to focus} ⁄ _{(As-maj - distance from centre to focus), where the distance from the centre to focus is given by c = √(}A_{s-maj² -} A_{s-min²) = √[(length of semi-major axis)² - (length of semi-minor axis)²], As-maj} is the length of the semi-major axis and A_s-min is the length of the semi-minor axis. [3][10]

This looks quite complex, and it certainly took a while for me to research my way around it, but I stumbled upon the idea of Kepler's Laws, one of which states that the centre of mass of a two-body orbit system is the focus of the elliptical orbit. Firstly, assuming that the centre of mass of the Earth-Moon system is the centre of the Earth (and mind that this is a simplification for now), the eccentricity of the ellipse can be calculated very easily. This is because the apogee distance (where the moon is farthest from the Earth) is the distance from the focus to the distal major-axis vertex, and the perigee distance (where the moon is closest) is the distance from the focus to the proximal major-axis vertex. According to Kepler's Law, the larger body becomes the focus of the elliptical orbit of the other so Earth is the focus of this ellipse.

The apogee distance is 251,968mi = (251968*1 609.344)m = 405503189m
The perigee distance is 225,804mi = (225804*1 609.344)m = 363396312.6m

Therefore the distance from the focus to the centre of the ellipse = (^{(405503189 - 363396312.6)} ⁄ ₂) = 21053438.2m
The distance from the centre to a vertex is 363396312.6 + 21053438.2 = 384449750.8m
The eccentricity = ^21053438.2 ⁄ _384449750.8  = 0.054762522 ≈ 0.05

However its not that simple, since the centre of the Earth is not the centre of mass of the orbital system. In fact, this "barycentre" has an average distance from the centre of the Earth of 4671km - although this is still within the Earth's radius of 6378km [4], this will change the value of eccentricity since the perigee and apogee are no longer so intuitive to use within the elliptical geometry. Since both objects rotate around the orbital barycentre of the system, this will cause the Earth to wobble on its orbit and hence change position at perigee and apogee - the only reason this didn't occur in the previous model was that we took the barycentre to be the centre of the Earth, so it was simply rotating around its axis. The new model looks like this:

This diagram is not in any way to scale - the size of the Earth has been increased, and the eccentricity of the barycentre's location too, to emphasise the displacement of the planet in space caused by its orbit around the Earth-Moon barycentre, whereas in actual fact it is a very small effect since the Earth has a much larger mass than the moon. The Earth at apogee and perigee is represented by the large grey circles, bordered by the corresponding colours.

The barycentre is the new focus of the elliptical lunar orbit.
The distance from the moon to the barycentre at perigee = (363396312.6 + 4671000) = 368067312.6m
The distance from the moon to the barycentre at apogee = (405503189 + 4671000) = 410174189m
The distance from the elliptical centre to a vertex is (405503189 + 363396312.6 + (2*4671000))/2 = 389120750.8m
The distance from the elliptical centre to the geometric centre at perigee = 389120750.8 - 363396312.6 = 25724438.6m
The distance from the elliptical centre to the barycentre = 25724438.6 - 4671000 = 21053438.6m
The eccentricity = 21053438/389120750.8 =0.054105155 ≈ 0.05

The internet, including [5], states that the most accurate calculation of average eccentricity places the moon at around 0.0549. However this takes into account other factors which have been ignored in my calculations, such as the 5° angle to the equatorial plane at which the moon orbits, so I'm pretty satisfied with my model getting to the same approximate answer of 0.05.

The importance of the perigee to increasing the drama of Monday morning's eclipse was that it made the moon appear larger in the sky, since it was at its closest to the Earth. However this effect will slowly wane over the coming thousands of years because the average radius of the moon's orbit is slowly increasing... but more about that later.

Bloodmoon

A bloodmoon, to reiterate, is the colloquial name for the effect on the moon's apparent colour caused by a lunar eclipse - this is when the Earth, Moon and Sun line up in such a way that the Earth casts a shadow over the surface of the satellite rock. [6] has a very good explanation of why eclipses do not happen often (i.e. at every new and full moon) - the angle of the moon's orbital plane means that the Earth's shadow from the Sun often misses it, leaving us without an event to observe.

The red colour was only possible because it was a total lunar eclipse - just as when we see a solar eclipse at totality and the Sun's corona becomes visible, this coronal light will impinge into the Earth's shadow to stain the moon a coppery red. There are other types of lunar eclipse which are less dramatic though: penumbral eclipses occur when the moon passes into the outer fringes of the Earth's shadow, producing a largely indiscernible effect; partial eclipses occur when the moon partially enters the darker area of the Earth's shadow, obscuring the surface in part [7]. As one might expect, more interesting eclipses are inversely proportional to their frequency - that's why this combination of bloodmoon and supermoon is so rare (the last one happened in 1982, and the next will happen in 2033 [8].

Even more interesting is the fact that Monday's spectacle was the last of a series of four total lunar eclipses, occurring in 6-month intervals in 2014-15 - this is called a tetrad [9]. This sequence of events occurs roughly every decade, but used to (and in some cases still do) have biblical connotations until its workings were more formally understood; a quick internet search of "bloodmoon" yields a wealth of religious rapture predictions!

Savour it while we can

  The moon and sun's gravitational fields work together to cause the tides. When the moon is in line with the sun, on the opposite side or the same side of the Earth, the oblateness of the planet is increased as the oceans are pulled by metres towards the celestial bodies (this is at full moon or new moon); conversely when the moon and sun are at right angles to the Earth, neap tides occur where the difference between high and low tides is least (this is at first quarter and third quarter). As the Earth rotates, an observer on its surface will experience high and low tide twice each day as their position passes through the areas of higher and lower planetary radius from the centre to the sea level.

This constant pulling of the oceans naturally creates friction on the sea bed between H₂O/salt/silt in the liquid and rock/sand on the floor. Since energy must be conserved within this closed Earth-Moon system, a loss of energy on Earth will result in an increase in the moon's kinetic energy. An orbiting object with more kinetic energy will move further from the object it orbits - now that the average orbital radius is greater, the moon experiences a slightly smaller acceleration due to gravity towards the Earth (gravitational fields follow the inverse square law, as F = ^Gm1^m2 ⁄ _r2) so a smaller velocity perpendicular to the centripetal force is required to stop the two bodies colliding. It is amazing that this friction is making the moon slow down and move further away, as well as slowing down the rotation of the Earth very slightly (the frictional force resists the rotational motion of the planet) - [11] quantifies this best, since the day in 100 years will be 2ms longer as a result of this effect.

Therefore the moon will slowly move away from the Earth over the coming millennia at an approximate rate of 3.8cm/year [11], or in SI units (3.8/(100*365.242*24*60*60) = 1.204173712 ≈ 1.20nm/s. This is not a hugely significant amount but, assuming that the elliptical shape of the moon remains constant as the average radius increases, the current apogee distance will become the perigee distance in (^{(405503189-363396312.6)} ⁄ _0.038) = 9002812537 years, or approximately 9 billion years!

Sources

[1]   Time and Date - "What are Solar Eclipses?" - here

[2]   Space - "Does the Moon Rotate?" - here

[3]   Maths Open Reference - "Ellipse Eccentricity" - here

[4]   Wikipedia - "Barycenter" - here

[5]   Wikipedia - "Orbit of the Moon" - here

[6]   Space - "'Blood Moons' Explained: What Causes a Lunar Eclipse Tetrad?" - here

[7]   NASA - "A Tetrad of Lunar Eclipses" - here

[8]   Telegraph - "Supermoon lunar eclipse 2015 live: Amazing pictures from the UK and around the world of the 'blood moon'" - here

[9]   Wikipedia - "Tetrad" - here

[10] 1728 - "Ellipse Calculator" - here

[11] Ask an Astronomer - "Is the Moon moving away from the Earth? When was this discovered?" - here

Personal musings and research arising from Serge Haroche's RI Discourse, 25/09/2015

My first visit to the Royal Institution

Friday's lecture from the 2012 Nobel Laureate was a highly interesting overview of quantum effects and their brief histories. I have had previous learning experiences with quantum mechanics at its most introductory level, with thoroughly enjoyable reads such as How to Teach Quantum Physics to your Dog by Chad Orzel and riveting science-to-the-masses documentaries from the likes of  Dr Jim Al-Khalili, but watching a lecture on the baffling topic, in the hallowed theatre of the birthplace of modern science, was a really unrivalled opportunity. It was heartening to know that my prerequisite knowledge allowed me to come away with a good understanding of what was presented, wholly justifying the 7-hour round trip from Suffolk in my finest (and only) smart suit; even more thrilling though was that I have so much to come away and look deeper into, and the purpose of this article will be to convey some of the understanding I think I have gleaned from the realm of the interweb, linking back to what Haroche discussed in his discourse.

[1]

Einstein's Slit argument

Haroche mentioned that there was perpetual disagreement between Einstein and Bohr over the fundamentals of quantum effects throughout their careers - the former was ironically to a large extent responsible for the birth of QM, stemming from his explanation of the photoelectric effect, yet he was one of its biggest critics in the 20th century.


One such criticism arose in 1927 where Einstein laid down a thought-experiment to challenge Bohr; he sought an alternative to the superposition explanation for single photons in a Young's Slits experiment continuing to form an interference pattern over time.

To do this he supposed that an alternative version could be established; instead of two fixed slits, the upper slit would be suspended on springs such that the slightest input of force would cause it to move. Einstein postulated therefore that one could tell whether the photon passes through the upper slit, for a collision would cause the slit to move by conservation of momentum as the particle is deviated vertically. Hence he would accurately be able to measure the position of the photon (by tracing the path back from the screen to the collision point) and the momentum too (by the magnitude of slit displacement), thus violating the principle of indeterminacy.

However, Bohr had several arguments in return.

Firstly, Einstein's model required an extremely precise knowledge of the slit's original position, at a much deeper precision than the measurement of how far it is displaced by the photon, a nearly massless entity.

Also Heisenberg's Uncertainty Principle would mean that having such a precise knowledge of the slit's velocity would reduce accuracy of the slit's position. Even a displacement by half a wavelength would shift the bright patches of the interference pattern towards darkness, by inducing partially destructive interference.

An ideal experiment would average over every possible position of the slit, Bohr argued, and so on the screen the perfect constructive and destructive fringes would be different for each position. This would fill the screen with a uniform grey colour, destroying the interference pattern. A modern explanation for the loss of the interference pattern is decoherence - the two paths of the photon are individually entangled to two macroscopic observational states:

|X> = |goes through top slit>⋅|top slit moves> + |goes through bottom slit>⋅|top slit doesn't move>

In reality this means that such fundamental environmental entanglement causes the wavefunction to collapse (or the universe to diverge!) extremely quickly, in a matter of femtoseconds or even less. Hence, in the usual style of QM, the measurement changes the outcome and ruins the quantum effect.

With reference to whether the wavefunction collapses or the universe diverges at the point of measurement, Haroche was very careful when asked about his opinion to state that it really is trivial in terms of the effects we are observing here - I must say that I agree, since I see no compelling evidence to support either Copenhagen or Many Worlds, so I feel that I can for now banish it to the realm of irrelevance. 

[2]


 Bell's inequalities and experimental disproving of Local Hidden Variable Theory
Einstein and other QM skeptics (including Podolsky and Rosen who collaborated with him on the EPR Paradox) were extremely concerned with the implications of quantum entanglement between particles - in the paradox Einstein referred to "spooky action at a distance" between two particles, produced by the decay of a single particle, such that a measurement of one's spin would immediately allow the observer to know the spin of the other (it would be opposite, since they must cancel to the 0 spin of the original particle). If Bohr's Copenhagen Interpretation was to be believed, neither particle's spin is definite until measured so the fact that the other's is determined immediately would imply that information is sent between the two faster than the speed of light, violating Einstein's relativity. The only explanation consistent with relativity would be that the spin is already defined, but remains a "local hidden variable" until one is measured. It is clear that the EPR Paradox was a serious challenge to QM, intending to expose its current inconsistencies with classical mechanics.

John Bell came up with a method of testing this paradox. Instead of measuring spin, the focus for his experiment was to use the polarization of photons, of which identical pairs would be generated by a decay process in an atom. Three polarizing filters would be available to use to test each entangled photon, and the results of each binary reading could be compared.

The key is to use this information to generate another table, where we can determine for a given permutation if the results will be the same or different for any two filters:

The probability of two filter readings being the same is always at least 1/3, and this is what Local Hidden Variable Theory would predict (since measuring each photon cannot affect the value of the other). However experimentally, the probability of getting the same reading is much lower, at around 1/4. Therefore, since the experiment violates the Bell Inequality,

⅓ ≤ P(same)

local hidden variables cannot explain away the "spooky action at a distance". In reality this is the simplest explanation of the more complex field of Bell Inequalities, which I will endeavour to explore in greater depth in the future, but I must thank DrPhysicsA for this excellent explanation in his youtube tutorial.


[3]

The concept of the universal wave function 

One of the questions put forward to Haroche was this: "Is all matter in the universe (or indeed multiverse) entangled to form a single universal wavefunction?". The answer the Nobel Laureate provided was very simple and quite sensible - yes, but it is of no mathematical use to consider it in its totality because the resolution of knowledge of microscopic physical systems would be lost, so it could not be applied to any laboratory experiments to improve our understanding of QM (like the experiment detailed in the above section).

It is curious to think that the space-time fabric of the entire universe plays out as the consequence of one giant probabilistic function, and does a certain justice to the idea of "God playing dice" - throughout the study of the history and development of QM , Einstein's objections continue to be thrown up and this refers to his famous quote "God doesn't play dice": current theories would suggest otherwise.

Rydberg atoms

A Rydberg state of an atom is where one or more electrons are excited enough to have a very large principle quantum number, many energy levels above the core electrons (which remain in their normal positions as defined on the periodic table). Since the excited electrons are in higher energy states, they experience a lesser attraction to the positive nucleus so occupy vastly wider orbits - this means, paraphrasing phys.org, exciting the outer electron of a rubidium atom from n=5 to n=18 would extend the atomic radius from 1nm to 700nm.

Rydberg atoms are a viable method for storing quantum information as qubits because they can be "sustained for a long time in a quantum superposition system", and interact strongly such that they would form stable and effective logic-gate-type systems.

[4]

Cavity quantum electrodynamics to count photons

Haroche's Nobel Prize winning paper was the source material for this section, which seems only fitting. The quantum cavity is based on the Bohr-Einstein photon box, yet another hypothetical piece of apparatus to constitute the battleground for thought experiments over quantum mechanics. As I understand it, the cavity consists of two mirrors which continually reflect photons inside the cavity until absorbed - the mirrors were constantly plagued by slight imperfections reducing the lifetime of the experiment, but a collaboration at the French Atomic Energy Commission led to precisely machined copper mirrors, covered in superconducting niobium, which form a quasi-spherical surface; as a result, a photon lifetime of 130ms was achieved in 2006.

To actually count the photons inside the cavity, the use of lasers produced rubidium Rydberg atoms with an outermost orbit diameter approximately 1000 times larger than the ground state version - a condition of a stable orbit is that the De Broglie wavelength divides as an integer into the circumference of the orbit, leading to the principle quantum number of 51 or 50 for this experiment. These circular Rydberg states allow a long lifetime of 30ms, on the same order of magnitude as the photon lifetime - this means that the production of photons from the decay of the excited electron orbits can initially be discounted from the uncontrollable variables of the experiment.

In the two states, e and g, the wave has uniform amplitude around the orbit, leading to a electron charge density centred on the atomic nucleus. However a pulse of resonant microwaves brings it the electron into a superposition of both e and g states, causing constructive interference on one side and destructive interference on the other. The result of this is that a net electric dipole is created, extremely sensitive to microwave radiation (i.e. the photons being counted).

Non-resonant microwave photons are not absorbed by the Rydberg atoms, making the process transparent to the entry of exterior photons. However tuning the cavity photons to very close to the phase-shift frequency of the Rydberg atoms allows them to exit the cavity with a dipole shift of up to 180 degrees. Such a shift is equivalent to a single photon, which allows them to be discretely counted.

[5]

The acceleration of light between media of different densities

My final talking point stems from yet another point raised by the audience during the question time: does light literally accelerate when it passes between two media of differing densities? We learn at school that light has different speeds in different materials, standardised in the form of refractive indices, yet it goes against classical mechanics to assume that there is an instantaneous change in the velocity of photons across this boundary - a change in velocity over ∆t ≈ 0 would involve a → ∞ (a = ^(v-u)⁄_t). Since F = ma, the resultant force on any mass-possessing body would also approach ∞N, tearing the object apart.

However in my opinion this is exactly what happens when light passes between two bodies. By Einstein's laws of special relativity, a massive object with a velocity approaching the speed of light will have a mass approaching infinity - since light cannot be weighed in the same way one might measure the weight of an apple or a car, it must be assumed that a photon is massless (this concept is, in retrospect, universally agreed across the physics community too). Since a photon is massless, it is not constrained by the limit placed on instantaneous acceleration for massive objects, so it seems completely reasonable that a photon will instantly change velocity at a boundary between media.

A related topic refers to the strangeness of the limit placed on universal speeds, relative or not, by Einstein. The confusion occurs when it is suggested that two spaceships are travelling very quickly towards each other. Ship A has a beam of light coming out of the front, naturally travelling at the approximate speed of 3.0e8ms^-1. Classical mechanics would tell us that the relative velocity of the light beam, according to an observer sitting in the pilot seat of the other spaceship, would be (c + v_A - v_B)ms^-1, where v_B is the velocity of ship B and v_A is the velocity of ship A. However this value will have a magnitude greater than c, and hence is not possible according to Einstein's axiom.

However special relativity has an explanation for this. The consequence of the principle of time dilation is the principle of length contraction at speeds close to the speed of light, so from the view of the observer the light beam is blueshifted towards the higher-frequency, lower-wavelength end of the visible light spectrum. This constitutes the physical manifestation of the exceeding of the speed of light, allowing the observable speed of the light to remain at the familiar constant c.

Perhaps this is why the surroundings of the Millennium Falcon shift towards the violet end of the visible light spectrum when Han and Chewie jump to lightspeed?

Sources

September 2015 Friday Night Royal Institution Discourse - Serge Haroche - "Light and the Quantum" (the recording can be found on the RI official youtube channel, here)

Wikipedia - "The Bohr-Einstein Debates" - here 

DrPhysicsA - "Bell's Inequality" - here

Phys.org - "Tuning up Rydberg atoms for quantum information applications" - here

Nobelprize.org - "Controlling Photons in a Box and Exploring the Quantum to Classical Boundary" - here