Average Orbital Distance

If a planet is orbiting the Sun with a semi-major axis, a, and orbital eccentricity, e, it is often stated that the average distance of the planet from the Sun is simply a.  This is only true for circular orbits (e = 0) where the planet maintains a constant distance from the Sun, and that distance is a.

Let’s imagine a hypothetical planet much like the Earth that has a perfectly circular orbit around the Sun with a = 1.0 AU and e = 0.  It is easy to see in this case that at all times, the planet will be exactly 1.0 AU from the Sun.

If, however, the planet orbits the Sun in an elliptical orbit at a = 1 AU and e > 0, we find that the planet orbits more slowly when it is farther from Sun than when it is nearer the Sun.  So, you’d expect to see the time-averaged average distance to be greater than 1.0 AU.  This is indeed the case.

The Earth’s current osculating orbital elements give us:

a = 0.999998 and e = 0.016694

Earth’s average distance from the Sun is thus:

Mercury, the innermost planet, has the most eccentric orbit of all the major planets:

a = 0.387098 and e = 0.205638

Mercury’s average distance from the Sun is thus:

Changing Solar Distance

Between January 2 and 5 each year, the Earth reaches orbital perihelion, its closest distance to the Sun (0.983 AU).  Between July 3 and 6 each year, the Earth reaches orbital aphelion, its farthest distance from the Sun (1.017 AU).  These dates of perihelion and aphelion slowly shift across the calendar (always a half year apart) with a period between 22,000 and 26,000 years.

These distances can be easily derived knowing the semi-major axis (a) and orbital eccentricity (e) of the Earth’s orbit around the Sun, which are 1.000 and 0.017, respectively.

perihelion
q = a (1-e) = 1.000 (1-0.017) = 0.983 AU

aphelion
Q = a (1+e) = 1.000 (1+0.017) = 1.017 AU

So, the Earth is 0.034 AU closer to the Sun in early January than it is in early July.  This is about 5 million km or 3.1 million miles.

How great a distance is this, really?  The Moon in its orbit around the Earth is closer to the Sun around New Moon than it is around Full Moon.  Currently, this difference in distance ranges between 130,592 miles (April 2018) and 923,177 miles (October 2018).  Using the latter value, we see that the Moon’s maximum monthly range in distance from the Sun is 30% of the Earth’s range in distance from the Sun between perihelion and aphelion.

How about in terms of the diameter of the Sun?  The Sun’s diameter is 864,526 miles.  The Earth is just 3.6 Sun diameters closer to the Sun at perihelion than it is at aphelion.  Not much!  On average, the Earth is about 108 solar diameters distant from the Sun.

How about in terms of angular size?  When the Earth is at perihelion, the Sun exhibits an angular size of 29.7 arcminutes.  At aphelion, that angle is 28.7 arcminutes.

Can you see the difference?

Stars Like Our Sun

The spectral type of our Sun is G2V, that is to say, a G2 main-sequence star.

Sun
Zodiacal Constellations
mv = -26.75, mb = -26.10, B-V = 0.65
Ecliptic
0.0000158 ly
Single star

Here are the brightest stars visible in the nighttime sky that have the same spectral type and therefore are, arguably, most like our Sun.  All have an apparent visual magnitude brighter than +6.00.

Rigil Kentaurus A, Alpha Centauri A (α Cen A)
Centaurus
mv = 0.01, mb = 0.72, B-V = 0.71
α2000 = 14h 39m 36s, δ2000 = -60° 50′ 02″
4.30 – 4.34 ly
Triple star system

Alula Australis B, Xi Ursae Majoris B (ξ UMa B)
Ursa Major
mv = 4.73, mb = 5.38, B-V = 0.65
α2000 = 11h 18m 11s, δ2000 = +31° 31′ 46″
28 – 30 ly
Quintuple star system

HR 4523 A
Centaurus
mv = 4.88, mb = 5.55, B-V = 0.67
α2000 = 11h 46m 31s, δ2000 = -40° 30′ 01″
30.0 – 30.1 ly
Binary star system; exoplanet

Eta Coronae Borealis A & B (η CrB A & B)
Corona Borealis
A: mv = 5.577, mb = 6.123, B-V = 0.546
B: mv = 5.95, mb = 6.48, B-V = 0.53
α2000 = 15h 23m 12s, δ2000 = +30° 17′ 18″
57 – 59 ly
Triple star system

HR 8323
Grus
mv = 5.58, mb = 6.18, B-V = 0.60
α2000 = 21h 48m 16s, δ2000 = -47° 18′ 13″
51.9 – 52.5 ly
Single star

Mu Velorum B (μ Vel B)
Vela
mv = 5.59, mb = 6.10, B-V = 0.51
α2000 = 10h 46m 46s, δ2000 = -49° 25′ 12″
116 – 119 ly
Binary star system

HR 7845 A
Capricornus
mv = 5.65, mb = 6.34, B-V = 0.69
α2000 = 20h 32m 24s, δ2000 = -09° 51′ 12″
79 – 80 ly
Binary star system

HR 3578
Hydra
mv = 5.86, mb = 6.39, B-V = 0.53
α2000 = 8h 58m 44s, δ2000 = -16° 07′ 58″
68 – 69 ly
Single star

HR 2007
Orion
mv = 5.97, mb = 6.61, B-V = 0.64
α2000 = 5h 48m 35s, δ2000 = -4° 05′ 41″
49.2 – 49.8 ly
Single star with exoplanet

The Eta Coronae Borealis system is noteworthy in that its two primary components are both G2V stars orbiting each other every 41.6 years.  The third component of this system is a distant infrared dwarf, spectral type L8V.

Two of these G2V stars host at least one exoplanet: HR 4523A in Centaurus and HR 2007 in Orion.

HR 4523A has a planet midway in mass between Uranus and Neptune orbiting every 122 days between 0.30 and 0.62 AU from the star (similar to orbital distance of the planet Mercury in our own solar system).  The other stellar component of this system. HR 4523B, is a distant M4V star currently orbiting about 211 AU from HR 4523A.

HR 2007, a single star like the Sun, has a planet about 78% more massive than Neptune, orbiting every 407 days, more or less.  If this planet were in our own solar system, it would range between the orbits of Venus and Mars, roughly.

Metallicity

No, it’s not the name of a rock band. Astronomers (unlike everybody else) consider all elements besides hydrogen and helium to be metals. For example, our Sun has a metallicity of at least 2% by mass (Vagnozzi 2016). That means as much as 98% of the mass of the Sun is hydrogen (~73%) and helium (~25%), with 2% being everything else.

Traditionally, elemental abundances in the Sun have been measured using spectroscopy of the Sun’s photosphere.  In principle, stronger spectral lines (usually absorption) of an element indicate a greater abundance of that element, but deriving the correct proportions from the cacophony of spectral lines is challenging.

A more direct approach to measuring the Sun’s elemental abundances is analyzing the composition of the solar wind, though the material blown away from the surface of the Sun that we measure near Earth’s orbit may be somewhat different from the actual photospheric composition.  The solar wind appears to best reflect the composition of the Sun’s photosphere in the solar polar regions near solar minimum.  The Ulysses spacecraft made solar wind measurements above both the Sun’s north and south polar regions during the 1994-1995 solar minimum.  Analysis of these Ulysses data indicate the most abundant elements are (after hydrogen and helium, in order of abundance): oxygen, carbon, nitrogen, magnesium, silicon, neon, iron, and sulfur—though one analysis of the data shows that neon is the third most abundant element (after carbon).

The elephant in the room is, of course, are the photospheric abundances we measure using spectroscopy or the collection of solar wind particles indicative of the Sun’s composition as a whole?  As it turns out, we do have ways to probe the interior of the Sun.  Both helioseismology and the flux of neutrinos emanating from the Sun are sensitive to metal abundances within the Sun.  Helioseismology is the study of the propagation of acoustic pressure waves (p-waves) within the Sun.  Neutrino flux is devilishly hard to measure since neutrinos so seldom interact with the matter in our instruments.  Our studies of the interior of the Sun (except for sophisticated computer models) are still in their infancy.

You might imagine that if measuring the metallicity of the Sun in our own front yard is this difficult, then measuring it for other stars presents an even more formidable challenge.

In practice, metallicity is usually expressed as the abundance of iron relative to hydrogen.  Even though iron is only the seventh most abundant metal (in the Sun, at least), it has 26 electrons, leading to the formation of many spectral lines corresponding to the various ionization states within a wide range of temperature and pressure regimes.  Of the metals having a higher abundance than iron, silicon has the largest number of electrons, only 14, and it does not form nearly as many spectral lines in the visible part of the spectrum as does iron.  Thus defined, the metallicity of the Sun [Fe/H] = 0.00 by definition.  It is a logarithmic scale: [Fe/H] = -1.0 indicates an abundance of iron relative to hydrogen just 1/10 that of the Sun.  [Fe/H] = +1.0 indicates an abundance of iron relative to hydrogen 10 times that of the Sun.

The relationship between stellar metallicity and the existence and nature of exoplanets is an active topic of research.  It is complicated by the fact that we can never say for certain that a star does not have planets, since our observational techniques are strongly biased towards detecting planets with an orbital plane near our line of sight to the star.

References
Vagnozzi, S. 2016, 51st Recontres de Moriond, Cosmology, At La Thuile