Venus as the evening “star”, in the lower left hand corner above the lake and to the right of the less-bright Jupiter. Picture by Shawn Malone(space.com)

From a comfortable position on Earth’s surface, Venus does look beautiful. It is one of the brightest objects in the sky; only the moon and the sun and some future nearby nova or supernova can outshine it. When Earth turns away from the Sun, it also turns away from Venus. Being an “inferior” planet, meaning that it is closer to the sun than we are, Venus can only be seen shortly before and after sunrise or sunset, depending on where it is on its journey around the sun.

There is one of a few special positions during such a revolution around the sun, which has led to a revolution in scientific thought and to another aspect of beauty, a beauty where observations, mathematics and astronomy combine to reveal simple relationships. One of those positions occurs when our line of sight from Earth to Venus form a tangent to the approximate circle of Venus’ orbit. As we know, a circle’s radius is always perpendicular to a tangent. In this case the radius is the distance between the sun and Venus. (In reality the orbit is slightly elliptical, but we’ll assume it’s a circle to simplify the math and to get the general ideas across.)

Shown to the right is what it looks like on paper, but in the sky, how do we recognise such an event? When Venus is at that position, it’s as far as it can be seen from the sun, here on Earth. We dub it the greatest eastern elongation because it’s stretching away from the sunset in the west. That we are indeed at that point is verified by measuring the maximum angle between Venus and the setting sun (easier said than done!)  On paper that would be the angle at the vertex of the earth, close to 46º . Since the tangent angle at Venus is 90° , we can also easily compute the angle at the sun.

With either angle, we could get the Venus-sun distance relative to Earth’s, which is set at 1 astronomical unit (AU). Using a simple ratio from trigonometry, specifically sine, we use the angle directly measured from Earth, and it yields a relative Venus-sun distance, rv, of 0.723 AU.

We will come back to relative distances. For now, let’s move on to another astronomers’ measurement from the distant past. How  long does it take for the greatest eastern elongation to reappear? It takes longer than a year on Earth, about 583.9 days. Dubbed the synodic period, this is not the length of the Venusian year. Venus has a shorter path to complete and it experiences a greater force from the sun, so why does sit take so long for the same alignment to reappear?

On Venusian year later, Venus has returned to its original spot, but Earth has not yet completed a revolution. As a result, Venus is not at greatest eastern elongation; in fact at this point in time it’s not even visible from Earth.

It’s because the earth keeps moving along its orbit while Venus does likewise, so Venus has to gain 360° on Earth, not merely move 360º. But from that realisation and synodic period-measurement we can deduce its period. How?

First lets’ figure out how many degrees, measured from the sun, Venus gains on earth. That’s 360 ° ÷ ( x ° /day) = 583.9 days, where x turns out to be 0.6165 °  /day.

About 584 days after the previous greatest eastern elongation we have the same alignment. Notice, however, that both Earth and Venus are forming the same angles in different positions in space.

Now of course the reason that Venus gains that many degrees on Earth daily is that it’s moving faster, so that number is the difference between Earth’s orbital angular velocity and that of Venus. The orbital angular velocity is 360 ° /P, where P is the period of a planet. Writing the previous sentence mathematically we obtain,

360 ° /P –  360 π /365.25 = 0.6165 °  /day

Solving the simple equation we obtain the period of Venus to be 224.7 days.

The period , P is also equal to the circumference of 2πr divided by the planet’s velocity, v, expressed in whatever units the radius has per second.

P = 2πr/v                                … call that equation (1)

This implies that the angular velocity = 360 ° /(2πr/v)= 180° v/ πr. But if we convert the degrees to radians to make the numerator dimensionless, given that π represents 180°  in a unit circle, angular velocity = v/r in days -1.

Venus has a mass, mv , and it has an approximately fixed speed vv. Since it’s moving in an almost circular ellipse, its direction constantly changes. Its speed is constant, but its velocity , a vector quantity is not. Thus Venus is accelerated by a force. How do we express that force?

Its momentum (product of mass and velocity) is equal to its impulse, expressed as the integral sum of the product of force and time, acting over that period of time. To simplify:

mv vv= Ft

Dividing by t,

the force , F = mv vv/t =mv vv(t-1). The (t-1) is in essence a frequency which in this context is none other than the dimensionless angular velocity of Venus, vv/r.

Substituting we obtain, F = mv vv 2/r, which is the expression for the centripetal force experienced by Venus—the force which keeps it moving around the sun.

Historically, Newton based his Law of Gravitation on Kepler’s laws, which in turn were based on observations of the planets. But to keep the narrative going, let’s time-travel backwards and arrive at Kepler’s laws, which will still reveal the consistency between the two.

The gravitational force between the sun and Venus is also given by the product of their masses(ms and mv) and the universal constant, G,  divided by the square of their separation distance, rv, which we determined by trigonometry earlier.  Equating centripetal force to Newton’s law, we obtain:

If we solve for the orbital velocity for Venus (vv)  we get vv = √(Gms /rv) . With a similar treatment Earth’s orbital velocity = ve = √(Gms /re). If we want to know how much faster Venus revolves relative to earth, we could simply divide the two expressions and we see that

vv / ve = √(re/rv)                                                     … call that equation (2)

Rearranging equation (1) , v = 2πr/P.  Now we could also express the ratio of velocities as the following:

vv / ve = (2πrv/Pv ) /  (2πre/Pe ), where Pv= period of Venus and Pe is the period of Earth.

Simplifying the above,  vv / ve =  rv Pe /re Pv.  …call that equation (3)

We first equate equations 2  and 3 , and after squaring both sides we get:

(re/rv) = (r² Pe² )/ ( re² Pv²)

Finally, we cross multiply and we see Kepler’s third Law emerge:

Pv² re³ = Pe² rv ³

If we set rto 1 astronomical unit (AU) and use our value of rv = 0.723 AU from trigonometry then

Pv = √ [(365.25)² (0.723) ³] = 224.5 days, fairly close to the value that we calculated from the synodic period.

Arrhenius was an imaginative scientist who gave us the electrolytic theory of dissociation, a correct prediction about the Earth’s climate. But his Venusian vision (1927) was fantasy.

Venus has a runaway greenhouse effect. Being too close to the sun, early in its evolution its water was dissociated by ultraviolet radiation. As a result the CO2 that was out-gassed was not kept in control by a cycle in which the majority of carbon dioxide could be dissolved. That made the planet unbearably hot. Interestingly, in the 19th century Svante Arrhenius, who  was able to to foresee that Earth could one day suffer from excess industrial CO2 output, imagined Venus as a lush, tropical planet in his book, Destinies of the Stars. Instead, due to oven-like surface temperatures (~460 ºC)  but mostly due to the high density of CO2, the pressure at the surface is a crushing 10 000 kPa. That in turn creates strong tidal forces, slowing its rotation to the point that its solar day is longer than its year. But doesn’t all that mathematical and physics- harmony compensate for the fact that Venus is not a green paradise filled with beautiful women?

In the silly movie Abbott and Costello Go to Mars(1953), the duo ends up on Venus inhabited by women, continuing a trend in fiction that reinforced the idea that its climate could support life.

# Neutrinos: the basics and why they matter

1. What is a neutrino?

A neutrino is a subatomic particle, grouped with other leptons that include the electron, muons and taus.The charged leptons and uncharged neutrino are all classified as leptons for a number of reasons, one being that neither is affected by the strong force, which keeps quarks together as either neutrons or protons. Due to their small size, leptons have an extremely weak interaction with gravity. But unlike charged leptons, neutrinos are not affected by the electromagnetic force. This is why they come to us straight from their source. We’re showered with them everyday from the sun and from the rest of the universe. If we are in the Northern hemisphere they also come at us from the Southern sky, through flesh and land without leaving a trace, and through the planet.

a) Part 1 : the theory

Before experimental evidence for a type of neutrino was obtained in 1956, their existence was proposed by Pauli in 1930 to secure energy, momentum and spin conservation in weak interactions. (His statement was preceded by a funny formal address “Dear radioactive ladies and gentlemen”. ) In beta decay, electrons are made in and emitted from the nucleus. But when the kinetic energy distribution, or spectrum, of beta particles was measured and compared to that predicted by relativistic energy, it seemed like energy was not conserved. But a small, neutral unknown particle, the neutrino, could be carrying off the unaccounted kinetic energy.

In Italian, the “one” suffix signifies that something is big. The plural becomes oni, as in spaghettoni. The opposite is ino (plural ini) as in thin spaghetti or spaghettini. Because the Italian word for neutron is neutrone, Fermi proposed the name neutrino for the smaller neutral particle.

What also confused physicists at the time is that for a while they did not immediately realise that another neutral particle, the neutron, existed. They believed that “nuclear electrons” along with protons were part of the nucleus. For example, the nitrogen nucleus, with a mass of fourteen and a charge of +7, was thought to consist of 14 protons and 7 nuclear electrons for a total of 21 particles, each known to have a fractional (1/2) spin number. Yet a proposed sum of 21 contradicted the evidence. The nuclear spin of nuclei with an even-numbered nuclear mass such as nitrogen-14 had been measured, and it was known to be an integer. Odd-numbered nuclei would have fractional spin numbers of 1/2, 3/2 etc. But if a neutron existed and its mass was respectively similar to that of a proton and its spin number was also fractional, then it implied that nitrogen actually had 7 protons and 7 neutrons. With an even number of particles, its spin number could indeed be +1.

But how is all this related to the neutrino?

Consider the beta decay of  carbon 14, a radioactive isotope in our bodies. If we assume that the even-numbered carbon-14 ( 14C ) isotope is only turning into the stable nitrogen isotope, 14N, and a beta particle -1e, not only is there kinetic energy missing, but we seemingly have an odd number of particles produced (14 for nitrogen and the beta particle for a total of 15), each having fractional spin numbers . That would not conserve spin number. So another particle must be carrying some of the energy and that particle must have a fractional spin number. All leptons do, and in this case, the other particle produced is the electron antineutrino ( ), a form of antimatter:

14C → 14N +-1e +

a) Part 2 : the experiment  Actually first, let’s dish out a little more theory. The lepton number is a conserved quantum number in an elementary particle reaction. All leptons have assigned a value of +1, antileptons −1, and non-leptonic particles like the neutron and proton, 0. In case you wondered why an antineutrino was produced in the  14C reaction, it was to preserve the overall lepton number of zero for the 14 nuclear particles of 14C .

The following reaction is sound in the sense that it also conserves lepton number.

Among the reactants we have the antineutrino with a lepton number of -1. And a positron should be produced, as opposed to a beta particle, which not only conserves the lepton number of -1 but also conserves charge. The neutron produced will be captured by some nucleus and in a second reaction, the positron will be annihilated by an electron, creating gamma. To identify the observed signal as neutrino-induced, Cowan and Reines used nuclear reactors to compare energies of the two pulses, their time-delay spectrum, the dependence of the signal rate on reactor power and its magnitude. They used two different detectors (a water based one and cadmium) for neutrons and a number of other experiments to make sure they were not fooling themselves. And it turned out that the data could only be explained by the presence of neutrinos.

3. Types of Neutrinos and the Solar Neutrino Problem Solved

Every flavour of neutrino has its antimatter counterpart. We’ve seen the electron antineutrino whose counterpart in matter is the electron neutrino. There are also muon and tau neutrinos and their respective antimatter counterparts. At one point theoretical physicists were questioning the neutrino detecting measurements because they did not confirm their theoretical models of the sun’s fusion reactions. Based on their understanding of fusion reactions, not enough electron neutrinos were being detected from the sun. ( Electron neutrinos are produced in the first part of the sun’s proton-proton chain reaction.

In this reaction a proton is being converted into a neutron,positron and neutrino. Notice how the lepton number of zero is conserved by having a neutrino (L= +1) accompanying the emission of the positron (L= -1).

Meanwhile, others were wondering if the experiments were revealing some flaw in the theory.

It turned out no one was actually wrong. They knew that the original chlorine-37 detectors were only sensitive to electron neutrinos — upon being captured by the unstable chlorine isotope they would be converted into a measurable argon-37 isotope. But this was done deliberately because the sun is not a direct source of either tau or muon neutrinos. Eventually measurements of other flavors of neutrinos, especially those from the Sudbury, Canada reactor, revealed that 2/3 of the sun’s electron neutrinos flipped flavors on their way to earth. Eventually investigators revealed a connection between the oscillations and mass differences of the different types of neutrinos and another neutrino-related Nobel was awarded in 2015.

4. High Energy Neutrinos

In Antarctica , there is a detector of high-energy neutrinos named IceCube. It’s a cubic kilometre of ice equipped with 5160 optical sensors that collect a shower of charged particles radiating blue light known as Cherenkov radiation. The shower is created after rare collisions between neutrinos and the nuclei of pure ultra-transparent ice, and the radiation can travel hundreds of meters to the detectors.  So far IceCube has detected high energy neutrinos with one to two billion times the energy of solar neutrinos. The suspected sources of these particles are supernovae, gamma ray bursts from large collapsing stars, supermassive black holes or even other exotic possibilities. It’s been postulated for example that the decay of heavy, long-lived dark matter particles, if they exist,  could also produce such signatures.

The largest pool of almost 100 000 photons was created by neutrinos dubbed Ernie and Bert, named after the Sesame Street characters. Their energies were 1.07 PeV and 1.24 PeV, respectively. When the number of such high-energy neutrinos totalled 54, the signals’ significance exceeded 5 standard deviations, implying that it’s highly unlikely that the observations are not atmospheric phenomena. (At sigma-5  the probability is 1 in 3.5 million that if the high energy-particles do not exist, the data that IceCube collected in would be at least as extreme as what was observed.) Some of the neutrinos were not from our galaxy, but Bert was within a degree of the galactic plane, which is rich in supernova remnants and is also the host of a giant black hole.

The signal from “Ernie”. Each sphere represents one optical sensor; the coloured spheres show modules that collected light from this event. The sizes of the spheres are proportional to how many photons each module recorded. Finally, the arrival time of the first photon is symbolised by colour from red (earliest) to blue (latest). Source: AntarcticaNeutrinos.blogspot.ca

They are hoping to build a larger collector, one that could conceivably collect GZK- neutrinos, made by interactions with the big bang’s afterglow. GZK could have energies 1000 times bigger than Bert’s. So far none have been found, and this blog from astronomer Spencer Kline,who is on site in Antarctica, provides updates.

Sources :

Detection of the Free Neutrino: A Confirmation Science Vol. 124, No. 3212, Jul. 20, 1956

NEUTRINO OSCILLATIONS compiled by the Class for Physics of the Royal Swedish Academy of Sciences

A Brilliant Darkness. Joao Magueijo. Basic books. 2009

Neutrinos at the Ends of the Earth. Francis Halzen (chief investigator of IceCube project). Scientific American. October 2015

# Cherry Juice-Chemistry in the Kitchen

What did I just photograph? It’s a “volcanic” island of baking soda in a sea of tart cherry juice. In a nutshell, there are acidic ions within tart cherry juice and they’re reacting with the baking soda’s bicarbonate ion, creating water and bubbles of carbon dioxide. But there’s more. Excess bicarbonate also changes the structure of the cherry’s anthocyanin-pigments. These natural indicators are pH-sensitive; they are red at low pHs and purple under slightly alkaline conditions. The movement of water molecules has not had a chance to spread the bicarbonate ions beyond the centre, and the rest of the “sea” remains red.

Found in young shoots, flowers and autumn leaves, an anthocyanin is a molecule with a variety of physiological and ecological roles. An anthocyanin consists of an anthocyanidin and a sugar component. Cherries along with apples, and strawberries and a few other fruits all contain a type of anthocyanidin called cyanidin. Delphinidin is another version found in flowering plants as diverse as violets, larspurs and certain grapes. And there are many more compounds, but delphinidin and cyanidin are two of the six most common ones. From Chemspider.com, here is the structure of cyanidin in 3D and 2D:

All anthocyanidins with cyanidin have the cyanidin salt-pigment(the negative ion attached is not shown), but the sugar on the cyanidin can vary. The table below reveals the ones found so far in a couple of cherry species; the numbers in the table represent concentrations in parts per million. Note that varieties of the two species of cherries analysed have at least two different sugars attached to the cyanidin, and both tart cherries and sweet cherries can also have peonidin (pn). Some sweet cherries can have a third anthocyanidin known as pelargonidin (pg).

from Recent Advances in Anthocyanin Analysis (see full reference below)

Anthocyanidins differ from one another in having different “R groups”, at the numbered positions of the flavylium core

The flavylium ion is the basic unit of all anthocyanidins, which when linked to a sugar are known as anthocyanins

The only difference between cyanidin, peonidin and pelargonidin is that attached to the carbon at the 3′ position, we find OH , OCH3 and H, respectively. If the ratios were reversed, it would impact the colour of cherries (pelargodinin for example is purple, while peonidin is purplish red). But as we can see from the table, all cherries have far more cyanidins than any other pigment, so both the different reddish hues of cherries and the changes due to pH result mostly from cyanidin.

Bu why is the cyanidin compound in cherries red at pH < 3, violet at pH 7-8,  blue around pH  11 and has other colours at higher pHs?

Shown is cyanidin and how its structure changes with pH. A higher pH leads to further changes in structure and bring about more colour changes.

The mechanism revealing electron flow in the conversion of the red form of cyanidin to the purple form.

At a pH above 7, we get an excess of hydroxide ions relative to the small amount of hydronium ions contributed by water. The former remove an H+ ion from the OH group at position 4′ in the structure of cyanidin (see step 1 in adjacent mechanism) responsible for a red colour. This forms water (HOH) and starts a cascade of moving pairs of electrons. First, one of the oxygen’s unbonded pairs attaches itself to the hexagonal ring to form a double bond (step 2). This displaces a double bond away from the first carbon to an adjacent carbon(step 3). That in turn shifts the double bond to the two hexagonal rings (step 4), finally eliminating a double bond from the heterocyclic ring and also getting rid of the oxygen’s positive charge in step 5 . For each of the red and blue structures we are showing just just one of the possible resonance structures, but the important thing is that the electron movement creates a new conjugated system (alternating double bonds). As a result the gap between π-bonding and π-antibonding molecular orbitals changes. Different frequencies of visible light are now absorbed, leading to the reflection of purple frequencies, hence the colour of the island in our picture. If another H+ ( a proton) is stripped from the violet-reflecting structure by higher concentrations of hydroxide found at a higher pH, we free up electrons from another oxygen, again changing the conjugated system. A bluish colour is the result.

If instead of adding baking soda to the cherry juice, we try spraying a little oven cleaner which contains sodium hydroxide, we will see green instead of purple (pictured above, on the left). Higher concentrations of hydroxide bring about more structural changes, resulting in a different colour. What’s more interesting is that if we add NaOH directly to a purple solution created with baking soda, we will witness a buffer at work and see different hues of purple. Stubbornly they do not shift to green very easily. (see my above picture on the right)

How is the buffer created? First there was excess bicarbonate ion(HCO3-) lingering after the neutralisation of malic acid from the cherry juice. Then we got carbonate created by the action of HCO3- on NaOH. HCO3- is amphoteric. In an acidic environment it acts as a base, but in NaOH’s presence HCO3- acts as an acid.

NaOH + HCO3(aq)  H2O(l) + CO32-(aq)

As long as the bicarbonate is still in excess, which is likely given that we added it in solid form,  we have the following buffer of a weak base (carbonate) and its conjugate acid ( bicarbonate).

HCO3(aq) + H2O(l)        H3O+(aq) + CO32-(aq)

Like all buffer systems, the bicarbonate-carbonate system can withstand small amounts of strong acid or strong base, as long as the pair of key buffer species is not consumed. For example, adding another 1.0 ml of 0.1 M NaOH to a buffer containing 0.0036 moles of bicarbonate and 0.0001 moles of carbonate will only move the pH from 8.8 to 9.2. This can be easily verified using stoichiometry and with HCO3- ‘s acid dissociation constant (KA) of 4.8 × 10-11. And a pH of 9.2 is not high enough to yield a blue colour, let alone green.

Interestingly, a similar buffer system is present in our blood and protects us from fluctuations in blood pH, which would adversely affect the functioning enzymes among other disruptions. The difference is that the bicarbonate buffer we discussed is based on bicarbonate ion and carbonate, whereas our blood uses a buffer at a lower pH by having a mix of bicarbonate and carbonic acid. In geochemistry both buffering systems are found in lakes near limestone and in all oceans. These are important in protecting ecosystems from dramatic pH-swings due to volcanic or industrial emissions.

Sources:

Recent Advances in Anthocyanin Analysis and Characterization
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2783603/

Anthocyanidins https://en.wikipedia.org/wiki/Anthocyanidin