A Rock Star and A Star’s Random Rocks

In 1970, Brian May, interrupted his PhD studies in astrophysics when his rock band Queen enjoyed success. Eventually he went back to school and completed his degree in 2007. He investigated the radial velocity of stars and later improved public awareness of asteroids by creating Asteroid Day.

There are 1.2 million of them, mostly between Jupiter and Mars. Most of the time they orbit the sun in very slightly elliptical orbits. But occasionally Jupiter’s powerful gravity and other effects disturb some of their orbits, allowing them to approach Earth too closely for comfort. There are currently several large known craters on Earth that have been created by asteroid impacts. The oldest, about 2 billion years old, is Vredefort Crater in South Africa; a much younger one, 35 million years old, is Chesapeake Bay Crater in the USA.

Of course, the most notorious one is Chicxulub Crater in Mexico. That asteroid was about 200 to 300 km in diameter, and it contributed to the demise of large dinosaurs and large marsupials. The survivors of this mass extinction were little placental mammals and the small-sized, close-relatives of the dinosaurs that evolved into today’s mammals and birds, respectively.

A moon or a spacecraft, such as Mission Galileo(1989-2003), allows astronomers to find an asteroid’s mass because of the asteroid’s gravitational effects on the orbit of its small neighbour. Radar gives an idea of size. Divide the two measurements and you get a good estimate of its density. This has led to the recent realization that not all asteroids are of the same composition. We not only have S-type asteroids, which are a mixture of rock and metals such as nickel, iron and magnesium. M-types seem to be entirely made of metal, while C-type asteroids are far less dense because they are probably rich in carbon and water-ice.

A few even have their own moons, like Florence, which has a pair. The Florence-trio came within 4 million miles of the Earth on September 1, 2017. Radar allowed astronomers to detect the two moons. The inner one takes about 8 hours to orbit Florence, while the outer moon’s period is estimated to be in the neighborhood of 22 to 27 hours. Is this consistent with Kepler’s 3rd Law?

Kepler based all of his laws on observations. Newton was the first to make sense of them. Using vector calculus, he showed that the first of Kepler’s laws—that bodies follow an elliptical path—is consistent with his inverse square law. The second law, which reveals that planets and moons sweep equal areas in equal times, is consistent with the conservation of angular momentum. If you have a central gravitational force, there is no torque, and since torque is the rate of change of angular momentum with respect to time, given that the derivative is zero, momentum must be a constant number. If you combine these concepts mathematically, it can be shown that the the square of a moon’s period is proportional to the cube of its semi-major axis. That’s Kepler’s 3rd law.


Using the radar image of Florence and its moons, I did a very rough calculation to see if the measurements are in agreement with Kepler’s 3rd law. I used the period of the inner moon which is known more accurately and set out to calculate the period of the outer moon. Based on the radar image of Florence on my desktop computer, I simply used a ruler to measure each moon’s respective distance from the center of Florence( its parent asteroid, not the city of Galileo’s museum. :)Then, since they orbit the same asteroid, you can use Kepler’s 3rd law to set up a ratio. The gravitational constant, Florence’s mass and the kilometer-to-centimetre-from-the-image ratio, all cancel out, leaving us with:

(x/8.0 hours)2= (20.9cm/9.2cm)3.

x = 27 hours, which is near the upper range of possibilities.


Back to the Moon and Why Its Orbit is Slanted

It’s important, at least periodically, to look at the night sky. It reminds us of basic astronomy and of some of the most noteworthy moments from the history of science.  And, most importantly, it leads to unsolved riddles.

Let’s start with the picture I took last night. It’s a far cry from what you’d get with professional equipment, but given that all I used was a little patience, binoculars on a tripod, and a shutter adjustment on a mirrorless digital camera, it’s not the worst picture of our satellite.

How can you tell that I took the picture soon after moonrise? At our latitude (45oN), at moonrise and for a couple of hours after, the Sea of Crisis is near the 12 o’clock position. Later at night, as the moon continues to move across the sky, that spot moves towards the 2 to 3 o’clock position. What’s really happening is the spherical earth’s rotation changes the perspective-angle we get on the moon. From mid latitudes in the southern hemisphere, given that the moon is “upside down”, the Sea of Crisis is initially at the six o’clock position and moves towards the

8 to 9 o’clock position, as in the following picture from Montevideo, Uruguay (36o S) . The same applies to the constellations. When Orion rises at mid northern latitudes, he on his side before eventually straightening out, with Betelgeuse at the upper left. But from Uruguay, Orion stands on his head, and the unstable red giant is on the lower right.

When both the moon and planets are visible, one observes that more often than not the moon is not exactly aligned with them. In other words, it does not perfectly move along the ecliptic, the way the sun does. It’s inclined to it at about 5o . But the plane of the moon’s orbit does cross does that of the earth’s orbit around the sun, and when it does, only then do we get a lunar eclipse if it’s a full moon, and only when it crosses the plane as a new moon is there a solar eclipse somewhere on Earth.

But why is it inclined? Does it reinforce the giant-impact hypothesis? It proposes that the Moon formed during a collision between the Earth and another small planet, about the size of Mars. The debris from this impact collected in an orbit around the Earth to form the Moon. Jupiter has 80 moons in all , which can be divided into three sets. The first set includes 8 moons, including the four that Galileo called the Medici stars, but which Mayr names Io, Europa, Callisto and Ganymede. By seeing them get eclipsed and then reappear, Galileo was the first to realize that these bodies orbited their parent Jupiter, shattering the Earth-centric view of Aristotelian thought. But he also observed that the moons were in line with the ecliptic. In fact they are also almost not inclined to the plane of Jupiter’s equator. The Galilean moon probably formed at the same time as Jupiter. They did not have the same origin as our moon, whose orbit is inclined. In contrast the 2nd set of Jupiter’s moons, which are believed to have been captured, have orbits that are inclined 25 to 56 degrees to Jupiter’s equator—-far more pronounced than that of our moon.

But such loose parallels can lead our thinking astray. Following the giant impact that eventually led to the formation of the Moon, as long “as the proto-lunar material disaggregated into a disk, the Moon is expected to have accreted within about one degree of the Earth’s equatorial plane.” So our satellite didn’t start off with much of an inclined orbit, despite its violent birth. The 5-degree inclination itself is not evidence for its origin story. It likely has another cause.

The quote is from planetary scientists, Kaveh Pahlevan and Alessandro Morbidelli, who used computer simulations to observe what could have happened when the early moon-earth system was almost struck by objects that ranged from 0.0075 to 0.015 Earth masses; that’s about 47 to 95 times more massive than Ceres, the largest existing asteroid. The large asteroid-like bodies were leftovers from the formation of Earth, Venus and the other inner planets. The simulation described in Nature in 2015 revealed that the perturbations from the near-misses could indeed have created a five-degree deviation from the ecliptic.

Postscript. Getting back to the Northern hemisphere, have you thought of what happens to the apparent position of the Sea of Crisis as the moon sets? You are observing it from a similar angle as the Earth rotates you away from the moon, except that you are on the other side. So it will continue to seemingly rotate clockwise and it should be close to the 5 to 6′ o’clock position. Here is a moon set at latitude 29 degrees North.

Euler’s Phi Function from Wells’ Curious and Interesting Numbers

In 1986, David Wells, a mathematician specializing in number theory, wrote a delicious little book, entitled The Penguin Dictionary of Curious and Interesting Numbers. If you enjoy dabbling in math, each numerical entry, although brief, is a gateway into all sorts of realms of an incredibly diverse field. Although it goes without saying that without mathematics, physics, chemistry, astronomy, computer science, engineering, banks and insurance companies would all be crippled, math is beautiful in its own right. It is a source of infinite puzzles to distract us from pain, taxes and death.

Under his entry for the number 87, Wells first wrote  σ[ϕ(87)]= σ(87). The neat thing is that σ[ϕ(n)]= σ(n) is only true for very few numbers. But what is σ(n) ? It is simply the sum of all the factors of n. In fact, capital sigma, Σ, is routinely used to designate sums of all sorts of things, but lower-case sigma, σ, has the specialized role of adding up only the factors of a number. So σ(87)=1+3+29+87= 120.

What is ϕ(n), the so-called, Euler’s phi function ? It’s the total number of positive integer values that are less than n and which are relatively prime to n. Relatively prime in turn means that the two numbers only have a common factor of 1. To obtain ϕ(87) the long, painful way, we begin by subtracting two from 86 ( 2 because of the two factors, 3 and 29, which are not relatively prime to 87; and 86 because the definition of ϕ(n) is based on integers less than 87). But then all other multiples of 3 and 29 are also to be excluded because they will have something in common with 87. Thus subtract the 27 other multiples of three that are less than 87, given that we’ve already counted the number 3; and finally the 1 other multiple of 29 < 87, which is 58. Hence ϕ(87) = 86 – 2 – 27 -1 = 56.

There are actually a few formulas, easily derived, that make the calculation of ϕ(n)less tedious. For any prime number, p, ϕ(p)=p-1. If you raise p to an integer power of k, the number of relatively prime numbers will be pk-1 lowered by (pk-1-1). Simplified that yields ϕ(pk)=pk(p-1)/p. For a non prime number, you can use the basis of the prime number formula to come up with:

p1, p2 and pk are the prime factors of n

To quickly evaluate ϕ(87),

Now that we know that ϕ(87)=56, we can get σ[ϕ(87)] = σ[56] = 1 + 2+ 4 +7+8+14+28+56 = 120. We saw earlier that σ(87)= 120, so indeed σ[ϕ(87)]= σ(87) =120. You can write a very simple computer program(see footnote) to discover that σ[ϕ(n)]= σ(n) is also true for 362, 5002 and 9374 and for two other numbers revealed by these images. This will give me an excuse to go off on artistic and scientific trivia-tangents. 🙂

The first image is a Delacroix painting of the great pianist-composer, Chopin. The artist, Eugene Delacroix, who believed that “nothing can be compared with the emotion caused by music” was born in 1798. σ(1798) = 2880, the year when asteroid 1950AD will come close to Earth. When it first made the news, the risk of an impact was reported to be higher than it is currently believed to be. Its trajectory can be difficult to predict due to the Yarkovsky effect in which unequal heating of an asteroid’s surface cause emitting photons to slightly propel it. ϕ(1798) = 1798[(2-1)/2*(29-1)/29*(31-1)/31]=1*28*30=840.

σ(840)=2880= σ(1798) , so σ(840)=σ(ϕ(1798)) = σ(1798)

The 3rd picture is that of the Indonesian volcano Mount Samalas, which erupted in 1257 in one of the world’s most intense eruptions in the last 12 millennia. The injection of aerosols high into the atmosphere cooled the earth for a few years. Ozone was even affected, given that halogen gases accompanied the release of the sulfur compounds by a ratio of almost 3:2. If you’ve ever wondered if global warming could have fluky, positive effects in being able to offset natural cooling from large volcanic eruptions, this is a reminder that the effects would only be very temporary. If you have mass volcanism in mind, such events are even rarer, with hundreds of thousands of years elapsing without them. If one would occur, the magnitude of the devastation would be so large, that it would overshadow anything else.

σ(ϕ(1257))=σ(836)=1680 =σ(1257), the year in which the long-tailed C/1680 V1 comet was observed. It’s also known as The Great Comet or Newton’s or Kirch’s Comet. It was bright enough to be seen during the day. The image is the 1680 Dutch painting, Staartster (komeet) boven Rotterdam(Tail star (comet) above Rotterdam) by Lieve Verchuier. The artist normally painted marine scenes; ships being built and the occasional landscape. But it’s no surprise that he chose to paint the unusual sighting of such a stunning comet, and imagine if he realized that σ(ϕ(1257))=σ(1257)=1680. 🙂


David Wells. The Penguin Dictionary of Curious and Interesting Numbers. Penguin. 1986

Vidal, C. M. et al. The 1257 Samalas eruption (Lombok, Indonesia): the single greatest stratospheric gas release of the Common Era. Sci. Rep. 6, 34868; doi: 10.1038/srep34868 (2016).

George Beekman. The Nearly Forgotten Scientist Osipovich Yarkovsky. J. Br. Astron. Assoc. 115, 4, 2005 https://articles.adsabs.harvard.edu/pdf/2005JBAA..115..207B

NASA. Sentry Earth Impact Monitoring. https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=2029075&view=OPC

Donald K. Yeomans. Great Comets in History. Jet Propulsion Laboratory. April, 2007

Maple Program

The phi and sigma functions are built-into Maple, which makes it easy. Also note that the syntax will be slightly different in newer versions of the software. Mine, version V, is 27 years old!


for w from 2 to 10000 by 1 do; if sigma(phi(w))= sigma(w) then print(w,sigma(w));fi;od;

And here is the output, where the first number of each pair is what we’re looking for, while the 2nd number is the sigma of that number.

                           362, 546
                          1257, 1680
                          1798, 2880
                          5002, 7812
                          9374, 14520

6 out of 10000, or 0.06% meet the criteria. How many solutions are there between 10 000 and 100 000?

for w from 10000 to 100000 by 1 do; if sigma(phi(w))= sigma(w) then print(w,sigma(w));fi;od;

The results:

21982, 34200
22436, 40320
25978, 40320
35306, 53760
38372, 68796
41559, 63360
50398, 76608
51706, 78624
53098, 80640
53314, 89280
56679, 86400
65307, 95040
68037, 90720
89067, 129600

There are only 14 out of 90 000 ( 0.016%). So do the solutions become consistently scarcer as the numbers grow? Between 100 000 and a million, σ[ϕ(n)]= σ(n) is true for only about 0.0093% of the numbers. So apparently yes.