## 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.

## Earth’s Fickle Companions

A small number of asteroids are currently in a temporary 1:1 orbital resonance with the Earth in their orbit around the Sun.  Unlike the Moon, which is in a stable orbit around the Earth, these much tinier “co-orbital” objects are “just passin’ through.”

3753 Cruithne (1986 TO)
Came relatively close to the Earth each November from 1994 to 2015.  This will next happen around 2292.
Wiki  JPL  Orrery

85770 (1998 UP1)
Passes close to Venus, too.  This next happens in 2115.
Wiki  JPL  Orrery

54509 YORP (2000 PH5)
This tiny asteroid, perhaps 492 × 420 × 305 feet across, is a rapid rotator, turning around once every 12m10s. It is named after the YORP effect, as it provided the first observational evidence of that effect speeding up its spin rate.  It’s day will be half as long in only 600,000 years, and it may eventually speed up to one rotation every 20 seconds!
Wiki  JPL  Orrery

2002 AA29
This near-Earth object has an orbit that is very similar to the Earth’s, and even more circular, though it is inclined a full 10.7° to the ecliptic.  This asteroid is a good candidate for an automated sample-return mission and then human exploration because it is relatively close to the Earth and the amount of energy needed to visit 2002 AA29 and return to Earth is relatively small.
Wiki  JPL  Orrery

164207 (2004 GU9)
Currently, this asteroid never strays far from Earth, sometime leading it and sometimes following it.
Wiki  JPL  Orrery

277810 (2006 FV35)
This asteroid is another good candidate for human exploration.
Wiki  JPL  Orrery

2006 RH120
This extremely tiny object (just 7 to 10 feet across) spins more rapidly than any other object on our list: once every 2m45s!  It may even be an old rocket booster from the Apollo era, but recent evidence indicates it is a bona fide space rock.  It is currently leading the Earth in a very similar orbit.
Wiki  JPL  Orrery

2009 BD
We’ve been able to observe orbital changes in this tiny object due to the Sun’s radiation pressure.  It is currently trailing the Earth.
Wiki  JPL  Orrery

419624 (2010 SO16)
This asteroid was discovered using an infrared space telescope (WISE) and is in an unusually stable orbit that will change little during the next several hundred thousand years.  It is currently trailing the Earth.
Wiki  JPL  Orrery

2010 TK7
Also discovered using WISE, about 1,000 ft. across.  The only known Earth trojan asteroid.  It currently orbits the Sun about the L4 Lagrange point (leading the Earth by 60°).
Wiki  JPL  Orrery

2013 LX28
This asteroid has the highest orbital inclination (50°) of all the objects on our list.
Wiki  JPL  Orrery

2014 OL339
Serendipitously discovered while observing asteroid 2013 VQ4.
Wiki  JPL  Orrery

2015 SO2
Discovered from Slovenia.  Currently leading the Earth.
Wiki  JPL  Orrery

469219 (2016 HO3)
Currently, a quasi-satellite of the Earth.  Always remains within 38 to 100 lunar distances from the Earth as it orbits the Sun.  Leads, then follows, then leads again.  Quite a do-si-do!
Wiki  JPL  Orrery

Acknowledgements
The orrery videos for each asteroid were generated using the Jet Propulsion Laboratory’s incredible Orbit Diagram Java applet on their Small Body Database Browser web site (https://ssd.jpl.nasa.gov/sbdb.cgi), and captured using the equally incredible ScreenFlow software from Telestream (https://www.telestream.net/screenflow/).  Kudos to both organizations!