Earliest Sunset, Latest Sunrise

Why does the Earliest Sunset come before the Winter Solstice and the Latest Sunrise after?


Why does the Earliest Sunrise come before the Summer Solstice and the Latest Sunset after?

Ever wonder? I have. And aside from some hand-wavy explanations, I’ve never been able to explain this very well. Here’s the best explanation I have seen yet, provided in the December 2007 issue of Sky & Telescope, p. 55:

You’d think the earliest sunset would come on the shortest day (or longest night) of the year, at the winter solstice. But in fact, the day-night cycle shifts back and forth a little with the seasons, due to the tilt of Earth’s axis and the ellipticity of Earth’s orbit. At the beginning of December, sunrise, midday, and sunset all happen a little earlier than they “should”, and in January they run a little late. So the earliest sunset ends up being two or three weeks before the solstice, and the latest sunrise is two or three weeks afterward. The exact dates depend on your latitude.

Continuing along that same line of thought…

At the beginning of June, sunrise, midday, and sunset all happen a little later than they “should” and in July they run a little earlier. So the earliest sunrise ends up being about a week before the solstice, and the latest sunset is about a week afterwards. The exact dates depend on your latitude.

I know, I know. You still have a question. “Why are the dates of earliest sunrise and latest sunset closer to the summer solstice than the dates of earliest sunset and latest sunrise to the winter solstice?” Good question. I think it has everything to do with the fact that the Earth is near aphelion at the time of the summer solstice, and thus moving most slowly in its orbit around the Sun (the Earth’s orbit is slightly elliptical and not circular). That means that the Sun is moving slowest against the background stars and thus the accumulated difference between the sidereal day and solar day is the smallest at that time of year. That means the spread of days between earliest sunrise and latest sunset is less. Conversely, at the winter solstice, Earth is near perihelion, and therefore it is moving most quickly in its orbit around the Sun. That means that the Sun is moving fastest against the background stars and thus the accumulated difference between the sidereal day and solar day is largest at that time of year. That means the spread of days between earliest sunset and latest sunrise is more.

Here in Dodgeville, Wisconsin, where the latitude is just shy of 43˚ N and the longitude is just a tad over 90˚ W, the earliest sunset this year is today, Tuesday, December 8, 2020, at 4:25:49 p.m.

Latest sunrise in 2021 will be on both Saturday, January 2 and Sunday, January 3 at 7:31:51 a.m.

Pause to consider that if we were on year-round daylight saving time, latest sunrise wouldn’t be until 8:31:51 a.m.

My preference would be to stay on standard time year-round, as Arizona does.

Comet Orbital Elements

The orbit of a comet can be defined with six numbers, called the orbital elements, and by entering these numbers into your favorite planetarium software, you can view the location of the comet at any given time reasonably near the epoch date. The epoch date is a particular date for which the orbital elements are calculated and therefore the most accurate around that time.

Different sets of six parameters can be used, but the most common are shown below. Example values are given for Comet Holmes (17P), which exhibited a remarkable outburst in October 2007, now almost 12 years ago.

Perihelion distance, q

This is the center-to-center distance from the comet to the Sun when the comet is at perihelion, its closest point to the Sun. For Comet Holmes, this is 2.05338 AU, well beyond the orbits of both the Earth and Mars.

Orbital eccentricity, e

This is a unitless number that is the measure of the amount of ellipticity an orbit has. For a circular orbit, e = 0. A parabolic orbit, e = 1. A hyperbolic orbit, e > 1. Many comets have highly elliptical orbits, often with e > 0.9. Short-period comets, such as Comet Holmes (17P), have more modest eccentricities. Comet Holmes has an orbital eccentricity of 0.432876. This means that at perihelion, Comet Holmes is 43.3% closer to the Sun than its midpoint distance, and at aphelion Comet Holmes is 43.3% further away from the Sun than its midpoint distance.

Date of perihelion, T

This is a date (converted to decimal Julian date) that the comet reached perihelion, or will next reach perihelion. For example, Comet Holmes reached perihelion on 2007 May 5.0284.

Inclination to the Ecliptic Plane, i

This is the angle made by the intersection of the plane of the comet’s orbit with the ecliptic, the plane of the Earth’s orbit. Comet Holmes has an inclination angle of 19.1143°.

Longitude of the ascending node, Ω

The intersection between the comet’s orbital plane and the Earth’s orbital plane forms a line, called the line of nodes. The places where this line intersects the comet’s orbit forms two points. One point defines the location where the comet crosses the ecliptic plane heading from south to north. This is called the ascending node. The other point defines the location where the comet crosses the ecliptic plane heading from north to south. This is called the descending node. 0° longitude is arbitrarily defined to be the direction of the vernal equinox, the point in the sky where the Sun in its apparent path relative to the background stars crosses the celestial equator heading north. The longitude of the ascending node (capital Omega, Ω) is the angle, measured eastward (in the direction of the Earth’s orbital motion) from the vernal equinox to the ascending node of the comet’s orbit. For Comet Holmes, that angle is 326.8532°.

Argument of perihelion, ω

The angle along the comet’s orbit in the direction of the comet’s motion between its perihelion point and its ascending node (relative to the ecliptic plane) is called the argument of perihelion (small omega, ω). For Comet Holmes, this angle is 24.352°.


If all the mass of the Sun and the comet were concentrated at a geometric point, and if they were the only two objects in the universe, these six orbital elements would be fixed for all time. But these two objects have physical size, and are affected by the gravitational pull of other objects in our solar system and beyond. Moreover, nongravitational forces can act on the comet’s nucleus, such as jets of material spewing out into space, exerting a tiny but non-negligible thrust on the comet, thus altering its orbit. Because of these effects, in practice it is a good idea to define a set of osculating orbital elements which will give the best positions for the comet around a particular date. These osculating orbital elements change gradually with time (due to gravitational perturbations and non-gravitational forces acting on the comet) and give the best approximation to the orbit at a given point in time. The further one strays from the epoch date for the osculating elements, the less accurate the predicted position of the comet will be.

For example, the IAU Minor Planet Center gives a set of orbital elements for Comet Holmes that has a more recent epoch date than the one given by the JPL Small-Body Database Browser. The MPC gives an epoch date of 2015 Jun 27.0, reasonably near the date of the most recent perihelion passage of this P = 6.89y comet (2014 Mar 27.5736). JPL, on the other hand, provides a default epoch date of 2010 Jan 17.0, nearer the date of the 2007 May 5.0284 perihelion and the spectacular October 2007 apparition. For the most accurate current position of Comet Holmes in your planetarium software, you’ll probably want to use the MPC orbital elements, since they are for an epoch nearest to the date when you’ll be making your observations.

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:

PlanetaeqQ
Mercury0.3870.206
Venus0.7230.007
Earth1.0000.017
Mars1.5240.093

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.

Planeta (AU)eq (AU)Q (AU)
Mercury0.3870.2060.3070.467
Venus0.7230.0070.7180.728
Earth1.0000.0170.9831.017
Mars1.5240.0931.3821.666

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 = 87.9691d = orbital period of Mercury

P2 = 224.701d = orbital period of Venus

P3 = 365.256d = orbital period of Earth

P4 = 686.971d = orbital period of Mars

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

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

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

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 0h UT 1 May 2019 through 0h UT 10 Jan 2045. Here are the results:

PlanetMean (AU)Median (AU)Min (AU)Max (AU)
Mercury1.0390221.0731480.5491441.451501
Venus1.1383831.2384530.2652601.735280
Mars1.7111761.8422600.3804202.675330

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.

Like Sun, Like Moon

The Earth orbits the Sun once every 365.256363 (mean solar) days relative to the distant stars.  The Earth’s orbital speed ranges from 18.2 miles per second at aphelion, around July 4th, to 18.8 miles per second at perihelion, around January 3rd.  In units we’re perhaps more familiar with, that’s 65,518 mph at aphelion and 67,741 mph at perihelion. That’s a difference of 2,223 miles per hour!

As we are on a spinning globe, the direction towards which the Earth is orbiting is different at different times of the day.  When the Sun crosses the celestial meridian, due south, at its highest point in the sky around noon (1:00 p.m. daylight time), the Earth is orbiting towards your right (west) as you are facing south. Since the Earth is orbiting towards the west, the Sun appears to move towards the east, relative to the background stars—if we could see them during the day.  Since there are 360° in a circle and the Earth orbits the Sun in 365.256363 days (therefore the Sun appears to go around the Earth once every 365.256363 days relative to the background stars), the Sun’s average angular velocity eastward relative to the background stars is 360°/365.256363 days = 0.9856° per day.

The constellations through which the Sun moves are called the zodiacal constellations, and historically the zodiac contained 12 constellations, the same number as the number of months in a year.  But Belgian astronomer Eugène Delporte (1882-1955) drew up the 88 constellation boundaries we use today, approved by the IAU in 1930, so now the Sun spends a few days each year in the non-zodiacal constellation Ophiuchus, the Serpent Bearer. Furthermore, because the Earth’s axis is precessing, the calendar dates during which the Sun is in a particular zodiacal constellation is gradually getting later.

Astrologically, each zodiacal constellation has a width of 30° (360° / 12 constellations = 30° per constellation).  But, of course, the constellations are different sizes and shapes, so astronomically the number of days the Sun spends in each constellation varies. Here is the situation at present.

Constellation

Description

Sun Travel Dates

Capricornus

Sea Goat

Jan 19 through Feb 16

Aquarius

Water Bearer

Feb 16 through Mar 12

Pisces

The Fish

Mar 12 through Apr 18

Aries

The Ram

Apr 18 through May 14

Taurus

The Bull

May 14 through Jun 21

Gemini

The Twins

Jun 21 through Jul 20

Cancer

The Crab

Jul 20 through Aug 10

Leo

The Lion

Aug 10 through Sep 16

Virgo

The Virgin

Sep 16 through Oct 31

Libra

The Scales

Oct 31 through Nov 23

Scorpius

The Scorpion

Nov 23 through Nov 29

Ophiuchus

Serpent Bearer

Nov 29 through Dec 18

Sagittarius

The Archer

Dec 18 through Jan 19

The apparent path the Sun takes across the sky relative to the background stars through these 13 constellations is called the ecliptic.  A little contemplation, aided perhaps by a drawing, will convince you that the ecliptic is also the plane of the Earth’s orbit around the Sun.  The Moon never strays very far from the ecliptic in our sky, since its orbital plane around the Earth is inclined at a modest angle of 5.16° relative to the Earth’s orbital plane around the Sun.  But, relative to the Earth’s equatorial plane, the inclination of the Moon’s orbit varies between 18.28° and 28.60° over 18.6 years as the line of intersection between the Moon’s orbital plane and the ecliptic plane precesses westward along the ecliptic due to the gravitational tug of war the Earth and the Sun exert on the Moon as it moves through space.  This steep inclination to the equatorial plane is very unusual for such a large moon.  In fact, all four satellites in our solar system that are larger than our Moon (Ganymede, Titan, Callisto, and Io) and the one that is slightly smaller (Europa) all orbit in a plane that is inclined less than 1/2° from the equatorial plane of their host planet (Jupiter and Saturn).

Since the Moon is never farther than 5.16° from the ecliptic, its apparent motion through our sky as it orbits the Earth mimics that of the Sun, only the Moon’s angular speed is over 13 times faster, completing its circuit of the sky every 27.321662 days, relative to the distant stars.  Thus the Moon moves a little over 13° eastward every day, or about 1/2° per hour.  Since the angular diameter of the Moon is also about 1/2°, we can easily remember that the Moon moves its own diameter eastward relative to the stars every hour.  Of course, superimposed on this motion is the 27-times-faster-yet motion of the Moon and stars westward as the Earth rotates towards the east.

Now, take a look at the following table and see how the Moon’s motion mimics that of the Sun throughout the month, and throughout the year.

 

——— Moon’s Phase and Path ———

Date

Sun’s Path

New

FQ

Full

LQ

Mar 20

EQ

EQ

High

EQ

Low

Jun 21

High

High

EQ

Low

EQ

Sep 22

EQ

EQ

Low

EQ

High

Dec 21

Low

Low

EQ

High

EQ

New = New Moon

near the Sun

FQ = First Quarter

90° east of the Sun

Full = Full Moon

180°, opposite the Sun

LQ = Last Quarter

90° west of the Sun

EQ

= crosses the celestial equator heading north

High

= rides high (north) across the sky

EQ

= crosses the celestial equator heading south

Low

= rides low (south) across the sky

So, if you aren’t already doing so, take note of how the Moon moves across the sky at different phases and times of the year.  For example, notice how the full moon (nearest the summer solstice) on June 27/28 rides low in the south across the sky.  You’ll note the entry for the “Jun 21” row and “Full” column is “Low”.  And, the Sun entry for that date is “High”.  See, it works!

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?