Numerical Astronomy

Time and Place

text and diagrams by Scott Robert Ladd

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"Time is an illusion -- lunch time, doubly so."
Douglas Adams (as Ford Prefect) in Hitchhiker's Guide to the Galaxy

Astronomy is a science of time and place. Time measures motion, and we base our clocks and calendars on the apparent movements of celestial bodies. We ask when the sun will rise tomorrow, or how high Arcturus will rise tonight, or when the next full Moon will occur. Questions of place invariably involve time, and vice versa -- for as Einstein showed us, time and space are mutually relative. And so, to understand the movements in the heavens, we need to know where we are and what time it is.

Latitude & Longitude

Our view of the sky is relative to our position on the spherical Earth. Identifying an Earth-based location begins at the equator, a plane that bisects the planet into northern and southern hemispheres. The equator is a great circle, meaning that its plane passes through the center of the Earth; the equator is also perpendicular to a line running through the north and south poles. Figure 1.1 shows the plane of the equator on a globe.

Figure 1.1 - The Great Circle of the Equator

Just two angular values can specify any location on the Earth's surface. The latitude is a measure of distance north or south of the equator, figured as the angle between the plane of the equator and a line drawn from a point to the center of the Earth. The equator is defined as latitude 0°; at the North Pole, the latitude is +90°, while the South Pole is at latitude -90°. In Figure 1.2, the latitude of point P is the angle θ.

Figure 1.2 Spherical Coordinates

You'll notice a dotted line running through P from north to south. This second position angle, identified by the Greek letter ψ, is the longitude, a measurement of distance along the equator from the 0°, or prime meridian, line. For consistency's sake, the universal convention places the prime meridian on a line running from the north to south poles through Greenwich, England. In most calculations -- including those in this book -- a longitude west of Greenwich is positive and an east longitude is negative. Longitude is measured on half circles, with 180° in both the east and west directions; the common 180° line runs through the center of the Pacific Ocean, and is known as the International Date Line. It is one day earlier east of the date line than it is to the west -- and I'll explain that rule when I look at time below.

The prime meridian could be any longitude, but consistency in navigation and maps requires an agreed-upon standard. But why is the zero mark set in Greenwich, and not, say, New York City or Tokyo? Beginning in the 17th century, England began a program to develop a system of navigation based on latitudes and longitudes; to that end, the English established a Royal Observatory at Greenwich for studying the positions of the moon and stars. When the final system of longitude was adopted in 1884, England's dominance at sea gave it the power to set the base meridian in Greenwich. I think it is quite remarkable that, in a world of varying cultures and political contentions, the system of latitude and longitude is universal.

Figure 1.3 shows a globe marked with longitude and latitude lines every 15°. To demonstrate the coordinate system, I've plotted the location of two cities: Buenos Aires, Argentine (at 58° 30'W, 34° 35'S) and Moscow, Russia (at 37° 35'E, 55° 45'N). You can see the reason why another name for latitude is parallel -- for all lines of latitude parallel each other in circles around the globe.

Figure 1.2.3 Latitude and Longitude

If you know the latitude and longitude of two places, you can find the distance between them. First, calculate the angle between the two points, using the following formula from spherical geometry:

Equation 1.1

Using the example cities of Buenos Aires and Moscow, from Figure 1.3, we get:

Once you know the angle between two points, it's a simple matter to use the following formula to calculate the distance between them.

Equation 1.2

The radius of the Earth, r, is approximately 6370 kilometers; setting A to 121.2° gives the distance between Buenos Aires and Moscow as 13,476 kilometers.

The Flattened Earth

The Earth is not a sphere; the equator bulges outward due to the planet's rotation. You can see this affect clearly when looking at the gaseous planets Jupiter and Saturn; they have a distinctly "squashed" look to them. In geometric terms, the Earth is a geoid, not a sphere, and its polar and equatorial radii differ slightly. According to the International Astronomical Union (IAU):

     polar radius rp = 6,378,140 meters
equatorial radius re = 6,356,755 meters

The difference isn't much, but it does affect our view of the stars by changing our latitude slightly. Most maps show geodetic latitudes, based on lines perpendicular to a tangent plane on the Earth's surface. Another type of type of latitude, geographic, is calculated as the angle of a plumb line with the equator; both geodetic and geographic latitudes measure similar angles. If the Earth were a sphere, a plumb line would point toward the center of the planet -- but the flattening of the planet makes this true only at the equator and poles. So neither geodetic nor geographic latitude is accurate from the perspective of astronomy, which calculates coordinates from the center of the Earth. For tracking the stars, we need to calculate a geocentric latitude: a line drawn from the point of observation to the center of the Earth.

Figure 1.4 - Geodetic (f) and Geocentric (f') Latitude

Figure 1.4 shows the relationship of geodetic (φ) and geocentric (φ') latitude. The difference is defined by the formula:

Equation 1.3

A geocentric latitude is slightly smaller then the equivalent geodetic latitude, except at the poles and equator. The greatest difference is about 11.55' at a geodetic latitude of 45° North or South.

Astronomical calculations give celestial coordinates in terms of the Earth's center; if we want to be truly accurate in plotting the stars, it is best to use the geocentric latitude of our observing position. My observing site is located at 37° 48' North latitude; I can calculate my geocentric latitude φ from Equation 1.3:

The impetus for developing latitude and longitude came from mariners, who couldn't use landmarks for navigation on the open seas. By studying the movements of the stars, sailors could locate themselves and tell time -- and now, it's to time that we turn our attention.

Time

Time "begins" at 0° longitude in Greenwich; locations west of the prime meridian have earlier times on their clocks, and locations east have later times. It takes the Earth 24 hours to rotate 360° on its axis, meaning that the sun traverses 15° of longitude for every hour of the day. Many calculations in this book refer to a longitudinal angle in terms of hours, with each 15° of longitude equal to one hour of angle.

Time steadily moves from day to night without much consideration for the requirements of human society. Consider, for example, an airplane leaving Lisbon, Portugal, at noon on a Monday. Lisbon is at about 39° north latitude; if the plane flies West at about 1300 km/hour, it will keep pace with the rotation of the Earth. Four-and-a-half hours later, when the plane flies over Washington, DC, the sun will still be at noon -- and noon it remain, from the aircraft's perspective, until the plane completely circles the globe and lands in Lisbon again. For the aircraft's passengers, it is still noon on Monday; to their friends and relatives in Lisbon, a night has passed and it is now Tuesday.

To accurately follow the apparent movement of the sun, your clock should be set based on your longitude west of Greenwich. If a clock in Greenwich reads noon, my watch should, according to strict solar time, read 4:50 AM, since my latitude of 107° 40' places me 7 hours and 10 minutes "behind" Greenwich. Someone in Moscow would be see the sun rise two-and-a-half hours before dawn appears in Greenwich. But setting time according to the movement of the sun is inconvenient and confusing; for example, if I traveled 65 kilometers east, I would need to set my watch ahead three minutes to match the sun. In a world of deadlines and schedules, it simply isn't practical to depict precise solar time on our clocks -- yet we need to account for differences in time between various places around the globe.

The compromise is to create time zones based on the average hourly position of the sun. Since the sun's position changes by 15° of longitude each hour, each time zone encompasses approximately 15° of longitude. Thus, my longitude of 107° 40' falls within the seventh time zone west of Greenwich; when it is noon in Greenwich, my watch reads 5:00 AM. But longitude alone cannot always determine your time zone, which has an irregular border based on political boundaries. And we can't forget the affects of Daylight Savings Time -- Benjamin Franklin's invention that adjusts clock time to reflect the longer daylight hours of summer.

What does all of this mean to an astronomer? First and foremost, it means that clock time has little or no relationship to the motions of the stars or even the sun. A hallmark of science is its reliance on standard frames of reference; an astronomer in Australia must use conventions held in common with a colleague in Arizona. So astronomers define their work in terms of Universal Time, or UT, which is the actual time in Greenwich of an event. The local clock time can be converted to UT through simple additions or subtracts for time zones and Daylight Savings Time. For example, when a clock in my time zone reads 2:00PM, the Universal Time is seven (six during Daylight Savings Time) hours later, or 9:00PM.

Scientists have added other time systems, too. International Atomic Time (TAI) is calculated from changes in the energy states of cesium atoms, allowing scientists to track time without relying on the inherently unstable movements of the Earth. The Terrestrial Dynamical Time, or TDT, is a replacement for UT in high-precision astronomical calculations. TDT is 32.184 seconds greater than TAI. Another value derived from TAI, the Coordinated Universal Time (UTC) is the basis for the call-up "atomic" clocks sponsored by observatories and government agencies. UTC never differs from TAI by more than a few integral seconds; in general, UT and UTC can be considered the same.

UT is the most applicable value for your observations, unless you'll be timing a pulsar or guiding the Hubble Space Telescope. Most references give the times for astronomical events in UT, and you need only include adjustments for time zone and daylight savings time at the observation site. If an event on 3 August 1997 occurs at 4:15UT, I will look for it six hours earlier, at 22:15, Mountain Daylight Time, on 2 August 1997.

Several caveats apply when comparing our calendar to the planet's movements. Figure 1.5 schematically shows the movement of the Earth around the sun for the year 1996. You'll note that the year begins three days before the planet arrives at perihelion, its closest approach to the sun. And the solstices and equinoxes correspond only somewhat with the Earth's position in orbit; the winter solstice, for example, is ten days before January first and two weeks before perihelion.

Figure 1.5 - The Earth's Orbit in 1996

The Earth's axis is tilted, by about 23.5°, to the plane of its orbit; as our planet moves around the sun, the angle of the sun changes. At the summer solstice, the northern hemisphere is pointed toward the sun, receiving overhead sunlight -- while the southern hemisphere views the sun low on the horizon during its winter. When the winter solstice arrives 6 months later, the northern hemisphere experiences winter while the southern hemisphere enjoys summer. The equinoxes fall midway between the solstices, and mark a time when the sun shines directly down at the equator.

Calendars

Astronomy is probably an outgrowth of timekeeping. People based their calendars on lunar phases and seasons, but it became apparent that the number of days and months in a year were not integers. The anomalistic year, in which the Earth orbits from perihelion to perihelion, is 365.2596 days long. It takes the Earth 365.2422 days to move from one vernal equinox to the next, and that is called the tropical year. Finally, the sidereal year measures 365.2564 days, the time required for one complete revolution of the earth about the sun, relative to the fixed stars. The tropical year is the one we use for tracking the seasons, and it will be my definition of year from hereon, unless otherwise specified.

A calendar containing exactly 365 days will become out of step with the tropical year, falling behind about one day every four years. The first Roman calendars tried to reconcile the 29� day lunar month with the year, creating a 354 day calendar that included periodic "leap months" to maintain consistency between dates and seasons. In 46 B.C, Julius Caesar mandated the use of a calendar that had twelve months totaling 365 days, with an extra "leap" day being added every four years. But the Julian calendar, while an improvement over its predecessor, slowly began falling out of synch with the seasons due to its 11 minute difference from the tropical year. Pope Gregorius XIII mandated calendar reform in 1582, when the Julian calendar had slipped more than ten days behind the tropical year. The Gregorian calendar, which we use today, adds a leap day every 4 years, except in years evenly divisible by 100 but not by 400. The year 1900 did not have a leap day, but 2000 will. Differing from the tropical year by only about a half-minute, the Gregorian calendar will be essentially accurate for three millennia.

Calendar dates do not lend themselves to calculations. Try finding, for example, the 1000th day after 12 July 1997; you'll need to keep track of leap years and varying numbers of days in different months. Astronomers (and others who manipulate dates) prefer the Julian date: a count of days since the beginning of the Julian period. French scholar Joseph Scaliger defined the Julian dating system in 1582, as part of his work on the Gregorian calendar; the period was named for his father Julius Caesar Scaliger, and not after the unrelated Roman calendar. Joseph Scaliger set the first Julian Day at noon on 1 January 4713 B.C., the most recent date on which three archaic chronological cycles coincided -- a 15-year Roman taxation cycle, a 19-year lunar cycle, and a 28-year solar cycle.

So how do we calculate the Julian Date? Several formulas exist; I use the following procedure, which assumes that y₀ is the four-digit year, m₀ is the month, and d₀ is the day number. The functions INT(x) and FRAC(x) return, respectively, the integer and fractional parts of x.

Equation 1.4

The formulas in Equation 1.4 compute the Julian Date, J, at midnight (0 hours UT) on the specified date. For example, to calculate the Julian Date for midnight, 16 January 1997:

To convert from a Julian Date to a calendar date, use the process shown in Equation 1.5.

Equation 1.5

The following example converts the Julian date 2450702.5 to a Gregorian calendar date:

A Julian date of 2450702.5 converts to the Gregorian calendar date of 11 September 1997.

Julian dates prove their utility in calculations. Looking back to the question of finding the thousandth day after 12 July 1997, we can find the answer by following these steps:

1. Find the Julian date for 12 July 1997, which is 2,450,642.5.
2. Add one thousand to get the required destination date, giving a Julian date of 2,451,642.5.
3. Convert the destination Julian date to a calendar value, which results in the date 7 April 2000.

To find the number of days between dates, simply subtract their Julian equivalents. Also note that Julian dates begin at noon; the 17 July 1997 Julian date at midday is 2,450,642, while the Julian Date at midnight of that day is 2,450,641.5.

Sidereal Time

Because everything in the universe is moving, we must declare when a celestial coordinate is being given, and relative to what. For astronomers, the vernal (Spring) equinox provides a consistent frame of reference for both time and place. At the moment of an equinox, a equatorial view parallel to the sun is also parallel to the plane of the Earth's orbit. In the astronomical coordinate system, the time Greenwich faces the vernal equinox is the zero point, just as the prime meridian defines the zero point of longitude.

Figure 1.6 illustrates the relationship of the Earth's orbit and the vernal equinox. At the time of the vernal equinox, the longitude of Greenwich also points to the center of the sun. As the year progresses, the positions of the Sun and vernal equinox diverge. The direction of the vernal equinox is fixed relative to the Earth, and the time between culminations of the vernal equinox is the sidereal day. The Sun, however, changes position relative to the Earth, by almost a full degree each solar day; thus the Earth must turn an extra degree between culminations of the sun.

This means that the solar day is 3 minutes and 56.56 seconds longer than the sidereal day. For astronomers to use the vernal equinox as a reference point, they must determine the current sidereal time from the solar time. In essence, when you calculate the sidereal time, you determine the direction of the last vernal equinox.

Figure 1.6 - Sidereal and Solar Time

The formulas given in Equation 1.6 show how a GMST (denoted by the Greek symbol Θ) is calculated from UT.

Equation 1.6

The last Equation 1.accounts for a fraction of a sidereal day that has passed in t hours. For example, the GMST at 14:00 UT on 16 January 1997 (Julian date 2450464.5) is:

For values of Θ greater than or equal to 24 hours, subtract 24. In astronomical calculations, it can often be useful to add the sidereal time to Julian dates as a fraction of a day.

Equation 1.7

So the Julian date for 14:00 UT on 16 January 1997 would be:

Thus far, I've discussed the mean sidereal time at Greenwich (GMST), using universal time. Greenwich Mean Sidereal Time measures the angle between the zero meridian and the last vernal equinox, without accounting for short-term perturbations such as nutation. The Greenwich Apparent Sidereal Time (GAST) includes the "Equation 1.of the equinoxes", a calculation I'll discuss in a later article.

Your local mean sidereal time (LMST) determines the position of the celestial sphere over your location. To calculate LMST, you must subtract the offset for your time zone from local clock time to obtain a UT for computing the GMST. Next, divide your longitude by 15 to calculate the number of actual hours your clock time differs from UT, and add that value to GMST to obtain your LMST.

At longitude L, the Local Mean Sidereal Time (denoted by Θ_L) can be computed from the GMST (Θ) by the formulas in Equation 1.8.

Equation 1.8

My usual observing location is at 107° 40' West longitude. If I have my telescope out at 07:00 on 16 January 1997, the universal time would be seven hours later, or 14:00. The GMST would be 21.7361074, as calculated above. My LMST would be:

When I refer to sidereal time in later articles, I'm talking about the LMST for the place of observation (unless specified otherwise.)

Onward

Now that we know how to locate ourselves in time and space, it's time to look at finding the apparent positions of objects on the celestial sphere. The next article delves into astronomical coordinate systems, a bit of spherical trigonometry, and some further understanding of the Earth's orbit.