Space and Time

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ephemeris - Latin, originally from the Greek "ephémeros, -on," daily. An almanac of the daily motions of the planets and stars.
ephemeris.com - A website devoted to information about time and motion in the universe.

Space and Time

An ephemeris charts the movement of celestial bodies (planets, moons, etc.) and predicts their positions at a given time. For a brief discussion of the history of ephemeris theories, see the Modern Theories section of the History page on this site.

Measures of Time

The fundamental unit of time is the SI second (see next section). Ordinarily we think of the second as defined by the length of a day, with 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute. However, we now know that the length of a day is gradually getting shorter. Because astronomers need a precise definition of the second that never changes, they now use atomic time (see SI Second, below). We can still use the ordinary definition for skywatching, where there are 86,400 seconds in a day.

SI Second
The Earth's rotation is not constant. It is affected by the gravitational pull of the Moon, Sun, and planets. This accelerates and decelerates the Earth's rotation sligtly over time. In the early 1900s, scientists found that the cesium atom had a very stable resonance. They built atomic clocks that detected transitions in these atoms. The SI second (defined in 1967) is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of Cesium 133. The SI day is 86,400 SE seconds, and the Julian year is 365.25 SI days. (See Julian Day below.)

Solar Day
The solar day is the time it takes the Sun to return to the same position in the sky. This is 24 hours.

Sidereal Day
If the Earth rotated exactly one time on its axis over a year, the same side of the Earth would always face the Sun. The same side of the Earth would always be in daylight, and the other side in darkness. Because of this, the Earth rotates one more time than it seems to rotate during a year: 366.24 times (not 365.24 times). A sidereal day is the time it would take for the stars to reappear in the same position, so it is a convenient measure of time when calculating stellar positions. The sidereal day is (365.24 / 366.24) times the length of a solar day of 24 hours. This is about 0.99727 days, which is about 23 hours and 56 minutes. Many observatories will have a clock that shows the local sidereal time. To find out why, see the section on Hour Angle below.

Julian Day
Astronomers make observations that span centuries. They need an easy method of finding the time between two dates regardless of leap days in a year, changes in how the calendar is calculated, etc. In 1583, Joseph Justus Scaliger proposed counting years in a continuous progression from 4713 B.C. onwards. This avoided the jump from 1 B.C. to 1 A.D. In the 1800s, the astronomer Herschel adapted this system to counting days. He defined Julian Day 0 as starting at noon on Monday, January 1, 4713 B.C. (Astronomers make their observations at night, so it was convenient for them to begin astronomical days at noon.)

In Herschel's time, the length of the tropical day was the constant of measure. Today, the standard unit of time is the SI second. A Julian day is 86,400 SI seconds.

Besselian Year
This measure was used 100 years ago, and referenced the "Besselian" tropical year. Besselian epochs are written staring with a 'B', for example, B1950.0. This began at January 0.923, 1950, which is Julian Date 2433282.423. Many older ephemerides and star catalogs were referenced to B1950, notably the Smithsonian Astronomical Observatory Star Catalog.

Julian Century
A Julian Century always has exactly 36525 Julian days. It is a costant measure of time, but does not reflect the solar calendar.

Julian Epoch
The first Julian Epoch noted was J1900.0, which is Julian Day 2415020.0. This corresponded to January 0.5, 1900 (midnight between 31 December 1899 and 1 January 1900). The year 1900 was not a leap year under the Gregorian Calendar, so the next Julian Century, J2000.0, is Julian Day 2451545.0 (2415020 + 36525). This is also written as January 1.5, 2000 (that is, noon on 1 January 2000).

Modified Julian Day
Rather than writing the entire Julian day, recent days can be written as follows. As an example, take the Julian Epoch J2000.0, which is JD 2451545.0. For recent days, we omit the "24" in the beginning. Because we are starting our days at midnight by convention, we also omit the ".0" at the end. We can write the J2000.0 epoch as a Modified Julian Day of 51545 for convenience.

Measurements of Space

Ephemerides commonly give positions in Right Ascension and Declination. For the stars, these values remain fairly constant. For planets, they change quickly. An observer at a Latitude and Longitude on the Earth's surface can use a compass to find the Azimuth of a celestial object, and a sextant or similar instrument to find its Elevation (Altitude).

Latitude and Longitude
Latitude is the number of degrees above or below the equator. In the northern hemisphere, latitude ranges from 0 to 90 degrees North. In the southern hemisphere, latitude ranges from 0 to 90 degrees South. The Equator has a latitude of exactly 0 degrees. The North Pole has a latitude of 90 degrees North. The South Pole has a latitude of 90 degrees South.

Longitude is the number of degrees east or west from the Prime Meridian (0 degrees longitude), at Greenwich, England. West of Greenwich, it ranges from 0 to 180 degrees West. East of Greenwich, it ranges from 0 to 180 degrees East. The International Date Line is at 180 degrees longitude (East or West).

Right Ascension and Declination
Right Ascension and Declination are positions of a celestial object as might be seen from the center of the Earth. These positions are common in almanacs, such as the U.S. Naval Observatory's The Astronomical Almanac. From this point in the center of the Earth, positions can be calculated for any place on the Earth's surface. Right Ascension is measured in hours, with 24 hours representing 360 degrees. Right Ascension is measured eastward from the First Point of Aries (0 degrees Aries, the Vernal Equinox). Declination is the elevation above or below the Equator (as seen from the center of the Earth). Declination ranges from 90 degrees North to 90 degrees South.

Azimuth and Elevation
Azimuth and Elevation are positions of a celestial object seen by an observer at one particular place on the Earth's surface. Typically you calculate the Azimuth and Elevation for your particular location from a celestial object's Right Ascension and Declination (see below). The Azimuth of a celestial object is the compass degrees above or below which it lies. Azimuth begins at 0 degrees for North, and continues clockwise to 360 degrees for North again, the same as the degrees on a compass. The Azimuth tells you what direction to turn in to look, and Elevation tells you how many degrees (-90 to +90) to look down or up from the horizon. Note: Sometimes formulas start Azimuth at the South. The formulas below start Azimuth at the North, and count degrees in a clockwise direction, just like a compass.

Hour Angle
We saw above that a sidereal day only lasts about 23 hours 56 minutes. The Local Hour Angle of the Vernal Equinox is the same as the local sidereal time. Whereas Right Ascension is measured eastward, the Sidereal Hour Angle is measured westward. The Hour Angle of a celestial object is the same as the Local Sidereal Time minus the Right Ascension of the object.

Other Effects

Obliquity of the Ecliptic
The Earth is slightly tilted compared to the plane in which it orbits the Sun. This tilt is called the Obliquity of the Ecliptic, and is defined for the epoch J2000.0 as being 23.4392911 degrees (approximately 23.5 degrees).

Precession of the Equinoxes
The Earth rotates around its axis in the course of a day. However, this axis is not fixed in one place. The axis itself is rotating, and makes a complete turn in about 26,000 years. While this might seem like a long time, it is (360 times 60 times 60) arcseconds / 26,000 years = about 50 arcseconds per year. Because the Earth's axis is rotating, our Pole Star (the star closest to North in the Northern Hemisphere) changes over time. Right now, our Pole Star is Polaris. Over time, that will change. Right now, the Earth is closest to the Sun (at perihelion) in January and furthest (at perihelion) in July. In 13,000 years, the Earth will be closest to the Sun in January and furthest in July. Several ancient civilizations knew of the Precession of the Equinoxes from careful observations of the stars.

The Earth also wobbles slightly around its axis. This is caused by the Moon and the Sun pulling on the Earth's "equatorial bulge" (the Earth bulges slightly, sort of like a beach ball that has pressure above and below it). This is a periodic wobble, and is known as nutation. The effects of nutation are slight (it wasn't discovered until the 1700s), but noticeable.

Using an Ephemeris (Almanac) to Find a Celestial Object

General Observing
These formulas are low-accuracy formulas to find the Azimuth and Elevation of an object, given its Right Ascension and Declination, and the observer's Latitude. They do not take into account the effects of Precession and Nutation. However, they are accurate enough for general observations.

  1. Determine the Greenwich Sidereal Time (GST).
    1. At midnight on January 1, 2004, the Greenwich Sidereal Time (Greenwich Hour Angle of the Equinox) was 06:39:58.
    2. For each additional day, subtract 00:03:56. (that is, 236 seconds) as a reasonable approximation.
    3. Add the time from midnight of your local time, minus 10 seconds for each hour (240 seconds for each 24 hours). This is the Greenwich Sidereal Time.
  2. Convert Greenwich Sidereal Time to Local Sidereal Time (LST).
    1. Convert your Longitude into a decimal number. West longitudes are negative, and East longitudes are positive. For example, 114 degrees 30 minutes West Longitude would be -114.5 (change minutes and seconds into a decimal).
    2. Divide the result by 15, keeping five decimal digits to keep tenth of second accuracy (we'll round off when we're done). In our example, -114.5 / 15 = -7.63333. This converts degrees of Longitude into hours. The Earth revolves 360 degrees in Longitude ever 24 hours, which is 360 degrees / 24 hours = 15 degrees/hour.
    3. Convert the hour offset into hours, minutes, and seconds. In our example, our local offset is 7 hours 38 minutes and 00 seconds earlier than Greenwich, so we'll subtract this from the Greenwich Sidereal Time. If we had been in an East longitude, we would be ahead of Greenwich and would add the offset.
    4. Subtract (for West longitude) or Add (for East longitude) the local offset from Greenwich Sidereal Time.
  3. Convert Right Ascension and Declination to Azimuth and Elevation at a Given Latitude and Sidereal Time.
    1. Let H = Object's Hour Angle, α = Object's Right Ascension, δ = Object's Declination, φ = Observer's Latitude, λ = Observer's Longitude, A = Object's Azimuth, e = Object's Elevation (Altitude).
    2. Look up the object's Right Ascension and Declination in an ephemeris.
    3. Compute H = LST - α
    4. Compute A = arctan( -(sin(H) cos(δ)) / (cos(φ) sin(δ) - sin(φ) cos(δ) cos(H)))
    5. Compute e = arcsin( sin(φ) sin(δ) + cos(φ) cos(δ) cos(H))
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