Adventures With Venus

venus-jupiter-conjunction-malone
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.)

venus1Shown 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?

venus1yrlater
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.

venussynodic
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:venusformula

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-us-300
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?

abbott-and-costello-go-to-mars-lou-costello-venus-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.

Cosmic Origins of Atoms in a Mineral

A mineral is more pure than its parent rock. But compared to food additives, industrial compounds and pharmaceuticals, a mineral’s compound often hosts more elements. As a result it isn’t difficult to find a mineral whose atoms have a variety of cosmic origins.

Only a small percentage of elements on Earth are created in and around the planet by nuclear reactions, and even at that, they are only derivatives of atoms made elsewhere. The secondary creations result from the atmosphere’s interaction with cosmic rays, from the lithosphere’s minority of radioactive elements, from nuclear reactors and from scientific research—my favorite being the tanks that sit deep in abandoned mines collecting neutrinos from our sun and supernovae.

pezzottaite-256889
pezzottaite

So where in space did the majority of constituents of the living-geological continuum originate and by what mechanism? Let’s look at the cosmic roots of the six elements of a mineral known as pezzottaite, discovered in Madagascar and only officially recognized as a distinct mineral in 2003. Its formula is Cs(Be2Li)Al2Si6O18

Oxygen

What’s the ultimate source of oxygen? Big or small, stars spend the bulk of their time on the main sequence, a hydrogen-fusing stage that actually lasts longer for smaller stars. This is because a star’s lifetime is proportional to its mass but inversely proportional to the fourth power of its core temperature. Although small stars have less hydrogen, the smallest of the chemical elements, they also fuse it at a lower temperature from the lower force acting on its core. The product of the sequence of reactions involved in the fusion of hydrogen is helium. While helium grows as an onion-like outer-layer during its residence as a main sequence star, the temperature isn’t high enough to fuse the helium into bigger elements. But when the hydrogen fuel runs out, the star is for a while no longer in equilibrium. The outward radiative pressure isn’t there to balance out gravity, so the large force towards the star’s center “ignites” the fusion of helium and the star becomes a red giant.

red-giant
from J. Chem. Educ., 1990, 67(9), p 726

When the star’s core temperature reaches 108 K, from the diagram we see a pair of helium nuclei fusing to form an unstable beryllium nucleus, which then fuses to give us the life-essential carbon. This in turn fuses with another helium to produce oxygen. Oxygen can continue to fuse, but there are enough nuclei that remain as such. When stars, in a later stage of their evolution, either shed their outer layers either as a planetary nebula or supernova, stellar dust receives these oxygen atoms, some of which ended up in our water , skin and in our pezzottaite.

Silicon

To get silicon we need a more massive star capable of generating a red-giant-temperature and density of 500 million K and 5 million g/cm3. Under these conditions two oxygens (atomic number 8) will combine to create a silicon nucleus(atomic number 16). Fittingly some of this product and progenitor are eventually reunited on planets as sand, sandstone, quartz, clay and a wide variety of minerals that contain either silica or some form of silicate— including that of pezzottaite.

Lithium and Beryllium

A neat thing about pezzottaite is that it has two(lithium and beryllium) of three light elements that are relatively rare in the universe. The presence of each of lithium, beryllium and boron is only one billionth that of hydrogen and about a millionth of that of carbon, nitrogen and oxygen. The reason for this is that the bulk of Li, Be and B do not survive any of the stages of stellar evolution. Their fragile nature suggested they were synthesized in low-density, low-temperature environments.

galactiv
The broken line represents solar system abundance of the elements Li, Be and B. The solid line shows enrichment found in galactic cosmic rays. From J Chem Ed 1990, 67(9)p 729

One area where a high concentration of these light elements occurred was in galactic cosmic rays. This suggested that perhaps they are not being carried from elsewhere but being synthesized on the spot by the nuclear reaction between alpha particles(helium nuclei) or protons of the cosmic rays and larger elements like carbon, nitrogen and oxygen. This process is now called spallation. While other genesis-models failed to predict the exact concentrations of the isotopes, the spallation hypothesis came closest to account for the relative ratios.

cyclotron
scan from scientific American May 1987, from an article by researchers Viola and Mattheson

In the 1980s Viola and Mattheson used a cyclotron to accelerate protons and helium nuclei to the energies of cosmic rays and aimed them at targets of He, N, C and O to generate new nuclei. When the particles’ energies and speeds were analyzed, their calculated masses allowed them to identify the isotopes. Their abundance was similar to that found in galactic cosmic rays. The three reactions that created a fair amount of the lithium and beryllium in our pezzottaite are:

4He + 12C → 7Li + 2  4He + 1H (main isotope of lithium, 92.5% of what’s found on Earth)

2 4He → 6Li  + 1H + 10n

1H+ 14 N  →9Be + 4He + 2 1H

lithium
from http://subarutelescope.org/Pressrelease/2015/02/18/index.html

One anomaly, however, was that the amount of 7Li ( the heavier isotope) made in the cyclotron was a little lower than what’s actually found in space, suggesting that a minority of 7Li was not made in the cosmic rays but originated elsewhere. Some of the discrepancy is partly accounted for by the small amount made in the Big Bang, but in 2013, analyses of the Subaru Telescope High Dispersion Spectrograph revealed that a more significant contribution comes from novae. Smaller stars eventually become white dwarfs after passing through the red giant stage. But these remnants, if part of a binary system, could suddenly brighten from explosive nuclear reactions when material from its partner-star is pulled onto the dwarf’s surface. The nuclear reactions create a different series of elements compared to those produced in stellar interiors or during supernova explosions. One of these atypical reactions is the conversion of beryllium-7 to lithium-7 by electron capture, which lowers the atomic number without affecting the atomic mass.

Cesium

To explain the origin of the final two elemental components of pezzottaite, Al and Cs we need to examine supernovae. The next avenue of evolution of large stars is a type II supernova, which briefly outshines its entire galaxy. When all fuel is spent in large stars iron is left at the stellar core and a gravitational collapse ensues. In a rapid process-set of reactions ( r-process), neutrons  are initially generated by the gravitational collapse during photodisintegration, a process where gamma causes the fission of heavier nuclei. For example here’s a sequence of reactions generating a total of 7 neutrons from a single iron nucleus:

56Fe + ϒ → 13  4He + 4 10n
4He + ϒ → 2 ‘H + 2 10n

‘H + e- →   10n + ν

Then in the actual r-process neutrons are captured to form unstable neutron-rich isotopes which then undergo beta decay and turn into elements of higher atomic number.How does this happen? A little background info: Being electrically neutral, neutrons can penetrate the positively charged nucleus, especially at low temperatures. But free-roaming neutrons are short-lived lasting only about ten minutes as one of their down-quarks becomes an upquark, a process that generates a proton, a beta particle and an antineutrino. This raises the atomic number by 1. To generate sufficient numbers of neutrons and provide a constant supply of these ephemeral neutral particles, the high-energy environment of something like a supernova is needed. For example with the provided energy, gamma will break down enough iron to generate  enough neutrons, which in turn can convert other iron atoms (atomic number 26) into heavier iron isotopes, one of which will beta-decay into cobalt (number 27). That isotope of cobalt can then absorb more neutrons and eventually undergo beta decay to create an even higher-numbered element, Ni. The isotopes created by the r-process are not the stable ones of the heavier elements. But they can later become stable ones by undergoing fission and beta decay.

Once the stellar material has been enriched with the ejection of these new atoms, subsequent generations of stars can generate other isotopes in the slow process (s -process), which also involves absorption of neutrons but at a slower rate. In a less violent environment such as that of a red giant, the absorbed neutron has time to decay into a proton so it tends to produce isotopes of medium to lower atomic numbers. For example some of  pezzottaite’s cesium 133 (atomic number 55) could have been directly produced by the breakdown of an isotope created by the r-process or it could have formed later in another generation of stars by the beta-decay of  xenon 133:

13354 Xe  →133 55 Cs +0 -1 β

As shown in the diagram below, the unstable xenon 133 isotope was in itself generated by s-process. A 5-step sequence of neutron-absorption beginning with xenon-128 took place. In an r-process environment, more neutrons would have been absorbed before beta decay would have been possible.

cesium
Illustration of the r and s processes operating in the vicinity of cesium’s neighbors. each square is a stable isotope, like that of 133 Cs. The horizontal solid arrows represent neutron capture, while the wavy diagonal arrows represent beta decay. The isotopes represented by white boxes result from either the s or r process. The blue boxes represent isotopes that result only from the r process, while the red boxes are s-only isotopes. The yellow boxes represent isotopes produced by proton capture. from https://answersingenesis.org/astronomy/solar-system/discussion-stellar-nucleosynthesis/

Aluminum

Finally we get to aluminum. Before becoming a type II supernova, there is an important set of reactions that occurs in the core of a star exceeding 8-11 solar masses. Silicon burning -reactions mainly begin with silicon(atomic number 14) and add on a helium nucleus, creating sulfur(16), argon(18), and so on until iron is formed. But above a critical temperature  explosive silicon burning photodisintegrates all nuclei and rebuilds them up during the expansion. In one of these reactions magnesium-26 captures a proton to form aluminum 27.

William Blake was right. There is indeed a world in a grain of sand—and, we may add, a universe in a mineral.

Sources:

Formation of the Chemical Elements and the Evolution &
of Our Universe, V. E. Viola Journal of Chemical Education, 1990, 67 (9)

Scientific American   Grant J. Mathews, Victor E. Viola  May, 1987

http://www.fen.upc.edu/users/jjose/thielemann.pdf

http://subarutelescope.org/Pressrelease/2015/02/18/index.html

http://www.fen.upc.edu/users/jjose/thielemann.pdf

http://www.aanda.org/articles/aa/full_html/2013/11/aa22599-13/aa22599-13.html

http://adsabs.harvard.edu/full/1971ApJ…166..153A

https://answersingenesis.org/astronomy/solar-system/discussion-stellar-nucleosynthesis/

https://en.wikipedia.org/wiki/Silicon-burning_process

http://abyss.uoregon.edu/~js/ast122/lectures/lec18.html

http://pages.uoregon.edu/jimbrau/astr122-2009/Notes/Exam2rev.html

A More Interesting Calendar

There are trite ways of obtaining the date: looking at a cell phone, watch or conventional calendar. But if one is willing to sacrifice a little precision for something more in tune with the living world, one can simply observe our natural surroundings for often-ignored cues.

Smooth_HawksbeardI know it’s about mid-August from seeing the first ripe blackberries in our garden. Plums are purple but still firm and green inside; it isn’t time to bite into them just yet if we’re seeking their eventual blend of aromas and  sweetness. On hot, sunny days, we hear cicadas’ crescendos. Even in the cracks of pavement stones, terpene-filled thyme is fully grown and flowering. So-called empty lots are colored with the violet splash of blooming chicories and with the yellow taraxanthins of hawksbeards, which people mistake for dandelions. The crickets have started to sing at night for about a week. Also at our latitude, an hour after sunset, the length of the Northern Cross is in the northeastern sky, almost parallel to the horizon.

More confirmation comes in a simple mathematical form, from the time elapsed between sunrise and sunset: 14 hours and 8 minutes on August 14th, at the 45th northern or southern latitude. This implies that the days are approximately as long here, 55 days prior to the summer solstice, as opposed to that same number of days after. But judging from the temperatures and state of the surrounding vegetation, no one would mistake April 27th for today’s date. ( The date is probably off by a few days because the earth does not move around the sun at a constant speed.)

It’s good to break out of the role of being exclusively a food consumer and participate instead as a producer. Next month when carotenoids and anthocyanins get exposed in poplars and maples respectively, signalling the presence of a new month, we will contribute to the smells of September by cooking and preserving tomato sauce and by crushing grapes and letting them ferment. Mid-September is also marked by the presence of the beautiful Orion above the southeastern horizon, just before the storybook blue of dawn gets bleached away by the rising sun.

P1090646bely_nebula
Orion’s belt and the star-forming region known as Orion’s Nebula. Of the three prominent stars making up the belt, the middle one, Alnilam is the most distant of the trio.

One of my early morning photos of the constellation focused on the belt and the star-forming region known as Orion’s Nebula. Contrary to what we may presume, the trio of stars comprising the belt don’t lie in an equidistant plane from us. The lowest one at the left is Alnitak and is closest to Earth at a distance of 250.6 parsecs. The middle one, Alnilam, is the most distant of the belt-stars at 411.5 parsecs or 1342 light years away. We see it now in 2016 as it was in the year 674 when the first glass windows were placed in English churches. We see the third star Mintaka as it was in the year 1100, which unfortunately marked the end of the golden age of Islamic science. It was also when people were relying a lot more on nature for identifying the months.

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