## 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.

## Thank the Sumerians

Over five thousand years ago, the Sumerians in the area now known as southern Iraq appear to have been the first to develop a penchant for the numbers 12, 24, 60 and 360.

It is easy to see why. 12 is the first number that is evenly divisible by six smaller numbers:

12 = 1×12, 2×6, 3×4 .

24 is the first number that is evenly divisible by eight smaller numbers:

24 = 1×24, 2×12, 3×8, 4×6 .

60 is the first number than is evenly divisible by twelve smaller numbers:

60 = 1×60, 2×30, 3×20, 4×15, 5×12, 6×10 .

And 360 is the first number that is evenly divisible by twenty-four smaller numbers:

360 = 1×360, 2×180, 3×120, 4×90, 5×72, 6×60, 8×45, 9×40, 10×36, 12×30, 15×24, 18×20 .

And 360 in a happy coincidence is just 1.4% short of the number of days in a year.

We have 12 hours in the morning, 12 hours in the evening.

We have 24 hours in a day.

We have 60 seconds in a minute, and 60 minutes in an hour.

We have 60 arcseconds in an arcminute, 60 arcminutes in a degree, and 360 degrees in a circle.

The current equatorial coordinates for the star Vega are

α2019.1 = 18h 37m 33s
δ2019.1 = +38° 47′ 58″

Due to precession, the right ascension (α) of Vega is currently increasing by 1s (one second of time) every 37 days, and its declination (δ) is currently decreasing by 1″ (one arcsecond) every 5 days.

With right ascension, the 360° in a circle is divided into 24 hours, therefore 1h is equal to (360°/24h) = 15°. Since there are 60 minutes in an hour and 60 seconds in a minute, and 60 arcminutes in a degree and 60 arcseconds in an arcminute, it follows that 1m = 15′ and 1s = 15″.

Increasingly, you will see right ascension and declination given in decimal, rather than sexagesimal, units. For Vega, currently, this would be

α2019.1 = 18.62583h
δ2019.1 = +38.7994°

Or, both in degrees

α2019.1 = 279.3875°
δ2019.1 = +38.7994°

Even though the latter three forms lend themselves well to computation, I still prefer the old sexagesimal form for “display” purposes, and when entering coordinates for “go to” at the telescope.

There is something aesthetically appealing about three sets of two-digit numbers, and, I think, this form is more easily remembered from one moment to the next.

For the same reason, we still use the sexagesimal form for timekeeping. For example, as I write this the current time is 12:25:14 a.m. which is a more attractive (and memorable) way to write the time than saying it is 12.4206 a.m. (unless you are doing computations).

That’s quite an achievement, developing something that is still in common use 5,000 years later.

Thank the Sumerians!

## 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!

## Pole Stars

Currently, Polaris (Alpha α UMi) shines at magnitude 2.0 and lies just 0.7° from the North Celestial Pole (NCP).  Precession of the Earth’s rotation axis will bring the NCP to within 0.5° of Polaris in March 2100, its minimum distance.

The situation for the South Celestial Pole (SCP) is not such a happy circumstance.  The nearest naked-eye star to the SCP at present is neither near nor bright.  Sigma Octantis at magnitude 5.5 is not easy to see with the unaided eye, and being 1.1 degrees away from the SCP doesn’t win it any awards.  Besides, precession is moving the SCP farther away from Sigma Oct, not nearer.

One wonders, will precession someday bring us a south celestial pole star worthy of the name?  Even, perhaps, comparable to Polaris?  Here’s what our stargazing descendants can look forward to:

So, around 8100 A.D. Iota Carinae and around 9220 A.D. Delta Velorum will serve admirably as southern pole stars every bit as good as Polaris does now in the northern hemisphere.

Now, for the northern hemisphere…

Up until the year 10,000 A.D., no northern pole star will be as good as Polaris is now, though 4.8-magnitude 9 Cephei will be very close to the north celestial pole around 7400 A.D.

Thought you might enjoy seeing what deep sky objects will come close to the celestial poles, so those are listed in the above tables as well.

## Epoch and Equinox

We use the term epoch (of a given date) to refer to the actual measured coordinates of a star that takes into account precession, nutation, and proper motion. The term equinox means that the coordinates have been precessed to a given date, but that other factors affecting a star’s position have not been applied. So, equinox 2000.0 is not the same as epoch 2000.0.

Example: Barnard’s Star

Epoch 2000.0 coordinates: α = 17h 57m 48.49803s, δ = +4° 41′ 36.2072″ (the actual position of Barnard’s Star at 0h UT on January 1, 2000, accounting for precession, nutation, and proper motion)

Equinox 2017.1 coordinates: α = 17h 58m 39.20689s, δ = +4° 41′ 33.5614″ (coordinates have been precessed from epoch 2000.0 above to today’s date, but nutation and proper motion have not been applied)

Epoch 2017.1 coordinates: α = 17h 58m 37.85s, δ = +4° 44′ 37.8″ (the actual position of Barnard’s Star on January 19, 2017, accounting for precession, nutation, and proper motion)

Sometimes, the epochal coordinates are further adjusted to account for aberration and atmospheric refraction.  The latter tends to “lift” stars towards the zenith—the closer to the horizon the greater the lift.

## Eugène Delporte and the Constellation Jigsaw

Belgian astronomer Eugène Joseph Delporte (1882-1955) discovered 66 asteroids from 1925 to 1942, but he is best remembered for determining the official boundaries of the 88 constellations, work he completed in 1928 and published in 1930.  The constellation boundaries have remained unchanged since then.

The International Astronomical Union (IAU), founded, incidentally, in Brussels, Belgium in 1919, established the number of constellations at 88—the same number, coincidentally, as the keys on a piano—in 1922 under the guidance of American astronomer Henry Norris Russell (1877-1957).  The IAU officially adopted Delporte’s constellation boundaries in 1928.

All the constellation boundaries lie along lines of constant right ascension and declination—as they existed in the year 1875. Why 1875 and not 1900, 1925, or 1930? American astronomer Benjamin Gould (1824-1896) had already drawn up southern constellation boundaries for epoch 1875, and Delporte built upon Gould’s earlier work.

As the direction of the Earth’s polar axis slowly changes due to precession, the constellation boundaries gradually tilt so that they no longer fall upon lines of constant right ascension and declination. Eventually, the tilt of the constellation boundaries will become large enough that the boundaries will probably be redefined to line up with the equatorial coordinate grid for some future epoch. When that happens, some borderline stars will move into an adjacent constellation. Even now, every year some stars change constellations because proper motion causes them to move across a constellation boundary. For faint stars, this happens frequently, but for bright stars such a constellation switch is exceedingly rare.