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!

Click on the map above for a high resolution view

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, the “Ocean of Storms”

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.

Mare Orientale, the “Eastern Sea”

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, the “Sea of Moisture”

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

Mare Nubium, the “Sea of Clouds”

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, the “Sea That Has Become Known”

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, the “Sea of Islands”

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, the “Sea of Vapors”

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, the “Sea of Rains”

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, the “Sea of Cold”

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.

Mare Serenitatis, the “Sea of Serenity”

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, the “Sea of Tranquility”

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, the “Sea of Nectar”

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

Mare Fecunditatis, the “Sea of Fertility”

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, the “Sea of Crises”

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, the “Serpent Sea”

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, the “Sea of Waves”

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, the “Foaming Sea”

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, the “Southern 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, “Smyth’s Sea”

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, the “Sea of the Edge”

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, the “Sea of Alexander von Humboldt”

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.

BepiColombo passing through the constellation Crater 10 Apr 2020 0600 UT as seen from Dodgeville, WI

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.

Comet ATLAS (C/2019 Y4) 10 Apr 2020 0224 UT 4 minute exposure 300mm f/5.6 Dodgeville WI