NASA News Releases

I receive dozens of emails each day, and chances are you do, too.  But one email list I think you should seriously consider subscribing to is the NASA News Releases.  There have been 115 news releases and 185 media advisories issued so far this year, so that averages to about one email a day.  The quality of these news releases is consistently high—they are far better written and information rich than most of what clutters up our inboxes or what you’ll find on a typical internet news site.

Take, for example, the two news releases that were issued on December 10:

RELEASE 18-114
NASA’s Newly Arrived OSIRIS-REx Spacecraft Already Discovers Water on Asteroid

RELEASE 18-115
NASA’s Voyager 2 Probe Enters Interstellar Space

Subscribing is easy:

NASA news releases and other information are available automatically by sending an e-mail message with the subject line subscribe to 
To unsubscribe from the list, send an e-mail message with the subject line unsubscribe to

Exoplanets with Deep Transits

The list above shows the 35 stars presently known to dip in brightness by 0.02 magnitudes or more due to a transiting exoplanet.

The change in the star’s magnitude during transit is given by

\Delta m = 2.5\log_{10}\left ( 1+\delta \right )

where Δm is the drop in magnitude, and δ is the transit depth

The time between transits for these exoplanets ranges between 0.79 and 5.72 days, with a median period of 2.24 days.  You can generate your own ephemeris for any of these transiting exoplanets at:

The transit duration for these exoplanets ranges between 1.08 and 3.11 hours, with a median duration of 2.11 hours.

The exoplanets with the deepest transits, HATS-6 b at 0.035 magnitudes and Kepler-45 b at 0.034 magnitudes, cross stars that are 15.2 and 16.9 magnitude, respectively, so these events might be out of reach for most amateur photometrists.  The only other star hosting a transiting exoplanet with a Δm ≥ 0.03m is Tycho 5165-481-1 in Aquila (WASP-80 b) which at visual magnitude 11.9 is a better candidate for smaller instruments.  The brightest star on our list (by far) is HD 189733 in Vulpecula, magnitude 7.7, with a drop in brightness that is almost as good at 0.026 magnitudes.

Fakhouri, O. (2018). Exoplanet Orbit Database | Exoplanet Data Explorer. [online] Available at: [Accessed 11 Dec. 2018].

Great Courses Launchpoint

The Teaching Company, LLC offers hundreds of video courses under the name “The Great Courses” on just about every subject imaginable, with more being added all the time.

Though offered for personal in-home viewing, these 30-minute lectures (or 45-minute in the case of Robert Greenberg’s engaging music courses) would make a wonderful centerpiece for a continuing education course.

As an instructor, what I would like to be able to do is show my class a Great Courses lecture, and then follow that with discussion and activities that reinforce and expand upon those  concepts during the remainder of a 60-minute or 90-minute class.

Not unlike what a good teaching assistant does in a college recitation section after a lecture by the professor, The Great Courses lecture would provide instructional scaffolding for both instructor and student.

I believe The Teaching Company has a great opportunity here.  Just by allowing an instructor to show a course to students (and charging a reasonable fee to do so), they would be opening up a new market for their products, and would no doubt bring in many new individual customers.

The Teaching Company could provide the courses “as is”, or could make available supplemental materials for the continuing education teacher and their students.

I even have a name for this new offering: Great Courses Launchpoint.

Currently, The Teaching Company doesn’t exactly encourage the use of their materials for face-to-face teaching:

My hope is that they will see the value of incorporating their video lectures into the classroom, and maybe Great Courses Launchpoint will roll out by the time I semi-retire in three or four years.  One of the frustrations of getting older is that my “day job” is taking a greater share of my time and available energy than ever before.  I love teaching, though, and semi-retirement will afford me the opportunity to begin teaching on a regular basis again.  Looking forward to it!

Satellite, Meteor, and Aircraft Crossings 2018

Edmund Weiss (1837-1917) and many astronomers since have called asteroids “vermin of the sky”, but since October 4, 1957 another “species” of sky vermin made their debut: artificial satellites.  In the process of video recording stars for possible asteroid occultations, I frequently see satellites passing through my ~¼° field of view.

I’ve put together a video montage and some individual videos of satellites I’ve recorded between March 10, 2018 and November 24, 2018.  All of the events are shown below, with the boldface events being presented chronologically in the first video.  Both the NORAD and International designations are given for each satellite.  The range is the distance between observer and satellite at the time of observation.

UT Date
10-21-2018 (2 satellites)

Target Star
UCAC4 459-002239
TYC 621-45742-1
UCAC4 497-035454
UCAC4 416-092784
UCAC4 385-061427
N Sct 2018
UCAC4 429-110724
UCAC4 384-149264
UCAC4 362-194694
UCAC4 526-007192
UCAC4 316-210974
UCAC4 418-144100
UCAC4 302-215969

SL-8 RB (Kosmos 726)
unknown space debris
unknown satellite
unknown satellite
unknown satellite
Ariane 5 RB (Payload A)
SL-8 RB (Kosmos 726)
Ariane 5 RB (VA209)
Kosmos 1092
SL-8 RB (Kosmos 80)
Galaxy 17 & NIMIQ 6
Sentinel 1B

SL-8 RB (Kosmos 726)
unknown space debris
unknown satellite
unknown satellite
unknown satellite
Ariane 5 RB (Payload A)
SL-8 RB (Kosmos 726)
Ariane 5 RB (VA209)
Kosmos 1092
SL-8 RB (Kosmos 80)
Galaxy 17
Sentinel 1B

7737; 1975-028-B
27946; 2003-043-B
7737; 1975-028-B
38780; 2012-051-C
14659; 1984-005-A
11326; 1979-030-A
1575; 1965-070-F
31307; 2007-016-B
38342; 2012-026-A
41456; 2016-025-A

Range & Direction
2,199.9 km SE
unknown SE
unknown SE
unknown NE
unknown NE
34,141.7 km NE
1,483.2 km SE
18,153.7 km NE
39,042.5 km NE
1,870.9 km NE
3,137.8 km NE
37,737.7 km E
37,736.3 km E
2,028.6 km NW

You’ll notice that sometimes the satellite crosses the field as a moving “dash”. That’s because sometimes I used longer exposure times to record a fainter target star.  A wind gust hit the telescope during the second event (3-25-2018).  The field is oriented North up and East to the left.  In this first video, you’ll notice that Sentinel 1B (the last event) has a unusual retrograde orbit (sun-synchronous) and is moving towards the NW.

In general, the slower the satellite is moving across the field, the higher is its orbit around the Earth.  One must also consider how much of the satellite’s orbital motion is along your line of sight to the satellite.

In the following video clip, you’ll see an unidentified piece of space debris, a very faint “dash” (due to integration) moving NE across the field from lower right to upper left, recorded on May 5, 2018 UT.

Next, we see a Ariane rocket body used to hoist SMART-1 towards the Moon and the Insat 3E and eBird 1 towards their geostationary orbits.  This recording was made on July 6, 2018 UT.  The rocket body is traveling NE (mostly east).  The light curve below the video suggests the possibility of some tumbling motion, but the satellite is faint and the photometry noisy.

And here is a rapidly tumbling (but low amplitude) Ariane rocket body, observed on July 31, 2018 UT and traveling NE.

Here is a no-longer-operational Japanese communications satellite named Yuri 2A, launched in 1984 and captured here on August 3, 2018 UT.  It is traveling NE (mostly east) and shows a beautiful long-period large-amplitude light curve.

Finally, we see not one but two geostationary communication satellites, Galaxy 17 (first and fainter) and NIMIQ 6 moving east across the field (as my telescope tracks westward to follow the Earth’s rotation), captured here on October 21, 2018 UT.  Galaxy 17 exhibits no discernible rotation, but NIMIQ 6 shows a low-amplitude long-period change in brightness.

Next we turn to three telescopic meteors I recorded on June 4, July 7, and September 11, 2018 UT.

UT Date

Target Star
UCAC4 408-094611
UCAC4 275-188730
UCAC4 399-093188

Constellation & Direction
Scutum, SSE
Sagittarius, SW
Scutum, NNE

Here these meteors are presented in a video montage.

I even captured an airplane crossing the field on August 22, 2018 UT:

Hughes, D. W. & Marsden, B. G. 2007, J. Astron. Hist. Heritage, 10, 21

Radio Quiet Zones

If you thought light pollution is bad (and it is!), radio pollution for radio astronomers is much worse.  Even years ago, terrestrial pollution of the radio spectrum tended to swamp faint celestial sources at many frequencies, and in 1958 the FCC established a 13,000 square mile rectangular region of West Virginia, Virginia, and Maryland as the National Radio Quiet Zone.  Two facilities within this protected region—whose natural topography helps to screen out many terrestrial radio emissions—are the Sugar Grove Station and the Green Bank Observatory near Green Bank, West Virginia.  The world’s largest fully-steerable radio telescope dish was built at Green Bank in 1956.  Though the original 300-ft. dish collapsed in 1988 due to a structural failure, it was rebuilt in 2000 as the Robert C. Byrd Green Bank Telescope, a leading facility for radio astronomy.

National Radio Quiet Zone

Counties wholly within the NRQZ, where many radio-emitting sources are regulated or banned outright, are Alleghany, Augusta, Bath, Highland, Nelson, and Rockbridge in Virginia, and Hardy, Pendleton, Pocahontas, Randolph, and Upshur in West Virginia.

The NRQZ isn’t the only radio quiet zone.  Here are some others:

  • Arecibo Observatory, Puerto Rico
  • Astronomy Geographic Advantage Act (AGAA), South Africa
  • Atacama Large Millimeter Array (ALMA), Chile
  • Australian Radio Quiet Zone WA (ARQZWA), Murchison Radio-astronomy Observatory (MRO)
  • Dominion Radio Astrophysical Observatory (DRAO), Canada
  • Five hundred meter Aperture Spherical Telescope (FAST), China
  • Institute for Radio Astronomy in the Millimeter Range (IRAM), Spain
  • Itapetinga Radio Observatory (IRO), Brazil
  • Large Millimeter Telescope (LMT), Mexico
  • Pushchino Radio Astronomy Observatory, Russia

The best place in the world to do radio astronomy is not on our world at all but instead on the far side of the Moon.  Radio telescopes deployed on the lunar farside could “listen” to the universe with absolutely no interference from Earth.  The solid body of the Moon (and its lack of an atmosphere) would completely block all radio signals and noise emanating from the Earth and Earth orbit.  And some radio telescopes could be quickly and easily deployed (think long-wire antennas rather than radio dishes).  Of course, the Moon itself will need to be designated as a radio quiet zone so that any lunar colonies, rovers, or satellites operate at frequencies and times that will not interfere with scientific work.  Maybe infrared or optical lasers would be a better way to communicate?

How would data from a lunar farside radio observatory be transmitted back to Earth?  One way would be to have a dedicated lunar satellite that receives data from the radio observatory while it is traveling over the lunar farside.  It would then re-transmit that data to Earth while it is traveling over the Earth-facing nearside.

Another (probably more expensive) approach would be to have a series of radio relay towers spaced at intervals from the radio observatory around to the lunar nearside where a transmitter could send the data back to Earth.

A third choice would be to locate the radio observatory in a libration zone along the border between the lunar nearside and farside.  At a libration zone radio observatory, data would be collected and stored until each time libration allows a direct line-of-sight to Earth.

The crater Daedalus, near the center of the lunar farside, has been suggested as the best location for a radio astronomy facility on the Moon (Pagana et al. 2006).

There is also a region above the farside lunar surface where radio emissions from Earth and Earth-orbiting satellites, would be blocked by the Moon, called the “Quiet Cone”, as illustrated in the diagram below.

The Earth-Moon L2 Lagrange point (EML2) is probably going to be within the lunar quiet cone.  Because L2 is an unstable Lagrange point, a radio telescope in the quiet cone would need to be in a halo orbit about EML2, and a tight one at that to avoid “seeing” any radio emissions from the highest Earth-orbiting satellites.

Antonietti, N.; Pagana, G.; Pluchino, S.; Maccone, C.
A proposed space mission around the Moon to measure the Moon Radio-Quiet Zone, 36th COSPAR Scientific Assembly. Held 16 – 23 July 2006, in Beijing, China.

Name That Comet

As of this writing, there are 3,635 comets named SOHO, over 300 comets named LINEAR, some 179 comets named PANSTARRS, 82 comets named McNaught, 62 comets named NEAT, and so on.

Except for the comets discovered by Scottish-Australian astronomer Robert H. McNaught (1956-), all of the above comets were discovered by various automated surveys.

SOHO = Solar and Heliospheric Observatory (spacecraft)

LINEAR = Lincoln Near-Earth Asteroid Research

Pan-STARRS = Panoramic Survey Telescope and Rapid Response System

NEAT = Near-Earth Asteroid Tracking

How do we distinguish between comets having the same name?  Each has a separate comet designation.  The first Comet LINEAR has a designation of P/1997 A2, and the most recent Comet LINEAR has a designation of C/2017 B3.

A comet designation starts with one of the following prefixes:

P/ – a periodic comet (orbital period < 200 years or confirmed observations at more than one perihelion passage)

C/ – non-periodic comet (orbital period ≥ 200 years and confirmed observations at only one perihelion passage)

X/ – comet for which no reliable orbit could be calculated (generally, historical comets)

D/ – a periodic comet that has disappeared, broken up, or been lost

A/ – an object that was mistakenly identified as a comet, but is actually a minor planet (asteroid, trans-Neptunian object, etc.)

I/ – an interstellar object that did not originate in our solar system

This is then followed by the year of discovery, a letter indicating the half-month of discovery, followed by the numeric order of discovery during the half-month.

So, we can see that the first Comet LINEAR, P/1997 A2, is a periodic comet discovered in 1997, between January 1 and January 15 of that year, and it was the second comet to be discovered during that period of time.  After the second perihelion passage, P/1997 A2 (LINEAR) was subsequently given the periodic comet number prefix of 230, so the full designation for this comet is now 230P/1997 A2 (LINEAR).

Likewise, the most recent Comet LINEAR (at the time of this writing), C/2017 B3, is a non-periodic comet discovered in 2017 between January 16 and January 31, the third comet discovered during that period of time.

Interestingly, if different periodic comets have the same name, they are sequentially numbered.  Perhaps the most famous example is Comet Shoemaker-Levy 9 that broke up and crashed into Jupiter during July 1994.  There are a total of nine periodic comets named Shoemaker-Levy.  They are:

192P/1990 V1   Shoemaker-Levy 1
137P/1990 UL3  Shoemaker-Levy 2
129P/1991 C1   Shoemaker-Levy 3
118P/1991 C2   Shoemaker-Levy 4
145P/1991 T1   Shoemaker-Levy 5
181P/1991 V1   Shoemaker-Levy 6
138P/1991 V2   Shoemaker-Levy 7
135P/1992 G2   Shoemaker-Levy 8
D/1993 F2      Shoemaker-Levy 9

However, four additional non-periodic comets were discovered by the Carolyn & Gene Shoemaker and David Levy team.  They have not received a numeric suffix and are all called “Comet Shoemaker-Levy”:

C/1991 B1      Shoemaker-Levy
C/1991 T2      Shoemaker-Levy
C/1993 K1      Shoemaker-Levy
C/1994 E2      Shoemaker-Levy

This strikes me as a bit strange.  Why afford a numeric suffix to a comet name only when it is a periodic comet?  Why not give all comets named “Shoemaker-Levy” a numeric suffix.  Normally, we would number them all in order of discovery, but since the nine periodic comets have already received a number, we would have to number the four non-periodic comets as C/1991 B1 (Shoemaker-Levy 10), C/1991 T2 (Shoemaker-Levy 11), C/1993 K1 (Shoemaker-Levy 12), and C/1994 E2 (Shoemaker-Levy 13).

I would like to see all comets, both periodic and non-periodic, receive a numeric suffix to their names whenever there is more than one.  So, instead of Comet LINEAR we would have Comet LINEAR 1, Comet LINEAR 2, Comet LINEAR 3, and so on.

By the way, the days of amateur astronomers discovering a new comet will probably soon come to a close.  Though this is a little sad, it does tell us that the entire sky is being monitored much more closely than in the past, by a number of automated surveys.  And that is a good thing, because we will be much less likely to miss anything “new” in the sky.

None One of the comets this year (so far) have has been discovered by amateurs.

UPDATE – November 20, 2018: California amateur astronomer and prolific comet hunter Don Machholz, along with Japanese amateur astronomers Shigehisa Fujikawa and Masayuki Iwamoto, independently discovered a new comet on November 7.  The new long-period comet has been named C/2018 V1 (Machholz-Fujikawa-Iwamoto).  Remarkable!

Here is the current tally of comet discoveries (or recoveries) this year:

Pan-STARRS (Panoramic Survey Telescope and Rapid Response System)
364P/2018 A2 (PANSTARRS)
372P/2018 P6 (McNaught) [recovery of P/2008 O2]

ATLAS (Asteroid Terrestrial-impact Last Alert System)
C/2018 A3 (ATLAS)
C/2018 E1 (ATLAS)
C/2018 K1 (Weiland) [H. Weiland, ATLAS]
C/2018 L2 (ATLAS)
C/2018 O1 (ATLAS)

MLS (Mt. Lemmon Survey)
C/2018 A6 (Gibbs) [A.R. Gibbs, MLS]
C/2018 B1 (Lemmon)
P/2018 C1 (Lemmon-Read) [M.T. Read, Spacewatch, Kitt Peak]
C/2018 C2 (Lemmon)
C/2018 EF9 (Lemmon)  [originally classified as an asteroid]
C/2018 F1 (Grauer) [A.D. Grauer, MLS]
C/2018 F3 (Johnson) [J.A. Johnson, MLS]
C/2018 KJ3 (Lemmon) [originally classified as an asteroid]
P/2018 L5 (Leonard) [G. Leonard, MLS]
C/2018 R3 (Lemmon)
C/2018 R5 (Lemmon)

SONEAR (Southern Observatory for Near Earth Asteroid Research)
C/2018 E2 (Barros) [Joao Barros, SONEAR]

NEOWISE (Near-Earth Object Wide-field Infrared Survey Explorer)
C/2018 EN4 (NEOWISE)  [originally classified as a Centaur asteroid]
C/2018 N1 (NEOWISE)

366P/2018 F2 (Spacewatch)

CSS (Catalina Sky Survey)
367P/2018 H1 (Catalina)
C/2018 M1 (Catalina)
C/2018 R4 (Fuls) [D.C. Fuls, CSS]

NEAT (Near-Earth Asteroid Tracking)
368P/2018 L3 (NEAT)
370P/2018 P2 (NEAT)

ASAS-SN (All Sky Automated Survey for SuperNovae)
C/2018 N2 (ASASSN)

OGS (ESA Optical Ground Station)
369P/2018 P1 (Hill) [recovery of P/2010 A1]
371P/2018 R1 (LINEAR-Skiff) [recovery of P/2001 R6]

373P/2018 R2 (Rinner)  [Jean-Francois Soulier, Maisoncelles, and Krisztian Sarneczky, University of Szeged, Piszkesteto Station (Konkoly), independently recovered P/2011 W2]

374P/2018 S1 (Larson) [Krisztian Sarneczky and Robert Szakats, University of Szeged, Piszkesteto Station (Konkoly), recovered P/2007 V1]

375P/2018 T1 (Hill) [Krisztian Sarneczky, University of Szeged, Piszkesteto Station (Konkoly), recovered P/2006 D1]

Cosmology: The History and Nature of Our Universe

Mark Whittle, Professor of Astronomy at the University of Virginia, has put together the most comprehensive and comprehensible treatment on the subject of cosmology that I have ever encountered.  Cosmology: The History and Nature of Our Universe, a series of 36 thirty-minute video lectures for The Great Courses (Course No. 1830), is a truly remarkable achievement.

Even though this course was released ten years ago in 2008, all of the material is still completely relevant.  This is the course on cosmology that I’ve always wanted but never had.  Enjoy!

Cosmology has come a long ways since I was a physics and astronomy student at Iowa State University from 1975-1980, and again in 1981, 1984, and 2000-2005.  I’m glad to see a course specifically about cosmology is now offered at a number of universities.  When I was an undergraduate student at ISU, it was unheard of.  The University of Wisconsin at Madison Department of Astronomy currently offers both an undergraduate and a graduate course in cosmology: Astronomy 335 – Cosmology, and Astronomy 735 – Observational Cosmology.  And the Department of Physics & Astronomy at Iowa State University now offers an undergraduate/graduate dual-listed cosmology course: Astro 405/505 – Astrophysical Cosmology.

When I retire in a few years, I would love to be a “fly on the wall” at the UW-Madison astronomy department.  Wonder if they could use an expert SAS programmer to help analyze the massive quantities of data they surely must have?  (Though the last time I interviewed for an astronomy job, at the McDonald Observatory in Texas, the interviewers had never heard of SAS but asked if I knew Python, which of course is what nearly everyone is looking for and using these days.  Tomorrow, it will be something else…).  In retirement, at the very least I would love to immerse myself in a few astronomy courses at UW-Madison.  Something to look forward to!

Blue Light Blues

One by one, all of our warm white lights are being replaced by cold, harsh, bluish-white LEDs.  And it is happening fast.

Everywhere.  In our streetlights, our workplaces, even our homes.  How do you like looking into those blue-white vehicle headlights as compared with the yellow-white ones we have been using since the automobile was invented?

LED lighting is the way of the future, don’t get me wrong, but we should be specifying and installing LED lights with a correlated color temperature (CCT) of 2700K or 3000K—with few exceptions—not the 4000K or higher that is the current standard.

Why is 4000K the current standard?  Because blue-white LEDs have a slightly greater luminous efficacy than yellow-white LEDs.  Luminous efficacy is the amount of light you get out for the power you put in, often measured in lumens per watt.  But should luminous efficiency be the only consideration?  What about aesthetics?  In addition to luminous efficacy, there are other, more significant ways to reduce power consumption and greenhouse gas emissions:

  • Use the minimum amount of light needed for the application; no need to overlight
  • Use efficient light fixtures that direct light only to where it is needed; near-horizontal light creates annoying and visibility-impairing glare and light trespass, and direct uplight into the night sky is a complete waste
  • Produce the light only when it is needed through simple switches, time controls, and occupancy sensors; or, use lower light levels during times of little or no activity

Even the super-inefficient incandescent light bulb (with a CCT of 2400K, by the way), operating three hours each night uses less energy than the light source with the highest luminous efficacy operating dusk to dawn.  Think about it.

In my town, as in most now, the soothing orange 1900K high pressure sodium (HPS) streetlights are being replaced with 4000K LEDs.  That’s a big change.  It will completely transform our outdoor nighttime environment.  Warm-white compact fluorescents are 2700K, and even tungsten halogen bulbs are 3000K.  Do we really want or need 4000K+ LEDs?

We are currently witnessing a complete transformation of our illuminated built environment.  Not enough questions are being asked nor direction being given by citizens, employees, and municipalities.  The lighting industry generally wants to sell as many lights as possible at the highest profit margin.  We as lighting consumers need to make sure we have the right kind of light, the right amount of light, and lighting only when and where it is needed.

Effective Diameter of an Irregularly-Shaped Object

A diameter of a circle in 2D is defined as any straight line segment that intersects the center of the circle with endpoints that lie on the circle.  Since all diameters of a circle have the same length, the diameter is the length of any diameter.

Likewise, a diameter of a sphere in 3D is defined as any straight line segment that intersects the center of the sphere with endpoints that lie on the surface of the sphere, and the diameter is its associated length.

But how do we define the diameter of an irregularly-shaped object such as a typical asteroid or trans-Neptunian object?

For a well-characterized object such as 951 Gaspra—the first asteroid to be photographed up close by a spacecraft—we’ll see the dimensions of the best fitting triaxial ellipsoid given in terms of “principal diameters”.  In the case of Gaspra, that is 18.2 × 10.5 × 8.9 km.

In certain circumstances, however, it would advantageous to characterize an irregularly-shaped object using a single “mean diameter”.  How should we calculate that?

There are two good approaches, provided you have enough information about the object.  The first is to determine the “volume equivalent diameter” which is the diameter of a sphere having the same volume as the asteroid.  This is particularly relevant to mass and density.

For purposes of illustration only, let’s assume Gaspra’s dimensions are exactly the same as its best-fitting triaxial ellipsoid.  If that were true, the volume of Gaspra would be

V = \frac{{4\pi abc }}{3}

where V is the volume, and a, b, and c are the principal radii of the triaxial ellipsoid.

Plugging in the numbers 9.1 km, 5.25 km, and 4.45 km (half the principal diameters), we get a volume of 890.5 km3.

The volume equivalent diameter is

d_{vol} = \left (\frac{6V_{obj}}{\pi } \right )^{1/3}

where dvol is the volume equivalent diameter, and Vobj is the volume of the object.

Plugging in the volume of 890.5 km3 gives us a volume equivalent diameter of 11.9 km.

The second approach is to determine the “surface equivalent diameter” which is the diameter of a sphere having the same surface area as the asteroid.  This is most relevant to reflectivity or brightness.

Once again using our triaxial ellipsoid as a stand-in for the real 951 Gaspra, we find that the general solution for the surface area of an ellipsoid requires the use of elliptic integrals.  However, there is an approximation that is more straightforward to calculate and accurate to within about 1%:

S\approx 4\pi\left ( \frac{a^{p}b^{p}+a^{p}c^{p}+b^{p}c^{p}}{3} \right )^{1/p}

where S is the surface area, p ≈ 1.6075 can be used, and a, b, and c are the principal radii of the triaxial ellipsoid.

Once again plugging in the numbers, we get a surface area of of 478.5 km2.

The surface equivalent diameter is

d_{sur} = \left (\frac{S_{obj}}{\pi } \right )^{1/2}

where dsur is the surface equivalent diameter, and Sobj is the surface area of the object.

Plugging in the surface area of 478.5 km3 gives us a surface equivalent diameter of 12.3 km.

You’ll notice that the surface equivalent diameter for 951 Gaspra (triaxial ellipsoid approximation) is 12.3 km which is larger than the volume equivalent diameter of 11.9 km.  The surface equivalent diameter is apparently always larger than the volume equivalent diameter, though I leave it as an exercise for the mathematically-inclined reader to prove that this is so.

Herald, David (2018, October 23).  [Online forum comment].  Message
posted to

Thomas, P.C., Veverka, J., Simonelli, D., et al.: 1994, Icarus 107The Shape of Gaspra, 23-26.