orbital eccentricity – Cosmic Reflections

Milutin Milanković

Serbian engineer, mathematician, and scientist Milutin Milanković was born 140 years ago on this date in 1879, in the village of Dalj on the border between Croatia and Serbia—then part of the empire of Austria-Hungary. He died in 1958 in Beograd (Belgrade), then in Yugoslavia and today in Serbia, at the age of 79.

Milanković is perhaps most famous for developing a mathematical theory of climate based on changes in the Earth’s orbit and axial orientation. There are three basic parameters that change with time—now known as the Milankovitch cycles—that affect the amount of solar energy the Earth receives and how it is distributed upon the Earth.

I. Orbital eccentricity of the Earth changes with time

The eccentricity (e) tells you how elliptical an orbit is. An eccentricity of 0.000 means the orbit is perfectly circular. A typical comet’s orbit, on the other hand, is very elongated, with an eccentricity of 0.999 not at all uncommon. Right now, the Earth’s orbital eccentricity is 0.017, which means that it is 1.7% closer to the Sun at perihelion than its semimajor axis distance (a), and 1.7% further from the Sun at aphelion than its semimajor axis distance.

The greater the eccentricity the greater the variation in the amount of solar radiation the Earth receives throughout the year. Over a period of roughly 100,000 years, the Earth’s orbital eccentricity changes from close to circular (e = 0.000055) to about e = 0.0679 and back to circular again. At present, the Earth’s orbital eccentricity is 0.017 and decreasing. We now know the Earth’s orbital eccentricity changes with periods of 413,000, 95,000, and 125,000 years, making for a slightly more complicated variation than a simple sinusoid, as shown below.

II. Tilt of the Earth’s axis changes with time

The tilt of the Earth’s polar axis with respect to the plane of the Earth’s orbit around the Sun—called the obliquity to the ecliptic—changes with time. The Earth’s current axial tilt is 23.4°, but it ranges between about 22.1° and 24.5° over a period of about 41,000 years. Greater axial tilt means winter and summer become more extreme. Presently, the axial tilt is decreasing, and will reach a minimum around 11,800 A.D.

III. Orientation of the Earth’s axis changes with time

The Earth’s axis precesses or “wobbles” with a period of around 26,000 years about the north and south ecliptic poles. This changes what latitude of the Earth is most directly facing the Sun when the Earth is closest to the Sun each year. Currently, the southern hemisphere has summer when the Earth is at perihelion.

Milanković used these three cycles to predict climate change. His ideas were largely ignored until 1976, when a paper by James Hays, John Imbrie, and Nicholas Shackleton in the journal Science showed that Milanković’s mathematical model of climate change was able to predict major changes in climate that have occurred during the past 450,000 years.

These Milankovitch cycles are important to our understanding of climate change over much longer periods than the climate change currently being induced by human activity. Note the extremely rapid increase of greenhouse gas concentrations (CO₂, CH₄, and N₂O) in our atmosphere over the past few decades in the graphs below.

https://www.noaa.gov/news-release/greenhouse-gases-continued-to-increase-rapidly-in-2022

The world population has increased by 93% since 1975. In 1975, it was about 4 billion and by 2020 it is expected to be 7.8 billion.

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.

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?