## A Case for Ten Planets

Clyde Tombaugh (1906-1997) spent the first fifteen years of his life on a farm near Streator, Illinois, and then his family moved to a farm near Burdett, Kansas (no wonder he got interested in astronomy!), and he went to high school there. Then, on February 18, 1930, Tombaugh, a self-taught amateur astronomer and telescope maker, discovered the ninth planet in our solar system, Pluto. It had been nearly 84 years since the eighth planet, Neptune, had been discovered, in 1846. And it would be another 62 years before another trans-Neptunian object (TNO) would be discovered.

Clyde Tombaugh made his discovery using a 13-inch f/5.3 photographic refractor at the Lowell Observatory in Flagstaff, Arizona.

Clyde Tombaugh was 24 years old when he discovered Pluto. He died in 1997 at the age of 90 (almost 91). I was very fortunate to meet Prof. Tombaugh at a lecture he gave at Iowa State University in 1990. At that lecture, he told a fascinating story about the discovery of Pluto, and I remember well his comment that he felt certain that no “tenth planet” larger than Pluto exists in our solar system, because of the thorough searches he and others had done since his discovery of Pluto. But, those searches were done before the CCD revolution, and just two years later, the first TNO outside the Pluto-Charon system, 15760 Albion (1992 QB1), would be discovered by David Jewitt (1958-) and Jane Luu (1963-), although only 1/9th the size of Pluto.

Pluto is, by far, the smallest of the nine planets. At only 2,377 km across, Pluto is only 2/3 the size of our Moon! Pluto has a large moon called Charon (pronounced SHAR-on) that is 1,212 km across (over half the size of Pluto), discovered in 1978 by James Christy (1938-). Two additional moons were discovered using the Hubble Space Telescope (HST) in 2005: Hydra (50.9 × 36.1 × 30.9 km) and Nix (49.8 × 33.2 × 31.1 km). A fourth moon was discovered using HST in 2011: Kerberos (10 × 9 × 9 km). And a fifth moon, again using HST, in 2012: Styx (16 × 9 × 8 km).

Pluto has been visited by a single spacecraft. New Horizons passed 12,472 km from Pluto and 28,858 km from Charon on July 14, 2015. Then, about 3½ years later, New Horizons passed 3,538 km from 486958 Arrokoth, on January 1, 2019.

Only one other TNO comparable in size to Pluto (or larger) is known to exist. 136199 Eris and its moon Dysnomia were discovered in 2005 by Mike Brown (1965-), Chad Trujillo (1973-), and David Rabinowitz (1960-). It is currently estimated that Eris is 97.9% the size of Pluto. Not surprisingly, in 2006 Pluto was “demoted” by the IAU from planethood to dwarf planet status. (Is not a “dwarf planet” a planet? Confusing…)

My take on this is that Pluto should be considered a planet along with Eris, of course. The definition of “planet” is really rather arbitrary, so given that Pluto was discovered 75 years before Eris, and 62 years before TNO #2, I think we should (in deference to the memory of Mr. Tombaugh, mostly) define a planet as any non-satellite object orbiting the Sun that is around the size of Pluto or larger. So, by my definition, there are currently ten known planets in our solar system. Is that really too many to keep track of?

There is precedent for including history in scientific naming decisions. William Herschel (1738-1822) is thought to have coined the term “planetary nebula” in the 1780s, and though we now know they have nothing to do with planets (unless their morphology is affected by orbiting planets), we still use the term “planetary nebula” to describe them today.

In the table below, you will find the eight “classical” planets, plus the five largest TNOs, all listed in order of descending size. (The largest asteroid, Ceres, is 939 km across, and is thus smaller than the smallest of these TNOs.)

You’ll see that the next largest TNO after Eris is Haumea, and that its diameter is only 67% that of Eris.

I’ve also listed the largest satellite for each of these objects. Venus and Mercury do not have a satellite—at least not at the present time.

It is amazing to note that both Ganymede and Titan are larger than the planet Mercury! And Ganymede, Titan, the Moon, and Triton are all larger than Pluto.

### Largest Objects in the Solar System

 Object Diameter (km) Largest Satellite Diameter (km) Size Ratio Jupiter 139,822 Ganymede 5,268 3.8% Saturn 116,464 Titan 5,149 4.4% Uranus 50,724 Titania 1,577 3.1% Neptune 49,244 Triton 2,707 5.5% Earth 12,742 Moon 3,475 27.3% Venus 12,104 N/A N/A N/A Mars 6,779 Phobos 23 0.3% Mercury 4,879 N/A N/A N/A Pluto 2,377 Charon 1,212 51.0% Eris 2,326 Dysnomia 700 30.1% Haumea 1,560 Hiʻiaka 320 20.5% Makemake 1,430 S/2015 (136472) 175 12.2% Gonggong 1,230 Xiangliu 200 16.3%

Should any other non-satellite objects with a diameter of at least 2,000 km be discovered in our solar system, I think we should call them planets, too.

## 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.

References
Herald, David (2018, October 23).  [Online forum comment].  Message
posted to https://groups.yahoo.com/neo/groups/IOTAoccultations/conversations/messages/65158

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