terrestrial planets – Cosmic Reflections

Mercury, Our Nearest Planetary Neighbor

If you’re an astronomy teacher that likes to put a trick question on an open book quiz or test once in a while to encourage your students to think more deeply, here’s a good one for you:

On average, what planet is closest to the Earth?

A. Mars

B. Venus

C. Mercury

The correct answer is C. Mercury.

Huh? Venus comes closest to the Earth, doesn’t it? Yes, but there is a big difference between minimum distance and average distance. Let’s do some quick calculations to help us understand minimum distance first, and then we’ll discuss the more involved determination of average distance.

Here’s some easily-found data on the terrestrial planets:

Planet	a	e	q	Q
Mercury	0.387	0.206
Venus	0.723	0.007
Earth	1.000	0.017
Mars	1.524	0.093

I’ve intentionally left the last two columns of the table empty. We’ll come back to those in a moment. a is the semi-major axis of each planet’s orbit around the Sun, in astronomical units (AU). It is often taken that this is the planet’s average distance from the Sun, but that is strictly true only for a circular orbit.¹ e is the orbital eccentricity, which is a unitless number. The closer the value is to 0.0, the more circular the orbit. The closer the value is to 1.0, the more elliptical the orbit, with 1.0 being a parabola.

The two empty columns are for q the perihelion distance, and Q the aphelion distance. Perihelion occurs when the planet is closest to the Sun. Aphelion occurs when the planet is farthest from the Sun. How do we calculate the perihelion and aphelion distance? It’s easy.

Perihelion: q = a (1 – e)

Aphelion: q = a (1 + e)

Now, let’s fill in the rest of our table.

Planet	a (AU)	e	q (AU)	Q (AU)
Mercury	0.387	0.206	0.307	0.467
Venus	0.723	0.007	0.718	0.728
Earth	1.000	0.017	0.983	1.017
Mars	1.524	0.093	1.382	1.666

Ignoring, for a moment, each planet’s orbital eccentricity, we can calculate the “average” closest approach distance between any two planets by simply taking the difference in their semi-major axes. For Venus, it is 1.000 – 0.723 = 0.277 AU, and for Mars, it is 1.524 – 1.000 = 0.524 AU. We see that Venus comes closest to the Earth.

But, sometimes, Venus and Mars come even closer to the Earth than 0.277 AU and 0.524 AU, respectively. The minimum minimum distance between Venus and the Earth in conjunction should occur when Venus is at aphelion at the same time as Earth is at perihelion: 0.983 – 0.728 = 0.255 AU. The minimum minimum distance between Earth and Mars at opposition should occur when Mars is at perihelion and Earth is at aphelion: 1.382 – 1.017 = 0.365 AU. Mars does not ever come as close to the Earth as Venus does at every close approach.

The above assumes that all the terrestrial planets orbit in the same plane, which they do not. Mercury has an orbital inclination relative to the ecliptic of 7.004˚, Venus 3.395˚, Earth 0.000˚ (by definition), and Mars 1.848˚. Calculating the distances in 3D will change the values a little, but not by much.

Now let’s switch gears and find the average distance over time between Earth and the other terrestrial planets—a very different question. But we want to pick a time period to average over that is sufficiently long enough that each planet spends as much time on the opposite side of the Sun from us as it does on our side of the Sun. The time interval between successive conjunctions (in the case of Mercury and Venus) or oppositions (Mars) is called the synodic period and is calculated as follows:

P₁ = 87.9691^d = orbital period of Mercury

P₂ = 224.701^d = orbital period of Venus

P₃ = 365.256^d = orbital period of Earth

P₄ = 686.971^d = orbital period of Mars

S₁ = (P₁^-1 – P₃^-1)^-1 = synodic period of Mercury = 115.877^d

S₂ = (P₂^-1 – P₃^-1)^-1 = synodic period of Venus = 583.924^d

S₄ = (P₃^-1 – P₄^-1)^-1 = synodic period of Mars = 779.946^d

I wrote a quick little SAS program to numerically determine that an interval of 9,387 days (25.7 years) would be a good choice, because

9387 / 115.877 = 81.0083, for Mercury

9387 / 583.924 = 16.0757, for Venus

9387 / 779.946 = 12.0354, for Mars

The U.S Naval Observatory provides a free computer program called the Multiyear Interactive Computer Almanac (MICA), so I was able to quickly generate a file for each planet, Mercury, Venus, and Mars, giving the Earth-to-planet distance for 9,387 days beginning 0^h UT 1 May 2019 through 0^h UT 10 Jan 2045. Here are the results:

Planet	Mean (AU)	Median (AU)	Min (AU)	Max (AU)
Mercury	1.039022	1.073148	0.549144	1.451501
Venus	1.138383	1.238453	0.265260	1.735280
Mars	1.711176	1.842260	0.380420	2.675330

As you can see, averaged over time, Mercury is the nearest planet to the Earth!

For a more mathematical treatment, see the article in the 12 Mar 2019 issue of Physics Today.

¹ See my article Average Orbital Distance for details.

Planets Without Satellites

It may be rare for terrestrial planets to be accompanied by satellites, especially large ones. It is far too early for us to draw any conclusions about terrestrial exoplanets (as no terrestrial exoplanet exomoons have yet been detectable), but in our own solar system, only two planets have no satellites, and they are both terrestrial planets: Mercury and Venus. Mars has two small satellites that are almost certainly captured asteroids from the adjacent asteroid belt rather than primordial moons, and that leaves only the Earth among the terrestrial planets to host a large satellite, though it, too, is almost certainly not primordial. Only the giant planets (Jupiter, Saturn, Uranus, and Neptune) have large systems of satellites, at least some of which may have formed while the planet itself was forming.

Though neither Mercury nor Venus has any natural satellites, Venus is known to have at least four transient quasi-satellites, more generally referred to as co-orbitals. They are:

322756 (2001 CK₃₂)
Comes close to both Earth and Mercury in its eccentric orbit (e=0.38).
Wiki JPL Orrery

2002 VE₆₈
Comes close to both Earth and Mercury in its eccentric orbit (e=0.41).
Wiki JPL Orrery

2012 XE₁₃₃
Comes close to both Earth and Mercury in its eccentric orbit (e=0.43).
Wiki JPL Orrery

2013 ND₁₅
Comes close to both Earth and Mercury in its very eccentric orbit (e=0.61), and is the only known trojan of Venus, currently residing near its L₄ Lagrangian point.
Wiki JPL Orrery

2015 WZ₁₂ is a possible fifth Venus co-orbital candidate. Observations during the next favorable observing opportunity in November of this year will hopefully better determine its orbit and nature.

2015 WZ₁₂
Possible Venus co-orbital.
Wiki JPL Orrery

There is concern that there may be many more Venus co-orbitals, as yet undiscovered (and challenging to discover) that pose risks as potentially hazardous asteroids (PHAs) to our planet.

There are no known Mercury co-orbitals. If any do exist, they will be exceedingly difficult to detect since they will always be in the glare of the Sun as seen from Earth.

Asteroids orbiting interior to Mercury’s orbit (a < 0.387 AU) would be called vulcanoids. I say “would be” because none have been discovered yet, though in all fairness, they will be extremely difficult to detect.

A spacecraft orbiting interior to Mercury’s orbit looking outward would be an ideal platform for detecting, inventorying, and characterizing all potentially hazardous asteroids (PHAs) that exist in the inner solar system. A surveillance telescope in a circular orbit 0.30 AU from the Sun would orbit the Sun every 60 days.

The Parker Solar Probe, scheduled to launch later this year, will orbit the Sun between 0.73 AU and an extraordinarily close 0.04 AU, though it will be looking towards the Sun, not away from it. The Near-Earth Object Camera (NEOCam) is a proposed mission to look specifically for PHAs using an infrared telescope from a vantage point at the Sun-Earth L₁ Lagrangian point.

References
de la Fuente Marcos, C., & de la Fuente Marcos, R. 2014, MNRAS, 439, 2970
de la Fuente Marcos, C., & de la Fuente Marcos, R. 2017, RNAAS, 1, 3
Sheppard, S., & Trujillo, C. 2009, Icarus, 202, 12

Separating Observer from Observed

One of the most difficult things to do in observational science is to separate the observer from the observed. For example, in CCD astronomy, we apply bias, dark, and flat-field corrections as well as utilize median combines of shifted images to yield an image that is, ideally, free of any CCD chip defects including differences in pixel sensitivity and zero-point.

We as observers are constrained by other limitations. For example, when we look at a particular galaxy, we observe it from a single vantage point in space and time, a vantage point we cannot change due to our great distance from the object and our existence within an exceedingly short interval of time.

Yet another limitation is a phenomenon that astronomers often call “observational selection”. Put simply, we are most likely to see what is easiest to see. For example, many of the exoplanets we have discovered thus far are “hot Jupiters”. Is this because massive planets that orbit very close to a star are common? Not necessarily. The radial velocity technique we use to detect many exoplanets is biased towards finding massive planets with short-period orbits because such planets cause the biggest radial velocity fluctuations in their parent star over the shortest period of time. Planets like the Earth with its relatively small mass and long orbital period (1 year) are much more difficult to detect using the radial velocity technique. The same holds true for the transit method. Planets orbiting close to a star will transit more often—and are more likely to transit—than comparable planets further out. Larger planets will exhibit a larger Δm than smaller planets, regardless of their location. It may be that Earthlike planets are much more prevalent than hot Jupiters, but we can’t really conclude that looking at the data collected so far (though Kepler has helped recently to make a stronger case for abundant terrestrial planets).

Here’s another important observational selection effect to consider in astronomy: the farther away a celestial object is the brighter that object must be for us to even see it. In other words, many far-away objects cannot be observed because they are too dim. This means that when we look at a given volume of space, intrinsically bright objects are over-represented. The average luminosity of objects seems to increase with increasing distance. This is called the Malmquist bias, named after the Swedish astronomer Gunnar Malmquist (1893-1982).