Limb Darkening and Luminosity

The Sun photographed on 8 May 2019 in white light by Matúš Motlo
showing sunspots, faculae, and limb darkening

The photosphere of our Sun and most other stars exhibit a phenomenon called limb darkening where the disk is brighter at the center than at the edges at optical wavelengths. This effect is more pronounced towards the violet end of the visible spectrum than it is towards the red end.

Limb darkening occurs because there is a strong temperature gradient within the photosphere (deeper is hotter) and we see deeper into the Sun at the center of the disk then we do toward the edges. The deeper, hotter regions of the photosphere produce more visible light than do the shallower, cooler regions.

Does this non-uniformity of light emitted from the disk of a star mean we are “missing” some light in measuring a star’s brightness that would then affect our ability to accurately calculate the star’s total luminosity? Not at all. Here’s why.

Stars are almost always isotropic emitters of light. That means they emit light uniformly in all directions. At a given distance from the star, an observer would measure the same brightness of the star no matter what their direction from it. Even though the edges of the stellar disk are darker, the center is brighter, and the total integrated brightness is the same as it would be if all parts of the disk were emitting uniformly.

We calculate the luminosity of the star by measuring the amount of light we receive across our collecting area (whether that be the human eye or the telescope aperture), and then dividing this collecting area into the total surface area of a sphere centered on the star and having a radius that is our distance from the star. We then take that quotient times the amount of light we detect in our small collecting area to get the total amount of light emitted by the star in all directions.

Dean Ketelsen (1953-2023): A Personal Remembrance

Dean Ketelsen at the Grand Canyon Star Party

A dear friend of mine passed away suddenly last week while mowing the lawn at his cottage in St. Charles, Illinois. Most of us are lucky to have maybe a dozen friends. Dean must have had hundreds. He was as generous and kind-hearted as anyone I have ever known. And incredibly knowledgeable about observational astronomy and optics.

I first met Dean while I was an undergraduate student at Iowa State University in the late 1970s. He was hired by Dr. Willet Beavers to make stellar radial velocity observations using the 24-inch telescope at ISU’s Erwin W. Fick Observatory. I was the primary data analyst reducing the data from the telescope, and all of us were amazed at how many stars Dean could observe in a night! I believe Dean was the most productive observer Fick Observatory ever had.

Dean and I were part of the ISU team that traveled to a farm near Riverton, Manitoba, Canada to observe the total solar eclipse on February 26, 1979.

Iowa State University Solar Eclipse Expedition – February 26, 1979
Front Row (left to right): Dan Peterson, Maria Meyers, Chuck Hoelzen, Ed Sexauer
Back Row (left to right): David Oesper, Jim Pierce, David Cook, Mike Andrews, Prof. Stan Williams, Prof. Willet Beavers, Dean Ketelsen, Joe Eitter

After graduation and working for Fick Observatory before the radial velocity grant money ran out (temporarily), I moved to Dell Rapids, South Dakota to work for the EROS Data Center near Sioux Falls. But, before I left, Dean gave me a Unitron refractor. One of many examples of his generosity.

Soon after I moved to South Dakota, Dean moved to Tucson, Arizona to become a telescope operator on the Mayall 4-meter telescope at Kitt Peak National Observatory. Then, as now, the 4-meter scope was heavily scheduled, but on Christmas and New Year’s he sometimes had the scope to himself for photography and visual observing. I asked him once, “What is the most impressive object you ever saw with the Kitt Peak 4-meter?” His reply: “The crescent moon!” I received some beautiful black & white large prints of galaxies and nebulae from Dean taken with one of the large Kitt Peak instruments. I framed and cherished these astrophotos.

Dean left the telescope operator position at Kitt Peak a few years later and began working at the University of Arizona Mirror Lab where he remained for the rest of his life. He was directly involved in fabricating several of the 8.4-meter mirrors—the largest monolithic telescope mirrors in the world—as well as smaller optics as well. Early in his career at the Mirror Lab, Dean was also working part-time on a Master’s degree in Optical Science at the University of Arizona, but he was never able to complete it before classes he took more than five years earlier no longer counted towards his degree. And I can see why. Dean led a rich and busy life, and his many friends and acquaintances were always his first priority.

Dean’s hospitality was legendary. My family regularly visited Tucson over the years, and Dean was always a most gracious host, transporting us to see all the good sights whenever we visited. A tour of the Mirror Lab was often included, so—thanks to Dean—I have been there many a time.

Dean’s generosity was also legendary. Besides the Unitron refractor and astrophotos, many years ago Dean “loaned” me a pair of Fujinon 16×70 binoculars, and after I moved to Tucson in 2022, he gave me a pair of Celestron 25 x 100 binoculars as a house-warming gift, no longer following any pretense that this would be a loan.

Over a several year period, Dean made a 24-inch mirror for the Ames Area Amateur Astronomers in Iowa, which they are still using today in a Dobsonian telescope built by club members. And, speaking of Dobsonians, Dean was a close acquaintance of John Dobson, and they often got together at star parties.

Dean Ketelsen with John Dobson at a star party

Here are some recent examples of Dean’s generosity. When Suzy and I came to Tucson to visit December 26-30, 2021, Dean and his dear friend Susan Yager picked us up at the Amtrak station and they both spent a lot of time with us as we were thinking about moving to Tucson. Ditto for our March 6-10, 2022 house-hunting trip. I was planning to take Amtrak back to Wisconsin with a stop in Alpine, TX to visit my daughter and her family while Suzy flew to Chicago to get back to work sooner, but Dean was driving from Tucson to St. Charles, Illinois so I rode with him. Though he didn’t have to, Dean went out of his way to drop me off in Dodgeville, Wisconsin and then went on to St. Charles.

Before that house-hunting trip, Dean had reached out to the relatives in charge of Derald Nye’s estate, knowing that I would be losing my backyard research observatory in Wisconsin and that it might be possible for me to purchase his home in Corona de Tucson, which would include an observatory. Unfortunately, that opportunity did not happen, but Dean subsequently put in a good word for me so that I could serve on the 16-inch Meade telescope committee which will add that telescope to the TAAA’s TIMPA observing site.

Before we moved to Tucson, Dean offered to transport my astronomical optics in his large van so that I didn’t need to entrust that delicate equipment to the movers. A week before moving, we drove from Dodgeville and he drove from St. Charles where we met up in Rockford at Lino’s for pizza (great restaurant!) and the transfer of optical equipment to his van afterwards. Needless to say, that equipment arrived safe and sound and in perfect condition at our new house just a few days after the movers when Dean made the trip back to Tucson.

After we moved to Tucson on May 1, 2022, besides restaurant get-togethers at Daily Mae’s and Bianchi’s, Dean & Susan joined John & Lana Gilkison at our house to observe the May 15, 2022 total lunar eclipse. Dean (and Susan) picked me up twice for dark-sky observing: once to watch the Tau Herculid meteor shower Memorial Day 2022 at his favorite observing spot along the road to the top of Kitt Peak, and once to observe from Empire Ranch SSE of Tucson. I was looking forward to many more observing sessions with Dean, but sadly that will not happen. I have lost my best observing buddy here.

No one person can relate all the accolades and experiences that Dean had, but I know of a few. Dean received the 2002 Las Cumbres Amateur Outreach Award from the Astronomical Society of the Pacific, and the asteroid 124075 Ketelsen (2001 GT1) was named after him.

Dean Ketelsen receiving the 2002 Las Cumbres Amateur Outreach Award
from the Astronomical Society of the Pacific

Dean was primarily responsible for reincarnating the Grand Canyon Star Party in 1991. He was a primary organizer for many years, and I believe he had attended every year since, including this year. I had the good fortune to attend in 2006, and gave one of the “Twilight Talks”. The most wonderful aspect of this star party that makes it very special and decidedly different from other star parties I have attended is that thousands of enthusiastic visitors to Grand Canyon National Park from all around the world are regaled by a twilight talk each night followed by observing through nearly 50 telescopes, binoculars, and green-laser-pointed constellations and satellites. The enthusiasm of the amateur astronomers sharing their love of astronomy with folks who are in an unusually good mood because they’re on vacation in a beautiful place is a winning combination. Dean had a lot to do with that vibe!

Dean was also an excellent public speaker, and frequently gave public astronomy talks and talks about the exciting things happening at the Mirror Lab.

Joan Oesper, Dean Ketelsen, Melinda Ketelsen, and David Oesper at Yerkes Observatory in 2008

Dean was an incredible photographer, whether the subject was astronomical, terrestrial, or people. He and his wife Melinda, who passed away after a long battle with cancer in 2016, have a blog called The Ketelsens! that includes many of his photos and descriptions of many of their experiences through the last posting on May 31, 2020—during the COVID-19 pandemic. I sincerely hope this blog will be moved to a permanent location on the internet before his blogspot account runs out. It would be a terrible shame to lose this treasure!

And, speaking of photography, Dean first suggested many years ago the idea of stereo photography of the aurora. To the best of my knowledge, this has seldom been done, though with cellular phones and digital cameras now it would be relatively easy to coordinate such a venture. Two observers separated by a hundred miles or more with identical cameras, lenses, and exposure times would need to take pictures of the aurora at exactly the same time and in exactly the same direction (centered on the same star or constellation). The results, I’m sure, would be spectacular!

I have found it difficult to capture all I want to say about Dean in this article, but I’d like to finish by sharing with you the recent email communications I had from Dean, right up to the day before he died. All but one of my emails to Dean are unimportant in the context of this article, so they are not included here.

June 4, 2023 email from Dean Ketelsen
Just got word from Elinor’s niece Cathy (Prescott) that Elinor died a couple weeks ago.  Evidently fell and broke her arm in several places, contracted pneumonia and died a week later.  So sad – about the last of that generation of friends.  She and David Levine, Derald Nye, Mike Terenzoni and I were the only folks (and Vicki!) at Grand Canyon #1.  No memorial is planned, but I’ve already asked thru Cathy for a vial of her ashes – maybe we can have our own at the Canyon next year!

July 26, 2023 email from Dean Ketelsen
Hi David-
How are you surviving the heat?  I’ve been up in St Charles coming up on 4 weeks and it has been delightful!  This week is the worst, supposed to be up over 90, I think for the first time, tomorrow and Friday before dropping to low 80s for the weekend.  I love those sunny days in the 70s, though we have been getting some smoke from the Canadian fires, some days worse than others.

The closer it comes, the less I’m excited about the annular eclipse.  Plus I’ve got a “Ketelsen reunion” on 8 October, and after driving to the Midwest, not sure I’m up for returning after less than a week!  So may watch the partial phase from here.  Still thinking about next April.  My first wife Vicki’s sister and her family live in Dallas and am welcome there.  I’ve sent them a map of the path and they are looking for a location closer to the center line for a small group.  Will see what they come up with.  Not sure I’m interested in trying to chase clear spots – again, will see what sort of a zoo it is!

Probably back in Tucson about the weekend of the 12th.  Hopefully temp will have dropped a little towards normal!

Hang in there!


August 5, 2023 email from Dean Ketelsen
Hi David-
This article caught my eye in NYT online site.  The hospital mixup was only a few miles from Riverton where we observed the ’79 solar eclipse…

Dean texted me about a Space X rocket launch from Vandenberg on August 7. I called him and we talked briefly on the phone. Little did I know it would be the last time I would hear his voice. Then, he sent me this email:

Good Luck! <> on behalf of Launch Alert <>
Sent: Monday, August 7, 2023 1:02 PM <>
Subject: [EXT][Launch Alert] Launch on Schedule

External Email

Tonight’s launch of a Falcon 9 rocket from Vandenberg SFB appears to be on schedule. The following is an update from SpaceX:

“SpaceX is targeting Monday, August 7 at 8:57 p.m. PT (03:57 UTC on August 8) for a Falcon 9 launch of 15 Starlink satellites to low-Earth orbit from Space Launch Complex 4 East (SLC-4E) at Vandenberg Space Force Base in California.”

For launch and countdown status, go to…

August 7, 2023 email from David Oesper to Dean Ketelsen
Hi Dean,
Thanks for letting me know about this.  We had partly cloudy skies tonight, which didn’t help, and I had to observe from my patio so if, as I suspect, the launch would only have been visible close to the WNW horizon, I wouldn’t have been able to see it.  I thought I might be able to see one of the stage separations as it was heading to our southern sky here, but no luck with that either.  Oh well, it was worth a try, anyway.



August 8, 2023 email from Dean Ketelsen
Hey David-
Watching the launch online, I could see the sunset from the onboard camera, but I don’t think it ever rose into bright sunset.  Still, Ben Bailly of TAAA captured the enclosed last night.  Still, not as spectacular as what it could be – the second taken be non-astronomer friend from Sabino Canyon area 10 months ago she noticed w/o advance warning…. Better luck next time!


Dean died the following day. Here is his obituary:

Dean’s obituary states that there will be a future gathering in Tucson to celebrate Dean’s life. As soon as that event is announced, I’ll post the information here in a comment.

Dean Ketelsen – Public Star Party at Sabino Canyon – April 29, 1989

I encourage you to share your personal remembrances of Dean by posting a comment here.

Otto Struve & Exoplanets, 1952

It’s too bad the remarkable Russian-born American astronomer Otto Struve (1897-1963) never lived to see the discovery of the first exoplanets, especially considering how he was probably the first to suggest the two main techniques by which they are now discovered.

The first discovery of something that could be called an exoplanet was announced in 1992 by the Polish astronomer Aleksander Wolszczan (1946-) and Canadian astronomer Dale Frail (1961-). They found two planets orbiting a neutron star 2,300 light years away in the constellation Virgo. This neutron star is the pulsar PSR 1257+12, which had only recently been discovered by Wolszczan (1990). The pulsar planets were detected using a variant of the Doppler (radial velocity) method, and a third planet was discovered by the same team in 1994. These planets likely formed from the debris disk formed when two white dwarf stars merged, so they could be considered “exotic” planets, quite unlike anything found in our solar system.

In 1995, the first exoplanet orbiting a “normal” star was announced by Swiss astronomers Michel Mayor (1942-) and Didier Queloz (1966-). Using the Doppler (radial velocity) method, they found a “hot Jupiter” orbiting the star 51 Pegasi at a distance of 51 light years (nice coincidence!).

In 1999, independent teams led by Canadian-American astronomer David Charbonneau (1974-) and American astronomer Gregory W. Henry (1972-) were the first to use the transit method to detect an exoplanet. They confirmed a hot Jupiter orbiting the star HD 209458 (also in Pegasus, another nice coincidence) 157 light years distant that had been discovered using the Doppler (radial velocity) technique only weeks earlier.

As you can see, the 1990s was the decade when exoplanetary science got its start!

Getting back to the prescience of Otto Struve—40 years prior to the discovery of the first exoplanets—Joshua Winn (1972-) in his newly-published The Little Book of Exoplanets writes:

Although the discovery of hot Jupiters came as a surprise, it’s not quite true that nobody foresaw them. In 1952, Otto Struve, an astronomer at the University of California at Berkeley, published a short paper pointing out that the precision of Doppler measurements had become good enough to detect planets—but only if there existed planets at least as massive as Jupiter with orbital periods as short as a few days. Setting aside the question of how such a planet might have formed, he realized there is no law of physics that forbids such planets from existing. In an alternate history, Struve’s paper inspired astronomers to launch a thousand ships and explore nearby stars for hot Jupiters. In fact, his paper languished in obscurity. None of the pioneers—neither Walker, Latham, Mayor, nor Queloz—were influenced by Struve’s paper. The planet around 51 Pegasi probably could have been discovered in the early 1960s, or surely by Walker in the 1980s, had the Telescope Time Allocation Committee allowed him to observe a larger number of stars.

Here is Otto Struve’s 1952 paper in its entirety (references omitted), published in the October 1952 issue of The Observatory.


By Otto Struve

With the completion of the great radial-velocity programmes of the major observatories, the impression seems to have gained ground that the measurement of Doppler displacements in stellar spectra is less important at the present time than it was prior to the completion of R. E. Wilson’s new radial-velocity catalogue.

I believe that this impression is incorrect, and I should like to support my contention by presenting a proposal for the solution of a characteristic astrophysical problem.

One of the burning questions of astronomy deals with the frequency of planet-like bodies in the galaxy which belong to stars other than the Sun. K. A. Strand’s discovery of a planet-like companion in the system of 61 Cygni, which was recently confirmed by A. N. Deitch at Poulkovo, and similar results announced for other stars by P. Van de Kamp and D. Reuyl and E. Holmberg have stimulated interest in this problem. I have suggested elsewhere that the absence of rapid axial rotation in all normal solar-type stars (the only rapidly-rotating G and K stars are either W Ursae Majoris binaries or T Tauri nebular variables, or they possess peculiar spectra) suggests that these stars have somehow converted their angular momentum of axial rotation into angular momentum of orbital motion of planets. Hence, there may be many objects of planet-like character in the galaxy.

But how should we proceed to detect them? The method of direct photography used by Strand is, of course, excellent for nearby binary systems, but it is quite limited in scope. There seems to be at present no way to discover objects of the mass and size of Jupiter; nor is there much hope that we could discover objects ten times as large in mass as Jupiter, if they are at distances of one or more astronomical units from their parent stars.

But there seems to be no compelling reason why the hypothetical stellar planets should not, in some instances, be much closer to their parent stars than is the case in the solar system. It would be of interest to test whether there are any such objects.

We know that stellar companions can exist at very small distances. It is not unreasonable that a planet might exist at a distance of 1/50 astronomical unit, or about 3,000,000 km. Its period around a star of solar mass would then be about 1 day.

We can write Kepler’s third law in the form V^{3} \sim \frac{1}{P}. Since the orbital velocity of the Earth is 30 km/sec, our hypothetical planet would have a velocity of roughly 200 km/sec. If the mass of this planet were equal to that of Jupiter, it would cause the observed radial velocity of the parent star to oscillate with a range of ± 0.2 km/sec—a quantity that might be just detectable with the most powerful Coudé spectrographs in existence. A planet ten times the mass of Jupiter would be very easy to detect, since it would cause the observed radial velocity of the star to oscillate with ± 2 km/sec. This is correct only for those orbits whose inclinations are 90°. But even for more moderate inclinations it should be possible, without much difficulty, to discover planets of 10 times the mass of Jupiter by the Doppler effect.

There would, of course, also be eclipses. Assuming that the mean density of the planet is five times that of the star (which may be optimistic for such a large planet) the projected eclipsed area is about 1/50th of that of the star, and the loss of light in stellar magnitudes is about 0.02. This, too, should be ascertainable by modern photoelectric methods, though the spectrographic test would probably be more accurate. The advantage of the photometric procedure would be its fainter limiting magnitude compared to that of the high-dispersion spectrographic technique.

Perhaps one way to attack the problem would be to start the spectrographic search among members of relatively wide visual binary systems, where the radial velocity of the companion can be used as a convenient and reliable standard of velocity, and should help in establishing at once whether one (or both) members are spectroscopic binaries of the type here considered.

Berkeley Astronomical Department, University of California.
1952 July 24.

Most Distant Human-Made Object

In 1895, Italian inventor and electrical engineer Guglielmo Marconi (1874-1937) produced the first human-made radio waves capable of traveling beyond the Earth, so radio evidence of the existence of human civilization has now traveled 128 light years from Earth. Assuming a stellar number density in the solar neighborhood of (7.99 ± 0.11) × 10−2 stars per cubic parsec1, Earth’s radio emissions have already reached about 20,000 star systems.

The most distant physical human-made object, however, is the Voyager 1 spacecraft, now over 160 AU from the solar system barycenter (SSB), a distance of almost 15 billion miles. That certainly sounds impressive by human standards, but that is only 0.0025 light years. As the distance of Voyager 1 from the solar system barycenter is constantly increasing, you’ll want to visit JPL Horizons to get up-to-date information using the settings below for your date range of interest. Delta gives the distance from the SSB to the Voyager 1 spacecraft in astronomical units (AU).

This still-functioning spacecraft that was launched on September 5, 1977, flew by Jupiter on March 5, 1979, and flew by Saturn on November 12, 1980, is now heading into interstellar space in the direction of the constellation Ophiuchus, the Serpent Bearer, near the Ophiuchus/Hercules border.

Given Voyager 1’s current distance (from Earth), a radio signal from Earth traveling at the speed of light would take 22 hours and 8 minutes to reach Voyager 1, and the response from Voyager 1 back to Earth another 22 hours and 8 minutes. So, when engineers send a command to Voyager 1, they won’t know for another 44 hours and 16 minutes (almost 2 days) whether Voyager 1 successfully executed the command. Patience is indeed a virtue!

Thanks to three onboard radioisotope thermoelectric generators (RTGs)2, Voyager 1 should be able to continue to operate in the bone-chilling cold of deep space until at least 2025.

In about 50,000 years, Voyager 1 will be at a distance comparable to the nearest stars.

1The Fifth Catalogue of Nearby Stars (CNS5)
Alex Golovin, Sabine Reffert, Andreas Just, Stefan Jordan, Akash Vani, Hartmut Jahreiß, A&A 670 A19 (2023), DOI: 10.1051/0004-6361/202244250

2At launch, the Voyager 1 RTGs contained a total of about 4.5 kg of plutonium-238, generating 390W of electricity.

The Dimmest Constellation

You are probably familiar with at least the names of the twelve constellations of the zodiac:


But are you familiar with the twelve constellations that have no stars brighter than 4th magnitude?

Coma Berenices
Corona Australis

All but two of these dim constellations are, at least in part, visible from southern Arizona; Chamaeleon and Mensa require a trip south to see.

The southern constellation Mensa, the Table Mountain (declination -70° to -85°) is a ghost of a constellation, exhibiting no star brighter than magnitude 5.1. That’s 17 times fainter than Polaris! In fact, that’s fainter than all the stars of the Little Dipper asterism! Mensa does have one claim to fame, however. The Large Magellanic Cloud, satellite galaxy of our Milky Way galaxy, straddles most of the border that Mensa shares with Dorado, the Swordfish.

Mensa is far and away the dimmest constellation. But Mensa is a small constellation, bested in size by 74 of the 88 constellations. So perhaps it is not too surprising that a small constellation is less likely to harbor a bright star. Another measure of faint, perhaps, is to determine which of these twelve constellations with no star brighter than 4th magnitude is largest. That might be more remarkable, because one is less likely to find no bright stars in a large area of sky than in a small area of sky. By this measure, Camelopardalis, the Giraffe, wins without a doubt. Camelopardalis is the 18th largest constellation, and yet contains no star brighter than magnitude 4.0. It is that empty region you might have not noticed midway between Capella and Polaris, best viewed at evening twilight’s end during the month of February each year.

Spectroscopic Parallax

For the nearest stars, the change in the position of the Earth in its orbit results in a tiny shift in the position of the nearby star relative to the distant background stars. This shift is called the trigonometric parallax. You can see the effect by holding your thumb up at arms length, closing your left eye, and lining up your thumb with something across the room. Now, alternate back and forth between having your right eye open and your left eye open and you’ll see the position of your thumb shift relative to an object further away. Move your thumb closer, and the shift is larger. That is the essence of trigonometric parallax.

Trigonometric Parallax

The distance to the star in parsecs (1 pc = 3.26 ly) is just

Now, a star’s distance, apparent brightness, and “true” (or intrinsic) brightness are related in the following way:

M = m + 5 (1 – log d)

where M = the absolute magnitude of the star

and m = the apparent magnitude of the star

and d = the distance to the star in parsecs

The absolute magnitude is the apparent magnitude the star would have if it were at a distance of 10 parsecs. Looking at it another way, the absolute magnitude is a proxy for the intrinsic brightness. The apparent magnitude is the star’s apparent brightness (as seen from Earth).

While the above equation is highly useful for general purpose calculations, to get the most accurate values astronomers must take into account atmospheric and interstellar extinction. And, anytime we deal with a star’s luminosity and its apparent brightness at some distance, d , we must specify the photometric system and optical filter that is being used. Or, less commonly (for practical reasons), we specify that the star’s luminosity and apparent brightness is to include all wavelengths of the electromagnetic spectrum, thus bolometric magnitudes are to be used.

Spectroscopic parallax is a bit of a misnomer, but here’s how it works for approximating the distance to main-sequence stars that are too far away to exhibit a measurable, reasonably certain, trigonometric parallax: measure the apparent magnitude of the star, and then using its spectrum to find its position on the H-R diagram, read off its absolute magnitude. Using your measured apparent magnitude and the star’s estimated absolute magnitude, you can solve for d the distance in the above equation.

Hertzsprung–Russell (H-R) diagram

The star’s color (the x-axis on the H-R diagram) is easy to measure, but a deeper analysis of the spectral lines is needed to determine whether the star is a main-sequence, giant, or supergiant star (or something else).

Using the Inverse Hyperbolic Sine

Image processing is both an art and a science, in equal measure, and I never cease to be amazed at the skill of the few people who are able to master it.

One tool in the ever-expanding workshop is the inverse hyperbolic sine, also known as the hyperbolic arcsine. Its use for image processing was described twenty years ago by Robert Lupton et al. (2003) in a paper entitled “Preparing Red-Green-Blue (RGB) Images from CCD Data.” In the abstract, the authors write:

We also introduce the use of an asinh stretch, which allows us to show faint objects while simultaneously preserving the structure of brighter objects in the field, such as the spiral arms of large galaxies.

Before we can know what a hyperbolic arcsine (asinh) is, we need to understand what a hyperbolic sine is. Just as a circle can be drawn out by the set of coordinates (x,y) = (cos θ, sin θ), the right half of an equilateral hyperbola (also known as a rectangular hyperbola) can be drawn using (x,y) = (cosh θ, sinh θ) where cosh is the hyperbolic cosine, and sinh is the hyperbolic sine. Just as the arcsine is the inverse sine function, i.e. if y = sin x, then x = asin y (also written as x = sin-1 y), so, too, the hyperbolic arcsine is the inverse hyperbolic sine function, i.e. if y = sinh x, then x = asinh y (or x = sinh-1 y).

If we consider the light intensity recorded by a pixel (say, a number between 0 and 65,536, where 0 is the darkest value and 65,536 the brightest) to be x, and then x′ to be the value of that pixel after passing through the hyperbolic arcsine function, we can map pixels using the following equation:

x'=sinh^{-1}\left ( \frac{x}{\beta } \right )=ln \left ( \frac{x+\sqrt{x^{2}+\beta ^{2}}}{\beta } \right )

where β is called the “softening parameter”, something you can tweak to bring out desired details.

If you play with this equation a little bit, you’ll quickly see that the smallest values of x (representing the darkest parts of your image) are pretty much left alone, but large values of x (representing the brightest parts of your image) are transformed to much smaller numbers. This then allows you to bring out the fainter details in your image without completely saturating the brighter parts of your image, since whether displayed on a monitor or the printed page, you have a limited dynamic range that can be rendered. Here is an example of an image that has benefited from a hyperbolic arcsine stretch.1

M17 with linear display (left) and after asinh stretching (right)

1IRIS Tutorial: Stretching levels and colors

IDA Information Sheets

I recently received a membership renewal notice from the International Dark-Sky Association (IDA) quoting Christopher Kyba that if light pollution continues to grow at the rate it currently is, “Orion’s belt will disappear at some point.”

This made me remember that I had written an IDA Information Sheet back in March 1997 that also had addressed how light pollution could erase much of the Orion constellation. I wrote,

Orion, arguably the most prominent of the constellations, begins to look more like “Orion, the Hunted” under a magnitude +4.0 sky. Under a magnitude +3.0 sky, Orion is on his deathbed. When light pollution is so bad that we have a magnitude +2.0 sky, only blazing Betelgeuse, regal Rigel, and Bellatrix and Alnilam remain to regale us.

Speaking of the IDA Information Sheets, I was the IDA Information Sheet Editor from 1996-1999, during which time I revised and edited most of the existing information sheets, edited and added many new ones from a number of contributors, as well as contributed many new ones that I authored, though I never credited myself as the author. One of the ones that I wrote was IDA Information Sheet 120, referenced above (and shown below). I have a complete hard copy set of IDA Information Sheets 1 through 175, the last of which was published in June 2000. I also have WordPerfect Macintosh source files for IDA Information Sheets 1 through 158, the last of which was completed on October 27, 1999.

Here’s IDA Information Sheet 120:

It is a shame that these IDA Information Sheets are no longer available anywhere on the Internet. At the very least, they are of historical interest, and I would say that much of the content is still relevant. Presumably, the IDA still has all of these information sheets, but after the Dave Crawford era, they have decided to remove access to them.

Finally, I want to express my disappointment that the International Dark-Sky Association has recently decided to change their name to DarkSky International. They are still in the process of changing everything over, but once that transition is complete, the IDA will be no more. The break with the Dave Crawford era will be complete. I, for one, will never forget how much Dave Crawford was able to accomplish during those early years, and how proud I was to have been a part of it.

The IDA/DSI is still a great organization, and I strongly encourage you to generously support it, as I do. It remains the most effective organization in the world addressing light pollution and the loss of our night sky and the natural nighttime environment.

Hidden Wonders of the Southern Sky

Here in southern Arizona, we can theoretically see 92.4% of the celestial sphere. I say “theoretically” because atmospheric extinction, light pollution, local topography, and obstructions limit the amount of the celestial sphere that we can see well. Also, far southern objects (down to δ = -58° at φ = 32° N) spend very little time above our horizon each day.

Practically speaking, then, we see somewhat less than 92% of all that there is to see from spaceship Earth.

Percent of the Celestial Sphere Visible

\% = 50\left [ 1-sin\left ( \left|\varphi \right| -90^{\circ}\right ) \right ]

where |φ| is the absolute value of your latitude in degrees

What are the most prominent objects we are missing, and what objects that we can see are they closest to?

Alpha Centauri

Never visible north of latitude 27° N, the nearest star system beyond our solar system is Alpha Centauri. Alpha Centauri A & B are bright stars, having a visual magnitude of 0.0 and +1.3, respectively, and in 2023 they are separated by just 8 arcseconds, about 1/4 of the angular separation between Albireo A & B. While Alpha Centauri A & B—which orbit each other once every 79.8 years—lie just 4.36 ly away, a faint red dwarf companion, Proxima Centauri (shining at magnitude +11.1), is even closer at 4.24 light years. It is not yet known whether Proxima Centauri, discovered in 1915, is gravitationally bound to Alpha Centauri A & B, or just presently passing through the neighborhood. Proxima is a full 2.2° away (over four moon-widths) from Alpha Centauri A & B.

When Arcturus (α Boo) and Zubenelgenubi (α Lib) are crossing our celestial meridian, so are Alpha & Proxima Centauri below the southern horizon.

Large Magellanic Cloud

The Large Magellanic Cloud (LMC), the largest satellite galaxy of our Milky Way galaxy and easily visible to the unaided eye, lies directly below our southern horizon when Rigel has crossed the meridian and Bellatrix is preparing to do so.

Small Magellanic Cloud

The Small Magellanic Cloud (SMC), the second-largest satellite galaxy of the mighty Milky Way lies underneath our southern horizon when M31, the Great Andromeda Galaxy, crosses the meridian near the zenith.

47 Tucanae

The 2nd brightest globular cluster in the sky (after Omega Centauri) is impressive 47 Tucanae. It is just 2.3° west and a little north of the Small Magellanic Cloud, so crosses the meridian below our horizon just as M31 is nearing the meridian.

Eta Carinae Nebula

Four times larger and brighter than the Orion Nebula, NGC 3372, the Eta Carinae Nebula, is a spectacular star-forming region containing a supermassive (130 – 180 M) binary star (Eta Carinae) that may go supernova at any time. When Leo the Lion is straddling the meridian, the Eta Carinae Nebula sneaks across as well.

Any other spectacular objects I should be including that are south of declination -58°? If so, please post a comment here.


George F. R. Ellis weighs in on the concept of infinity in his excellent paper, Issues in the Philosophy of Cosmology, available on astro-ph at He writes:

9.3.2 Existence of Infinities

The nature of existence is significantly different if there is a finite amount of matter or objects in the universe, as opposed to there being an infinite quantity in existence. Some proposals claim there may be an infinite number of universes in a multiverse and many cosmological models have spatial sections that are infinite, implying an infinite number of particles, stars, and galaxies. However, infinity is quite different from a very large number! Following David Hilbert, one can suggest these unverifiable proposals cannot be true: the word “infinity” denotes a quantity or number that can never be attained, and so will never occur in physical reality.38 He states:

Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought . . . The role that remains for the infinite to play is solely that of an idea . . . which transcends all experience and which completes the concrete as a totality . . .

This suggests “infinity” cannot be arrived at, or realized, in a concrete physical setting; on the contrary, the concept itself implies its inability to be realized!

Thesis I2: The often claimed physical existence of infinities is questionable. The claimed existence of physically realized infinities in cosmology or multiverses raises problematic issues. One can suggest they are unphysical; in any case such claims are certainly unverifiable.

This applies in principle to both small and large scales in any single universe:

The existence of a physically existing spacetime continuum represented by a real (number) manifold at the micro-level contrasts with quantum gravity claims of a discrete spacetime structure at the Planck scale, which one might suppose was a generic aspect of fully non-linear quantum gravity theories. In terms of physical reality, this promises to get rid of the uncountable infinities the real line continuum engenders in all physical variables and fields40. There is no experiment that can prove there is a physical continuum in time or space; all we can do is test space-time structure on smaller and smaller scales, but we cannot approach the Planck scale.

Infinitely large space-sections at the macro-level raise problems as indicated by Hilbert, and leads to the infinite duplication of life and all events. We may assume space extends forever in Euclidean geometry and in many cosmological models, but we can never prove that any realised 3-space in the real universe continues in this way—it is an untestable concept, and the real spatial geometry of the universe is almost certainly not Euclidean. Thus Euclidean space is an abstraction that is probably not physically real. The infinities supposed in chaotic inflationary models derive from the presumption of pre-existing infinite Euclidean space sections, and there is no reason why those should necessarily exist. In the physical universe spatial infinities can be avoided by compact spatial sections, resulting either from positive spatial curvature, or from a choice of compact topologies in universes that have zero or negative spatial curvature. Machian considerations to do with the boundary conditions for physics suggest this is highly preferable; and if one invokes string theory as a fundamental basis for physics, the “dimensional democracy” suggests the three large spatial dimensions should also be compact, since the small (“compactified”) dimensions are all taken to be so. The best current data from CBR and other observations indeed suggest k = +1, implying closed space sections for the best-fit FL model.

The existence of an eternal universe implies that an infinite time actually exists, which has its own problems: if an event happens at any time t0, one needs an explanation as to why it did not occur before that time (as there was an infinite previous time available for it to occur); and Poincaré eternal return will be possible if the universe is truly cyclic. In any case it is not possible to prove that the universe as a whole, or even the part of the universe in which we live, is past infinite; observations cannot do so, and the physics required to guarantee this would happen (if initial conditions were right) is untestable. Even attempting to prove it is future infinite is problematic (we cannot for example guarantee the properties of the vacuum into the infinite future—it might decay into a state corresponding to a negative effective cosmological constant).

It applies to the possible nature of a multiverse. Specifying the geometry of a generic universe requires an infinite amount of information because the quantities necessary to do so are fields on spacetime, in general requiring specification at each point (or equivalently, an infinite number of Fourier coefficients): they will almost always not be algorithmically compressible. All possible values of all these components in all possible combinations will have to occur in a multiverse in which “all that can happen, does happen”. There are also an infinite number of topological possibilities. This greatly aggravates all the problems regarding infinity and the ensemble. Only in highly symmetric cases, like the FL solutions, does this data reduce to a finite number of parameters, each of which would have to occur in all possible values (which themselves are usually taken to span an infinite set, namely the entire real line). Many universes in the ensemble may themselves have infinite spatial extent and contain an infinite amount of matter, with all the problems that entails. To conceive of physical creation of an infinite set of universes (most requiring an infinite amount of information for their prescription, and many of which will themselves be spatially infinite) is at least an order of magnitude more difficult than specifying an existent infinitude of finitely specifiable objects.

One should note here particularly that problems arise in the multiverse context from the continuum of values assigned by classical theories to physical quantities. Suppose for example that we identify corresponding times in the models in an ensemble and then assume that all values of the density parameter and the cosmological constant occur at each spatial point at that time. Because these values lie in the real number continuum, this is a doubly uncountably infinite set of models. Assuming genuine physical existence of such an uncountable infinitude of universes is the antithesis of Occam’s razor. But on the other hand, if the set of realised models is either finite or countably infinite, then almost all possible models are not realised. And in any case this assumption is absurdly unprovable. We can’t observationally demonstrate a single other universe exists, let alone an infinitude. The concept of infinity is used with gay abandon in some multiverse discussions, without any concern either for the philosophical problems associated with this statement, or for its completely unverifiable character. It is an extravagant claim that should be treated with extreme caution.

38An intriguing further issue is the dual question: Does the quantity zero occur in physical reality? This is related to the idea of physical existence of nothingness, as contrasted with a vacuum. A vacuum is not nothing!

40To avoid infinities entirely would require that nothing whatever is a continuum in physical reality (since any continuum interval contains an infinite number of points). Doing without that, conceptually, would mean a complete rewrite of many things. Considering how to do so in a way compatible with observation is in my view a worthwhile project.

So, given this discussion of infinities, the answer to the doubly hypothetical question, “Can God make a rock so big he can’t pick it up?” is likely a “Yes”! – D.O.