Just like geographical coordinates on the Earth each star has a celestial address.
● This address is impermanent because ➔ ➔ Stars move steadily as they randomly drift in the Galaxy The coordinate system (tied to the Earth) shifts as the Earth precesses like a top.
● Precession is slow (26,000 years/cycle) but even over a decade its effects are significant.
Coordinates are the analog of latitude and longitude, called Declination and Right Ascension respectively.
● Declination is straightfoward and is simply the angular distance a star lies above or below the celestial equator measured in degrees.
➔ ➔ The north celestial pole is at a declination of +90 degrees The declination of the bright star Vega is +38:47:01.9 (at least in the year 2000 it was – more on that later), so +dd:mm:ss.s in general.
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Right Ascension (longitude) is trickier ● If you point your finger at a particular Declination the declination value remains unchanged, but Right Ascension ticks away as the sky (actually the Earth) rotates.
● Right Ascension is thus naturally measured in units of time – hh:mm:ss.s
➔ ➔ ➔ One hour of right ascension is 15 degrees The sky rotates by at 15 arcseconds per second at the Equator Since lines of RA converge toward the pole – 1 minute of RA spans a different angle depending on Declination – a factor of cos(Dec) comes into play.
Right Ascension/Longitude needs an arbitrary zeropoint (Greenwich on Earth, the “First Point of Aries” on the sky).
This reference point is the intersection celestial equator and ecliptic at of the location of the Sun at the Spring Equinox.   3

The Sun rises and sets on a slightly different schedule than the stars.
● The difference arises from the changing perspective as the Earth orbits the Sun.
● While the Earth completes a rotation it moves 1/365th of the way around its orbit.
➔ It must turn for an extra 24 hours/365.25 (= about 4 minutes) to get the Sun back to “Noon” The Solar Day, by definition, is exactly 24.0000 hours long.
● The Sidereal Day – defining the rising and setting of the stars - is 3m 56s shorter and represents the true rotation period of the Earth.
A Sidereal clock keeps star time – it keeps 24 hour time, but completes a cycle in 23h 56m 4s of Solar time ● The time on a Sidereal clock equals the meridian of Right Ascension that is overhead at the moment.
● At Noon on the Spring Equinox R.A.=00:00:00.0 is overhead by definition.
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Every line of celestial longitude is a meridian of longitude, but we recognize the line of longitude, or simply the great circle line, running overhead as “THE” meridian. 5

The Hour Angle of a star is the time until (East) or since (West) it crosses or has crossed the meridian.
The Hour Angle is simply the Right Ascension of the star minus the current sidereal time.
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The complement of the altitude angle, the “zenith angle”, measures the angular separation of a star from the point overhead.
● A star that is rising or setting has a zenith angle of z=90.
● A star overhead has a zenith angle of zero.
The “airmass” of a star measures the number of atmospheric thicknesses a star's light is passing through on its way to the observer and equals the secant of the zenith angle.
● Since the atmosphere attenuates starlight, knowing the airmass is critical to precision stellar photometry.
* one atmosphere 7

Angle c is the altitude Angle b is the zenith angle, z Airmass = 1 cos( z) = sec(z) For a plane parallel atmosphere  (not what is pictured at left, but a  good approximation for small  zenith angles).
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airmass = sec(z ) = (( sin (lat)∗sin (dec)) + cos(lat )∗cos(dec)∗cos( H.A)) − 1 9 Airmass = 1 cos( z) = sec(z) For a plane parallel atmosphere  (not what is pictured at left, but a  good approximation for small  zenith angles).

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Because the Earth is a spinning top, the direction of its pole in the sky is fixed (at least over short timescales).
➔ Of course, this consistency of tilt is related to the cause of the seasons.
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Like any top under torque, however, the spin axis precesses slowly over time.
Because the celestial poles and equator define the RA/Dec “equatorial” coordinate system the “of-date” coordinates of a star change slightly from day to day and significantly over decades.
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Over long periods of time, the direction of the Earth's pole drifts in the same way it does for a child's top or gyroscope.
➔ ➔ ➔ The drifting of the axis of a top is called “precession” The rotation axis traces out a cone with an apex angle of twice the obliquity of the rotation axis – 47 degrees for the Earth.
The precession period of the Earth's rotation axis is 26,000 years.
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Mars completes a precession period once every 170,000 Earth years (93,000 Mars years). Why so slow?
Venus has a precession period that is difficult to measure (slow rotation, “tilted” only 3 degrees – actually 177 degrees as it rotates backwards), best estimate is ~29,000 years.
Saturn precession period varies around an average of a little less than 2 million years.
● ● Here torques from passing planets – Jupiter in particular – drive the precession more than the distant Sun.
Calculation of these effects is a graphic illustration of the field of gravitational dynamics – tracking the gravitational interplay of all objects in the solar system and the changing eccentricities and obliquities and precession rates – not just the influence of the Sun.
For all of these worlds this interplay of precession and elliptical orbits leads to climate effects (changing whether perihelion happens in Summer or in Spring for example).
● On Earth this effect is the primary driver of Milankovitch climate cycles, thought responsible for recent repeating ice ages.
Each planet name on this page is linked to a related technical paper 15

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Precession combined with  variation in the Earth's other  orbital parameters (eccentricity,  obliquity,...) lead to changes in net  insolation that can be as large as  20% at high latitudes.   It is  possible these changes are  connected to recent climate  swings (i.e. the ice ages).
Precession is a major player as it  affects the timing of when  Northern hemisphere summer  occurs relative to Earth's  perihelion.
Similar effects might be  responsible for climate cycles  revealed by the layered structure  of the  Martian  icecaps or by the  distribution of lakebeds on  Saturn's moon  Titan .
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It takes 26,000 years for the Earths pole to trace out a full circle on the sky.
➔ That circle is 47 degrees in diameter (2 x 23 ½) 19

It takes 26,000 years for the Earths pole to trace out a full circle on the sky.
➔ ➔ That circle is 47 degrees in diameter (2 x 23 ½) That point is the pole of the equatorial coordinate system. 20

Different stars occupy different positions above the Earth's pole over time.
➔ Polaris is currently getting further from the pole every year. Just how long will we hang on to it as our pole star???
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Different Stars are circumpolar at different times.
➔ 3000 years ago the Big Dipper was circumpolar at our latitude. It's not anymore.
Stars that currently never rise above our Southern horizon will be visible.
➔ The Southern Cross will be visible from Charlottesville in 10,000 years.
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Stellar celestial coordinates must be constantly updated to account for precession.
● Telescope control systems automatically precess coordinates so that the telescope correctly points to the “of date” position of the star given proper input of current date, R.A., Dec, and epoch of the coordinates.
Star catalogs must be tied to a particular “epoch”. Typically the default epoch changes every 50 years as even over this timescale the coordinate change can become significant.
Once again for the star Vega the coordinates are ● 18:36:56.3 +38:47:01.9 J2000.0 (J for Julian) ● 18:35:15.5 +38:44:24.7 B1950.0 (B for Besselian)   ➔ small differences, but large compared to many instrument fields-of-view.
Now in the computer age (and given the juicy J2000.0 round number epoch) it is likely that catalog coordinates will stick to J2000.0 for centuries to come.
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The orbit of the Earth's Moon can be thought of as a spinning top (it's all just angular momentum in the end). The Moon's orbit has an angular momentum vector perpendicular to the Moon's orbital plane. ● ● ● The Moon's orbit is tilted about 5 degrees from the Ecliptic. As a result the torque from the Sun causes the Moon's orbital plane/angular momentum to precess.
It takes the Moon's orbit 18.6 years to complete a precession cycle.
The line of intersection between the Earth's orbital plane and the Moon's orbit (known as the “line of nodes”) drifts along the ecliptic in analogy to the drift of the Spring equinox.
➔ Eclipse “seasons” drift by a bit less than a month a year as a result.
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The number of days in a year (Spring to the next Spring) is 365.2422
➔ Therein lies the problem.... a calendar can't have a fraction of a day.
Before looking at the solution(s), consider some subtle aspects of the definition of the year.
● The time it takes the Earth to complete an orbit around the Sun is 365.2564
days – a Sidereal Year

The number of days in a year (Spring to the next Spring) is 365.2422
➔ Therein lies the problem.... a calendar can't have a fraction of a day.
Before looking at the solution(s), consider some subtle aspects of the definition of the year.
● The time it takes the Earth to complete an orbit around the Sun is 365.2564
days – a Sidereal Year 27 Since the Earth's orbit precesses the intersection point between the ecliptic and celestial equator shifts year-to-year drifting all the way around the ecliptic in 26,000 years

The time it takes the Earth to complete an orbit around the Sun is 365.2564
days – a Sidereal Year During that year, however, precession moves the location of “Spring” a little ways along the orbit. ➔ ➔ ➔ The Tropical Year , the time from Spring to Spring, is 365.2422
days.
The crossing point on the celestial sphere between the ecliptic and celestial equator – the Vernal Equinox – shifts 1/26000 th of the way around the sky each year.
● ● This shift is the difference between the Tropical and Sidereal year.
The direction serves to make the Sidereal year shorter than the Tropical year.
Now the crossing point lies in the zodiac constellation of Pisces.
● ● Two thousand years ago it was in Aries (thus the “First Point of Aries” being Zero for R.A.) It will cross the formal constellation boundary into Aquarius around the year 2600 - beginning the astrological “Age of Aquarius”   28

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Each year the calendar misses accounting for the full year by ¼ of a day...almost: one year = 365.2422
days.
➔ After four years the calendar is running ahead by one full day – 4 x 0.2422 = 0.9688 – close enough to one to ...
Insert a “leap day” into the calendar every four years (roughly) and you can make up the difference.
30 What if you lived on a planet that has a year that is 397.10 days long?

If your name is Julius Caesar, quite a bit...
➔ ➔ ➔ In 46 B.C Caesar instituted the first formal calendar that included a leap year to keep it in sync.
Good idea... however, the “Julian” calendar included a leap year every four years without fail ● The calendar's average year was 365.25 days, not quite 365.2422
days.
● 0.0078 days per year doesn't sound like a lot, but it adds up.
By the 1500's the calendar was 11 days out of sync with the Seasons ● People began to notice....
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With the calendar increasingly out of sync with the seasons the Catholic church became concerned.
➔ Certain church holidays, Easter for example, are tied to astronomical events.
Pope Gregory XIII instituted a slightly revised calendar that was better matched to the Tropical Year – the Gregorian calendar.
● Skip leap year if it is a century year (1700, 1800, 1900), but not if that century year is divisible by 400.
● ➔ So the year 2000 was a once in 400 year special occasion This calendar, with 97 leap days ever 400 years, has an average year length of 365.2425 days compare with the 365.2422
day year ● It falls out of sync one day every 3300 years – easy to fix/adjust with an extra leap day every few thousand years.
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Not entirely   ● ● The seasonal shift was corrected by making the day after October 4, 1582 .... October 15, 1582.
➔ Landlords got to collect rent nearly 2 weeks early.... renters were not so happy.
The initiative came from the Catholic church in Rome ➔ ➔ The Protestants, for example, refused to adopt the new calendar.
It took 350 years before the world all agreed to the same calendar.
● The US (colonies) and England did not switch until 1752...
– George Washington was born on Feb 11 by his calendar, but his official birthday is Feb 22 ● 33

What people don't like is that 365 factors very poorly – 5 x 73 ➔ ➔ so there is no easy way to break up the year into constant sized months, weeks 7 is a pretty lousy number as well – although note that 7*52 = 364.
What constitutes “better”?
● ● ● No more leap years – every year is the same Every month the same length Every month starts on the same day of the week ➔ The 3 rd of the month, for example, is the same day of the week every month every year.... forever.
How can you accomplish this ➔ ➔ Days “outside” the calendar – they are just days, but not days of the week.
Fiddle with the length of the week – 5 or 6 day weeks help. ➔ 34

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Wait a while....
● As we will learn in detail later, the Moon's tug on the Earth is gradually slowing down Earth's rotation.
● ➔ ➔ The day gets about 1 second longer every 50 thousand years.
Over time, given longer days, fewer days will fit into a year.
● The year itself is not changing nearly as much, but that, too, is variable.
Specifically, in about 50 million years there will be exactly 360 days in a year and the calender will be quite simple.
➔ ➔ It is unlikely that a 7-day week would survive ● ● 360 = 2 * 2 * 2 * 3 * 3 * 5 Lots of options for weeks and months, but 7 isn't one of them.
Why the fascination with seven ?? – Moon + Sun + 5 planets – Sunday, Monday, Tuesday (mardi), Wednesday (mercredi), Thursday (jeudi), Friday (venredi), Saturday 36