## The Extraordinary Music of Joaquín Rodrigo

Joaquín Rodrigo was born in Sagunto, Valencia, Spain on November 22, 1901. At the age of three, a diphtheria epidemic ravaged his community and he was not spared. His eyes were damaged and he soon lost his eyesight. Despite his blindness, he went on to become Spain’s greatest composer of the 20th century. After immersing myself in his music for the past several weeks, this avid music listener would like to suggest that Joaquín Rodrigo was one of the greatest composers of the 20th century. I believe his acclaim has not yet reached its peak, and that many of his works that to the present day have seldom been played will soon become part of the standard repertory.

Rodrigo is primarily known for his wonderful guitar concertos: Concierto de Aranjuez (1939), Fantasía para un gentilhombre (1954), Concierto Madrigal (1966), and Concierto Andaluz (1967), but have you heard his purely orchestral work A la busca del más allá (In search of the beyond)? Or his piano version of Zarabanda lejana?

There is no better introduction to the music of Joaquín Rodrigo than the four-CD set from EMI Classics, The Rodrigo Edition. One of the foremost interpreters of Rodrigo, Mexican conductor Enrique Bátiz skillfully conducts the London Symphony Orchestra, the Orquesta Sinfónica del Estado de México, and the Royal Philharmonic Orchestra in these completely satisfying performances.

Joaquín Rodrigo died in 1999 at the age of 97. When he was 90, a loving and insightful documentary was produced, titled Shadows and Light. Please seek it out! It is well produced and inspiring. You can view this documentary on medici.tv (much of it is in English, but for the parts that aren’t you have the option to select English subtitles), or purchase the DVD through Amazon.

## The Lunar Equator

The equator of the Moon is defined by its rotational axis. It is the great circle contained by the plane that is perpendicular to the line connecting the north and south poles of the Moon.

Here is a list of named lunar features through which the Moon’s equator passes, from lunar east to lunar west on the part of the Moon visible from the Earth.

Remember, lunar longitude is opposite the direction in the sky. East longitude is the west/right side of the moon, and west longitude is the east/left side of the moon as viewed from the northern hemisphere of the Earth.

The diameter of each feature is included, followed by the depth of the feature, where available.

Wyld
Center: 98.101˚ E, 1.416˚ S
Range: 96.395˚ - 99.806˚ E, 3.121˚ S - 0.289˚N
Crater; libration zone
58 mi
James Hart Wyld (1913-1953), American rocket engineer
Dorsum Cloos
Center: 90.410˚ E, 1.149˚ N
Range: 90.387˚ - 91.144˚ E, 0.268˚ S - 2.576˚ N
Wrinkle ridge; libration zone
64 mi
Hans Cloos (1885-1951), German geologist
Mare Smythii
Center: 87.049˚ E, 1.709˚ S
Range: 80.941˚ - 92.719˚ E, 7.456˚ S - 4.496˚ N
Mare; libration zone; Smyth's Sea
232 mi, 3.1 mi
William Henry Smyth (1788-1865), English naval officer and astronomer
Schubert J
Center: 78.935˚ E, 0.012˚ S
Range: 78.579˚ - 79.292˚ E, 0.344˚ S - 0.320˚ N
Crater
12 mi
Friedrich Theodor von Schubert (1758-1825), German astronomer & geographer 
Jenkins
Center: 78.041˚ E, 0.372˚ N
Range: 77.418˚ - 78.663˚ E, 0.251˚ S - 0.994˚ N
Crater
24 mi, 1.9 mi
Louise Freeland Jenkins (1888-1970), American astronomer
Schubert X
Center: 76.750˚ E, 0.310˚ N
Range: 75.940˚ - 77.561˚ E, 0.501˚ S - 1.121˚ N
Crater
32 mi
Friedrich Theodor von Schubert (1758-1825), German astronomer & geographer
Nobili
Center: 75.949˚ E, 0.166˚ N
Range: 75.260˚ - 76.638˚ E, 0.523˚ S - 0.855˚ N
Crater
26 mi, 2.4 mi
Leopoldo Nobili (1784-1835), Italian physicist
Maclaurin X
Center: 68.708˚ E, 0.091˚ N
Range: 68.403˚ - 69.014˚ E, 0.214˚ S - 0.397˚ N
Crater
15 mi
Colin Maclaurin (1698-1746), Scottish mathematician
Maclaurin O
Center: 67.557˚ E, 0.135˚ S
Range: 66.873˚ - 68.241˚ E, 0.819˚ S - 0.548˚ N
Crater
23 mi
Colin Maclaurin (1698-1746), Scottish mathematician
Mare Spumans
Center: 65.303˚ E, 1.302˚ N
Range: 63.613˚ - 66.733˚ E, 1.062˚ S - 3.722˚ N
Mare
87 mi
The Foaming Sea
Webb C
Center: 63.833˚ E, 0.149˚ N
Range: 63.267˚ - 64.398˚ E, 0.247˚ S - 0.544˚ N
Crater
21 mi
Thomas William Webb (1807-1885), English astronomer
Sinus Successus
Center: 58.520˚ E, 1.124˚ N
Range: 56.519˚ - 60.188˚ E, 0.861˚ S - 2.872˚ N
Bay
82 mi
Bay of Success
Mare Fecunditatis
Center: 53.669˚ E, 7.835˚ S
Range: 40.771˚ - 63.340˚ E, 21.695˚ S - 6.112˚ N
Mare
429 mi, 1.1 mi
Sea of Fertility
Taruntius P
Center: 51.585˚ E, 0.060˚ N
Range: 51.473˚ - 51.696˚ E, 0.051˚ S - 0.172˚ N
Craterlet
5 mi, 0.9 mi
Lucius Tarutius Firmanus (fl. 86 B.C.), Roman philosopher, mathematician, and astrologer
Dorsum Cayeux
Center: 51.220˚ E, 0.763˚ N
Range: 50.922˚ - 52.000˚ E, 0.598˚ S - 2.113˚ N
Wrinkle ridge
59 mi
Lucien Cayeux (1864-1944), French sedimentary petrographer
Dorsa Cato
Center: 47.701˚ E, 0.213˚ N
Range: 46.605˚ - 49.599˚ E, 1.165˚ S - 2.233˚ N
Wrinkle ridges
87 mi
Marcus Porcius Cato (234-149 B.C.), Roman soldier, senator, and historian
Rima Messier
Center: 44.545˚ E, 0.756˚ S
Range: 43.357˚ - 45.581˚ E, 1.561˚ S - 0.015˚ N
Rille
62 mi
Charles Messier (1730-1817), French astronomer
Lubbock R
Center: 40.453˚ E, 0.167˚ S
Range: 40.060˚ - 40.845˚ E, 0.559˚ S - 0.225˚ N
Crater
15 mi
Sir John William Lubbock (1803-1865), English banker, barrister, mathematician, and astronomer
Maskelyne T
Center: 36.593˚ E, 0.040˚ S
Range: 36.507˚ - 36.678˚ E, 0.125˚ S - 0.046˚ N
Craterlet
3 mi
Nevil Maskelyne (1732-1811), English astronomer
Maskelyne A
Center: 34.089˚ E, 0.032˚ N
Range: 33.603˚ - 34.574˚ E, 0.453˚ S - 0.517˚ N
Crater
18 mi
Nevil Maskelyne (1732-1811), English astronomer
Mare Tranquillitatis
Center: 30.835˚ E, 8.349˚ N
Range: 16.924˚ - 45.490˚ E, 4.051˚ S - 19.375˚ N
Mare
544 mi
Sea of Tranquility
Rimae Hypatia
Center: 22.777˚ E, 0.340˚ S
Range: 19.690˚ - 25.975˚ E, 1.406˚ S - 0.672˚ N
Rilles
128 mi
Hypatia (c.370-415), Alexandrian philosopher, mathematician, and astronomer
Lade A
Center: 12.726˚ E, 0.161˚ S
Range: 11.773˚ - 13.680˚ E, 1.114˚ S - 0.793˚ N
Crater
35 mi
Heinrich Eduard von Lade (1817-1904), German banker and amateur astronomer
Lade B
Center: 9.796˚ E, 0.016˚ N
Range: 9.412˚ - 10.180˚ E, 0.368˚ S - 0.399˚ N
Crater
15 mi
Heinrich Eduard von Lade (1817-1904), German banker and amateur astronomer
Rhaeticus F
Center: 6.438˚ E, 0.060˚ S
Range: 6.134˚ - 6.742˚ E, 0.364˚ S - 0.244˚ N
Crater
11 mi
Georg Joachim Rheticus (1514-1574), Austria-born astronomer & mathematician
Rhaeticus
Center: 4.924˚ E, 0.032˚ N
Range: 4.192˚ - 5.657˚ E, 0.701˚ S - 0.764˚ N
Crater
30 x 27 mi, 1.0 mi
Georg Joachim Rheticus (1514-1574), Austria-born astronomer & mathematician
Rhaeticus L
Center: 3.484˚ E, 0.205˚ N
Range: 3.257˚ - 3.711˚ E, 0.022˚ S - 0.432˚ N
Crater
9 mi
Georg Joachim Rheticus (1514-1574), Austria-born astronomer & mathematician
Sinus Medii
Center: 1.027˚ E, 1.634˚ N
Range: 3.371˚ W - 5.551˚ E, 2.048˚ S - 4.641˚ N
Bay
178 mi
Bay of the Center
Mösting E
Center: 4.591˚ W, 0.178˚ N
Range: 5.189˚ - 3.992˚ W, 0.421˚ S - 0.777˚ N
Crater
27 mi
Johan Sigismund von Møsting (1759-1843), Danish banker, finance minister, and astronomy enthusiast
Sömmering
Center: 7.526˚ W, 0.193˚ N
Range: 7.987˚ - 7.065˚ W, 0.268˚ S - 0.654˚ N
Crater
17 mi, 0.8 mi
Samuel Thomas von Sömmering (1755–1830),German physician and anatomist
Lansberg
Center: 26.627˚ W, 0.312˚ S
Range: 27.266˚ - 25.988˚ W, 0.951˚ S - 0.327˚ N
Crater
24 mi, 1.9 mi
Philippe van Lansbergen (1561-1632), Dutch astronomer and mathematician
Mare Insularum
Center: 30.640˚ W, 7.792˚ N
Range: 39.195˚ - 22.153˚ W, 0.596˚ S - 16.345˚ N
Mare
318 mi
Sea of Islands
Oceanus Procellarum
Center: 56.677˚ W, 20.671˚ N
Range: 81.084˚ - 26.850˚ W, 16.266˚ S - 57.433˚ N
Mare
1611 x 353 mi
Ocean of Storms
Lohrmann D
Center: 65.273˚ W, 0.141˚ S
Range: 65.442˚ - 65.104˚ W, 0.310˚ S - 0.028˚ N
Crater
7 mi
Wilhelm Gotthelf Lohrmann (1796-1840), German selenographer
Rimae Hevelius
Center: 66.377˚ W, 0.809˚ N
Range: 67.849˚ - 63.582˚ W, 1.284˚ S - 2.956˚ N
Rilles
113 mi
Johannes Hevelius (1611-1687), Polish astronomer
Lohrmann
Center: 67.383˚ W, 0.440˚ S
Range: 67.898˚ - 66.867˚ W, 0.955˚ S - 0.075˚ N
Crater
19 mi, 1.0 mi
Wilhelm Gotthelf Lohrmann (1796-1840), German selenographer
Rimae Riccioli
Center: 73.071˚ W, 1.515˚ S
Range: 76.809˚ - 68.566˚ W, 4.754˚ S - 1.247˚ N
Rilles
249 mi
Giovanni Battista Riccioli (1598-1671), Italian astronomer
Schlüter P
Center: 85.208˚ W, 0.054˚ N
Range: 85.550˚ - 84.865˚ W, 0.289˚ S - 0.397˚ N
Crater; libration zone
12 mi
Heinrich Schlüter (1815-1844), German astronomer

Now that we’ve taken a tour of nearside features along the equator, let us turn to the lunar north and south celestial poles. As you know, the Earth’s north celestial pole (NCP) is currently located quite close to Polaris. However, on the Moon, the NCP is located in Draco near the Cat’s Eye Nebula (NGC 6543), about two-thirds of the way between Polaris and the center of the Head of Draco.

The Moon’s south celestial pole (SCP) is located in the constellation Dorado inside of the Large Magellanic Cloud (LMC). If you were stationed at the south pole of the Moon, you would see the Large Magellanic Cloud directly overhead at all times!

The Moon has many fascinating places, tempting us to explore. Some of them have quite interesting names. One of my favorites is Lacus Perseverantiae, Lake of Persistence. Its location is 62.0˚ E and 8.0˚ N. See if you can find it here. (Hint: under Layers : Overlays select Nomenclature, and under Settings select Show Graticule.) Have fun exploring!

References

Cocks, Elijah E.; Cocks, Josiah C. (1995). Who’s Who on the Moon: A Biographical Dictionary of Lunar Nomenclature. Tudor Publishers. ISBN978-0-936389-27-1.

1:1 Million-Scale Maps of the Moon, IAU/USGS/NASA. https://planetarynames.wr.usgs.gov/Page/Moon1to1MAtlas.

Virtual Moon Atlas 6.0 Pro. Computer software. https://ap-i.net/avl/en/start.

## Another COVID-19 Map

As long as Americans continue to suffer and die from the coronavirus pandemic, we will need to exercise an abundance of caution, regardless of what some might tell us. In the map below, you will find which counties in the United States reported new coronavirus deaths (shown in red) and, if there were no additional deaths, which counties reported new coronavirus positive cases (shown in orange) during the most recent reporting day. I will update this map each day until the pandemic has ended. Be safe!

## Mahler’s Farewell

Lewis Thomas (1913-1993) wrote in his essay Late Night Thoughts on Listening to Mahler’s Ninth Symphony,

“I cannot listen to the last movement of the Mahler Ninth without the door-smashing intrusion of a huge new thought: death everywhere, the dying of everything, the end of humanity…How do the young stand it? How can they keep their sanity? If I were very young, sixteen or seventeen years old, I think I would begin, perhaps very slowly and imperceptibly, to go crazy…If I were sixteen or seventeen years old…I would know for sure that the whole world was coming unhinged. I can remember with some clarity what it was like to be sixteen…I was in no hurry…The years stretched away forever ahead, forever…It never crossed my mind to wonder about the twenty-first century; it was just there, given, somewhere in the sure distance.”

Thomas was referring to the threat of nuclear war, which is still very much with us. Now, can you imagine as bad as the COVID-19 pandemic has been, what a nuclear war would be like? We need to rid our planet of these weapons, now.

As I was listening to the final movement of Gustav Mahler’s Symphony No. 9, the Adagio, this past Monday, I was also thinking, of course, about the frightening ravages of COVID-19, but also climate change and the precipitous decline in biological diversity caused by humans. All of this is driven by the fact that there are too many people on the planet, and the answer is not to kill (by whatever means) people who are already here, but to bring fewer children into the world so we can lower human population to a sustainable level in the coming generations. We could all have a higher standard of living without trashing the planet.

On Wednesday, the 50th anniversary of Earth Day, PBS aired a new BBC documentary, Climate Change: The Facts. I was riveted by the program, presented by Sir David Attenborough, who will turn 94 next month the day before I turn 64. David Attenborough is an international treasure. Watching him so expertly present, as he always does, the urgency of this climate crisis and remembering his many outstanding documentary series such as Life on Earth and The Living Planet, I became teary eyed knowing that he will not be with us for very much longer. You wish someone like David Attenborough or Carl Sagan could live for hundreds of years. Because, when our life is over, we will cease to exist as a conscious entity, for all eternity. I am now certain of that. Realizing that this is our one and only life gives one a very different perspective on what we are doing to this world—and to each other. Humanists value the sanctity of each human life more than anyone who believes in an afterlife. Humanists fully understand the enormous responsibility each of us living in this current generation has to ensure that our civilization does not descend into a dystopian existence. There will be no salvation, just unimaginable pain, suffering, and destruction of all that is good, if we fail.

I am so inspired by young Greta Thunberg, who features prominently in the documentary. Greta and the many other young activists around the world give me hope for the future. Her words and conviction brought more tears to my eyes. I may be 63, but I’m with you 100%, Greta. Sign me up!

In 1908 and 1909, Gustav Mahler finished his last completed work, the Symphony No. 9. There was much turmoil and tragedy in Mahler’s life prior to the writing of this symphony. His beloved oldest daughter, Maria Anna Mahler, died of scarlet fever and diptheria on 5 July 1907 at the age of 4. Immediately after Maria’s death, Mahler learned that he had a defective heart. And his relationship with his wife Alma had become strained. Gustav Mahler died on 18 May 1911. He never heard his Symphony No. 9 performed. It received its premiere on 26 June 1912 in Vienna with Bruno Walter conducting the Vienna Philharmonic Orchestra.

The final movement of Mahler’s Symphony No. 9, the Adagio, is one of the most moving pieces of music I have ever heard. While listening to it, one thinks of all the beauty that was and is in the world, and how terribly much we have lost.

The most expressive recording of the Adagio I have heard is by the Chicago Symphony Orchestra, conducted by Sir Georg Solti (Decca 473 274-2). If this movement of 24:37 does not lead you to weep, I don’t know what will.

## Lunar Maria

António Cidadão, of Oeiras, Portugal, many years ago produced a wonderful set of images showing the location of each mare on the Moon. His website has not been updated since 1999 and the contact email address provided there is no longer valid, and even after a thorough Google search I can find no way to contact him to ask permission to link images here to his site. Even worse, because his hosting site is not secure (http: instead of https:), WordPress does not allow me to link directly to his images so I had to put copies into my media library. Please know that the images shown below are all copyrighted by António Cidadão.

Each image shows north is up and west is to the left. This is direction of increasing longitude and therefore west on the Moon, but in our sky, east is to the left. In other words, these annotated images of the Moon are correctly oriented as they would appear to the unaided eye in the sky in the northern hemisphere. In the rest of this article, we will use the moon-centric east-west convention that Cidadão indicates in his image diagrams.

Let’s take a look at each of the lunar maria from moon-west to moon-east. Their fanciful names were mostly given (and codified in 1651) by the Italian astronomer Giovanni Battista Riccioli (1598-1671). Riccioli chose names related to weather, as it was then believed that the Moon, the closest celestial body to the Earth, exerted an influence on the Earth’s weather. This is perhaps not at all surprising given that the phenomenon of tides had been known since antiquity.

Most of the nearside west portion of the Moon is covered by a mare that is so large that it is given a unique designation: Oceanus for “ocean”.

Oceanus Procellarum contains the famously bright crater Aristarchus and the associated Aristarchus Plateau. In the image above you will notice what appears to be a tiny mare close to the limb of the Moon west of the southern part of Oceanus Procellarum. This is the lava-flooded crater Grimaldi.

South of Grimaldi and straddling the lunar limb is Mare Orientale. It is difficult to see because most of it is on the lunar farside, though libration can sometimes bring its oblique visage into view. The name Orientale, meaning “eastern”, describes its location on the eastward-facing limb of the Moon as seen from Earth, rather than its westward direction as seen from the surface of the Moon.

Mare Humorum is located just south of Oceanus Procellarum. It is round and inviting, though no spacecraft has ever landed there.

Mare Nubium is east of Mare Humorum. The large crater Bullialdus flanks the western edge of Mare Nubium, and Rupes Recta (the “Straight Wall”) flanks its eastern edge.

Mare Cognitum lies between Mare Nubium and Oceanus Procellarum. It was named in 1964 after the Ranger 7 probe took the first U.S. close-up pictures of the Moon’s surface prior to crashing there.

Mare Insularum is north of Mare Cognitum. Its current name was bestowed upon it in 1976 by lunar geologist Don Wilhelms (1930-). The crater Kepler on its western edge separates Mare Insularum from Oceanus Procellarum. The crater Copernicus is on the northeast side of its western lobe.

Mare Vaporum is the mare closest to the center of the Moon’s nearside. The bright crater Manilius lies towards its northeastern edge and the volcanic crater Hyginus and its associated rille (Rima Hyginus) are immediately to its south.

Mare Imbrium was created 3.9 billion years ago when an asteroid some 150 miles across crashed into the Moon. This ancient feature is so large that it forms the right eye of the “Man in the Moon” we see when looking at a full or nearly full moon with our unaided eyes.

Mare Frigoris lies north and northeast of Mare Imbrium. The dark crater between them is Plato. It is the mare closest to the north pole of the Moon.

Now we begin our tour of the eastern hemisphere of the Moon’s nearside. Mare Serenitatis has the distinction of being the landing site of the last human mission to the Moon, Apollo 17, in 1972. It was also the landing site of the Soviet unmanned spacecraft Luna 21 just one month later.

Mare Tranquillitatis is perhaps the most famous of the lunar maria, as it was there that humans first set foot on the surface of the Moon in 1969. The Apollo 11 landing site is located near its southwest corner.

Mare Nectaris lies south of Mare Tranquillitatis. This small, isolated, and nearly circular mare sports a prominent crater, Theophilus, at its northwest corner.

East of Mare Nectaris lies Mare Fecunditatis. Superposed upon Mare Fecunditatis is the striking crater pair Messier and Messier A with two prominent rays evocative of a comet’s tail. Named after the famous French comet hunter Charles Messier (1730-1817), these craters and their associated rays were formed from a grazing impact from the east.

Mare Crisium is a round and isolated mare that makes it easy to remember why it is called the “Sea of Crises”. The Soviet Luna 24 unmanned sample return mission landed there in 1976. The six ounces of lunar materials it brought back to Earth are the last lunar samples scientists have received.

Mare Anguis lies just northeast of Mare Crisium and is called the “Serpent Sea” for its serpentine shape rather than the more fanciful name “Sea of Serpents” referred to by some science fiction authors.

Mare Undarum lies southeast of Mare Crisium. Its uneven texture and lack of uniform smoothness appears to justify its name as “the sea of waves”.

Mare Spumans lies south of Mare Undarum and east of Mare Fecunditatis. The bright crater Petit on the western side of this tiny mare evinces a bit of foam on “the foaming sea”.

Mare Australe hugs the southeastern limb of the lunar nearside. Though obliquely viewed from Earth and wrapping around to the lunar farside, favorable libration makes it visible in its entirety on occasion.

Mare Smythii on the eastern limb of the Moon is one of two lunar maria named after people. The lucky honoree is English hydrographer and astronomer William Henry Smyth (1788-1865). The lunar equator passes through Mare Smythii.

Mare Marginis lies east of Mare Crisium, right along the lunar limb. The crater Goddard on the northeast side of Mare Marginis exhibits bright deposits on its northeastern side. This crater and its associated deposits can only be seen from Earth during favorable librations.

Mare Humboldtianum lies along the northeastern limb of the Moon and is the other lunar mare named after a person. The German astronomer Johann Heinrich von Mädler (1794-1874) named this feature after German geographer and explorer Alexander von Humboldt (1769-1859).

This completes our tour of the 21 maria on the nearside of the Moon.

References

António Cidadão’s Home-Page of Lunar and Planetary Observation and CCD Imaging, Moon-“Light” Atlas.  Retrieved 22 April 2020.
http://www.astrosurf.com/cidadao/moonlight_mare_oceanus.htm

Ewen A. Whitaker, Mapping and Naming the Moon: A History of Lunar Cartography and Nomenclature (Cambridge University Press, 2003).

## BepiColombo Passes Earth

The BepiColombo spacecraft flew by the Earth last night, the first of nine gravity-assist maneuvers it will make to slow it down so that it can go into orbit around the planet Mercury on 5 December 2025. This was the only Earth gravity assist. There will be a Venus flyby later this year and next year, and six Mercury flybys from 2021-2025.

BepiColombo passed 7,877 miles over the South Atlantic Ocean at 0425 UT on 10 April 2020 at its closest approach to Earth, and I was able to image it from my backyard observatory in Dodgeville, Wisconsin at 0600 UT at a distance (range) of 21,760 miles.

North is up and East to the left in the video frame, so BepiColombo is moving in a northwesterly direction. The two stars in the field are 3UC 145-134561 (12.2m, north) and 3UC 144-138354 (12.7m, south). The predicted equatorial coordinates (epoch of date) at 0600 UT from JPL Horizons were α = 11h 38m 03.90s, δ = -18° 08′ 25.4″. Please note when using JPL Horizons to generate ephemerides for spacecraft and minor planets passing close to the Earth that you should use the ICRF coordinates (astrometric) and not the apparent coordinates. They can be significantly different!

The integration time in the video above is 7.5 frames per second, or 0.13 second per frame. The field size is 17 x 11 arcminutes.

Here’s the video light curve of BepiColombo as it passed through the field. It was fairly constant in brightness with no obvious variability amidst the noisy measurements.

## A New Infinite Series, Convergent and Irrational

Infinite series are a log of func. All kidding aside, you may have heard of the sum of reciprocal squares.

$\sum\limits_{n=1}^{\infty}\frac{1}{{n}^2}=\frac{1}{{1}^2}+\frac{1}{{2}^2}+\frac{1}{{3}^2}+\frac{1}{{4}^2}+\frac{1}{{5}^2}+\frac{1}{{6}^2}+\frac{1}{{7}^2}+\cdots$

The sum of this slowly convergent series is approximately equal to 1.644934. Is there anything special about this number? Italian mathematician Pietro Mengoli (1626-1686) first posed the question in 1644 (published 1650), what is the exact sum of this infinite series? This problem was not solved until 90 years later by Swiss mathematician Leonhard Euler (1707-1783) in 1734 (published 1735). Euler proved that the exact sum is

$\sum\limits_{n=1}^{\infty}\frac{1}{{n}^2}=\frac{{\pi}^2}{6}$

There’s that number pi, the most famous of the irrational numbers, showing up once again in mathematics, ostensibly having nothing at all to do with circles. What’s an irrational number? It is any real number that cannot be expressed as a ratio of two integers. The decimal digits of an irrational number neither terminate nor end in a repeating sequence (e.g. 1/3 = 0.3333… or 9/11 = 0.81818181…).

Determining the exact value of the sum of reciprocal squares infinite series is known as the Basel Problem, named after the hometown of Euler (who solved it) and the Bernoulli family of mathematicians (who were not able to solve it).

My admired colleague in England, Abdul Ahad, has come up with a variant of the sum of reciprocal squares where every third term starting with n = 4 is subtracted rather than added.

$\frac{{\pi}^2}{6} -2\sum\limits_{n=1}^{\infty}\frac{1}{{\left (3n+1 \right )}^2}=\frac{1}{{1}^2}+\frac{1}{{2}^2}+\frac{1}{{3}^2}-\frac{1}{{4}^2}+\frac{1}{{5}^2}+\frac{1}{{6}^2}-\frac{1}{{7}^2}+\cdots$

Ahad has shown that this new series is convergent and sums to an irrational number, approximately equal to 1.40146804. The infinite sum portion of the above expression is approximately equal to 0.12173301 and is also an irrational number. Multiplying by 2 gives us an irrational number, and subtracting from π2/6, which is itself irrational, results in our final result being irrational.

Interestingly, almost all real numbers are irrational, strange as they are.

References

Ahad, Abdul. “An interesting series.” M500 Magazine 278, 8-9 (2017). http://m500.org.uk/wp-content/uploads/2018/11/M278WEB.pdf.

Ahad, Abdul. “A New Infinite Series with Proof of Convergence and Irrational Sum.” Res Rev J Statistics Math Sci, Volume 4, Issue 1 (2018). http://www.rroij.com/open-access/a-new-infinite-series-with-proof-of-convergence-and-irrational-sum.pdf.

## Comet ATLAS (C/2019 Y4)

Comet C/2019 Y4 ATLAS was discovered on December 28, 2019 and is named after the observational program that discovered it: Asteroid Terrestrial-impact Last Alert System (ATLAS). It could become a naked-eye comet—if it doesn’t disintegrate as it gets closer to the Sun. Here’s an ephemeris for the remainder of April and May.

## Jean Sibelius: An Introduction

The young Finnish composer Jean Sibelius (1865-1957) wanted to be a virtuoso violinist but it was in composition that his greatest talent lay. All his life, he was deeply connected to the natural world, and this love of Nature is expressed in much of his music.

I know of no better introduction to the music of Jean Sibelius than the two CD set of his Symphonies No. 1, 2, and 4, and Finlandia and the Karelia Suite by Vladimir Ashkenazy conducting the Philharmonia Orchestra.

The earliest composition featured on this recording is the Karelia Suite, completed in 1893; the latest is the pensive Symphony No. 4, completed in 1911. All of the music on these discs is splendid, the performances inspired, and the recordings immersive.

Ashkenazy seems to have an innate understanding of Sibelius, and his conducting and interpretations shine here throughout.

As with many (most?) of the greatest composers, Sibelius faced a number of challenges and personal demons throughout his life. Though he lived a long and productive life, he wrote almost no new music after his brilliant tone poem Tapiola in 1926, 31 years before his death. He did complete a Symphony No. 8, but threw the score into his fireplace in 1945. Sibelius once remarked, “If I cannot write a better symphony than my Seventh, then it shall be my last.”

To find out more about the life and music of Jean Sibelius, I’d like to direct your attention to an excellent two-part documentary film by Christopher Nupen, completed in 1984. It is available through the classical music streaming channel medici.tv (highly recommended) and Amazon.

## Shadows Cast by Starlight

Henry Norris Russell (1877-1957) received his Ph.D. at Princeton in 1899 at just 21 years of age. Three years later—in 1902 when he was 24 years old and years before his discovery of the color-luminosity relationship now known as the Hertzsprung-Russell (H-R) diagram—Russell had an interesting article published in the journal Popular Astronomy that shows him already to be a meticulous and perspicacious observational astronomer. This article, completed 118 years ago this day, is reprinted below.

SHADOWS CAST BY STARLIGHT.

HENRY NORRIS RUSSELL.

FOR POPULAR ASTRONOMY.

It has long been known that Venus casts a distinct shadow; and the same thing has sometimes been observed in Jupiter’s case. More recently, it has been stated in the daily press* that shadows cast by Sirius have been seen at the Harvard Observatory in Jamaica, though it was then said that they could probably be seen only where the air is exceptionally clear.

The writer began to investigate this subject, quite independently, last November, and has found that the shadows cast by a number of the brighter fixed stars can be seen without difficulty under ordinary circumstances, provided proper precautions are taken to exclude extraneous light, and to secure the maximum sensitiveness of the observer’s eyes.

* Interview with Professor W.H. Pickering, New York Tribune, Jan. 18, 1902.

The most convenient method of observation is as follows: Choose a window from which the star is visible, while as little light as possible enters from terrestrial sources. Darken the room completely, with the exception of this window. Open the window, and screen down its aperture to an area of a square foot or less. Hold a large piece of white paper in the path of the star’s rays, as far from the opening as possible. The image of the opening will then appear on the paper.

It cannot, however, be well seen until the observer has spent at least ten minutes in the dark, (to rest his eyes from the glare of ordinary lights). The paper should be held within a foot or so of the eyes, as the faint patch of starlight is most easily visible when its apparent area is large. The shadow of any convenient object may now be made to fall on the screen, and may be observed. By holding the object near the window and noticing that its shadow is still sharp, the observer may convince himself that the light which casts the shadow really comes from the star.

By the method above described, the writer has succeeded in distinguishing shadows cast by the following stars, (which are here arranged in order of brightness):

 Mag. Mag. α Canis Majoris (Sirius) – 1.4 ζ Orionis 1.9 α Bootis (Arcturus) 0.0 β Tauri 1.9 α Aurigae (Capella) 0.2 γ Geminorum 2.0 β Orionis (Rigel) 0.3 β Canis Majoris 2.0 α Canis Minoris (Procyon) 0.5 α Hydrae 2.0 α Orionis* (Betelgeuse) 0.8? α Arietis 2.0 α Tauri (Aldebaran) 1.0 κ Orionis 2.2 β Geminorum (Pollux) 1.1 β Leonis 2.2 α Virginis (Spica) 1.2 γ Leonis 2.2 α Leonis (Regulus) 1.4 δ Orionis 2.4 ε Canis Majoris 1.5 η Canis Majoris 2.4 α Geminorum (Castor) 1.6 ζ Argus 2.5 ε Orionis 1.8 α Ceti 2.7 δ Canis Majoris 1.9 15 Argus 2.9 γ Orionis 1.9

* Variable

The groups of stars comprised in the Pleiades and the sword of Orion also cast perceptible shadows. With a wide open window the belt of Orion should be added to this class.

Most of the observations on which this list is based were made at Princeton on February 7th, and 8th, and March 6th, 1902. The first of these nights is recorded as not remarkably clear, the others as very clear. Whenever there was any doubt of the reality of an observed patch of starlight, it was located at least three times, and it was verified each time that the star was really visible from the spot where its light had been located. Many more stars might have been added to the 29 in the foregoing list, had not unfriendly street lamps confined the observations to less than half the sky.

As many of the stars observed were at a low altitude, it may be concluded that a star of the 3rd magnitude, if near the zenith, would cast a perceptible shadow.

In attempting to get a shadow from these faint stars, the opening of the window should be narrowed to a width of a few inches, so as to cut off as much as possible of the diffused light of the sky. Care should be taken not to look at the sky while observing, as it is bright enough to dazzle the eyes for some little time.

By observing these precautions, the writer has been able to detect shadows cast by Sirius, Arcturus and Capella on moonlight nights,—in the case of Sirius, even when the Moon shone into the room.

The actual brightness of the screen, even when illuminated by Sirius, is very small in comparison with that of the “dark” background of the sky, as seen by the naked eye. White paper reflects about 80 per cent of the incident light. From photometric considerations, a disk of this material 1° in apparent diameter, illuminated perpendicularly by Sirius, should send us about 1/16,000 as much light as the star.

But, according to Professor Newcomb’s determination*, an area of sky 1° in diameter, remote from the Milky Way, sends us 9/10 as much light as a 5th magnitude star, or about 1/400 of the light of Sirius. Hence the sky is about 40 times as bright, area for area, as the paper illuminated by Sirius. The illumination of the paper by a 1st magnitude star is about 1/400 as bright, and by a 3d magnitude star less than 1/2000 as bright, area for area, as the “dark” background of the sky.

* Astrophysical Journal, December 1901.

This faint light, as might be anticipated, shows no perceptible color. The light of the white stars β and γ Orionis and the red star α Orionis does not differ sensibly in quality; but the light of the red star appears much fainter than the star’s brightness, as directly seen, would lead one to anticipate. On the screen, the light of α Orionis is much fainter than that of β, and only a little brighter than that of γ, while by direct vision α is much nearer to β than to γ in brightness. As β is 1 ½ magnitudes brighter than γ, it appears that, as measured by the intensity of its light on a screen, α Orionis is at least half a magnitude, perhaps a whole magnitude, fainter than when compared with the neighboring white stars by direct vision.

Such a result might have been anticipated à priori, since, in the ease of such faint lights as are here dealt with, the eye is sensitive to the green part of the spectrum alone, and this is relatively brighter in the spectrum of a white star than of a red one.

A much more interesting example of the accordance of theoretical prediction with observation is afforded by another phenomenon discovered by the writer, which is not hard to observe.

A surface illuminated by a planet—Venus for example—appears uniformly and evenly bright, but in the case of a fixed star, there are marked variations in brightness, so that the screen appears covered with moving dark markings.

This was predicted many years ago by Professor Young, in discussing the twinkling of the stars. He says*: “If the light of a star were strong enough, a white surface illuminated by it would look like the sandy bottom of a shallow, rippling pool of water illuminated by sunlight, with light and dark mottlings which move with the ripples on the surface. So, as we look toward the star, and the mottlings due to the irregularities of the air move by us, we see the star alternately bright and faint; in other words, it twinkles.”

General Astronomy, page 538 (edition of 1898).

It would be difficult to give a better description of the observed phenomenon than the one contained in the first part of the above quotation. It need only be added that the dark markings are much more conspicuous than the bright ones. This agrees with the fact that a star more frequently seems to lose light while twinkling than to gain it.

Sirius is the only star whose light is bright enough to make these light and dark mottlings visible without great difficulty, though the writer has seen them in the light of Rigel and Procyon. With Sirius they have been seen every time the star’s light has been observed on a moonless night. They are much more conspicuous when the star is twinkling violently than on nights when the air is steady. In the latter case there are only faint irregular mottlings, whose motion produces a flickering effect. More usually there appear also ill-defined dark bands, two or three inches wide. These are never quite straight nor parallel but usually show a preference for one or two directions, sometimes dividing the screen into irregular polygons. On some nights they merely seem to oscillate, but on others they have a progressive motion, which may be at any angle with their own direction. The rate of motion is very variable, but is greatest on windy nights,—another evidence of the atmospheric origin of the bands.

The best nights for observing these bands occur when the stars are twinkling strongly, and there is not much wind. The directions given above for observing shadows should be somewhat modified in this case.

If the room is not at the same temperature as the outer air, the window should be kept closed, as otherwise most of what is seen will be due to the air-currents near it. It is also desirable to have an area of star-light at least a foot square to see the bands in, so that a good sized part of the window should be left clear.

If Sirius is unavailable, Arcturus and Vega are probably the best stars in whose light to attempt to see the bands.

PRINCETON, N. J., March 24, 1902.