Stellar Cartography

text and diagrams by Scott Robert Ladd

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"Astronomy... compels the soul to look upwards, and draws it from the things of this world to the other."
Plato, The Republic, bk. 7, sec. 529.

Watch the stars for any length of time, and you'll notice that they seem to travel circular paths centered on a point above your horizon. In the northern hemisphere, one star appear motionless at the center of its circling companions. That star is Polaris, and it lies less than a degree from the actual celestial pole. The southern hemisphere lacks a pole star -- and in a few thousand years, changes in the Earth's orbit will leave the northern hemisphere's pole pointing toward nothing.

Movement in the Heavens

It isn't the stars that are moving -- it's you. Polaris just happens to be located in the direction pointed to by Earth's North pole, and the stars circle it every sidereal day due to the Earth's rotation. The stars closest to the pole turn on the tightest path, while stars far from the pole will dip below the horizon. The height of the celestial pole is equal to the latitude of the observer. The celestial poles lie on the horizon when you stand at the equator, and Polaris will be directly overhead when you visit the North Pole.

Figure 2.1 - Movement of the Stars

While the stars may seem to turn from our perspective, they maintain their positions relative to each other. As Figure 2.1 shows, the stars all move the same angular distance during the same period of time -- and the same stars will follow the same circular paths, no matter where you are on Earth. If we are going to track the stars, we need a system of reference that reflects the mutually-relative positions of stars while also being adaptable to observers at different locations.

Geometry in the Round

Scientists developed just such a system, based on an obsolete view of the Universe. Ancient astronomers believed that the stars and planets were attached to a rotating celestial sphere, which surrounded the Earth. While we know that the celestial sphere is a myth, it provides a useful model for determining the positions of heavenly bodies.

Astronomers use spherical coordinates for locating objects in the sky. In Time and Place, I showed how any point P on the surface of a sphere can be located through two angular values: ψ measured counterclockwise around the xy-plane, and θ as an angular distance above or below the xy-plane. For locating something on the surface of the Earth, ψ and θ correspond respectively to longitude and latitude. Figure 2.2 shows a point P on a sphere; three equations define the xyz-frame coordinates of P:

Formula 2.1

Figure 2.2 also shows an alternative frame of reference for P, based on a rotation of the y and z axes around the x axis.

Figure 2.2 - Rotating Spherical Coordinates

Look straight down the unchanged x axis, and evaluate the angular changes in z and y in a two-dimensional frame. Figure 2.3 shows how to construct a set of equations that calculate x'y'z'-frame coordinates from the original xyz-frame coordinates and the angle of rotation χ.

Figure 2.3 - Rotated Frame, Rectangular View

The x'y'z'-frame coordinates of P can be calculated using these formulas:

Formula 2.2

Since we know how to calculate x, y, and z from the angles ψ and θ, we can substitute the equations for the xyz-frame coordinates into the formulas for x'y'z'-frame coordinates, creating a set of equations in which the position of P is expressed entirely in terms of angles.

Formula 2.3

Understanding a rotated frame of reference is important because our view of the heavens is, in fact, rotated. To locate an object in the celestial sphere, we must map coordinates defined in terms of the celestial sphere to the actual view we have of the horizon.

Equatorial Coordinates

The equatorial coordinate system defines it's xy-plane as extending from the Earth's equator, out to the abstract celestial sphere; the z axis is a line running through the terrestrial poles. You can view equatorial coordinates as another system of latitude and longitude: The prime meridian becomes the direction of the vernal equinox, longitude is replaced by an angle called right ascension, and latitude on the celestial sphere is termed declination. Figure 2.4 shows the celestial sphere surrounding the Earth, and the orientation of the poles, equator, and angles.

Figure 2.4 - The Equatorial Coordinate System

The celestial equator and lines of declination form great circles. As described in Chapter 2, a great circle passes through the center point of a sphere, defining two equal hemispheres. A spherical triangle is formed by the arcs of three intersecting great circles, as shown in Figure 2.5.

Figure 2.5 - A Spherical Triangle

The length of arc AB equals the sphere's radius r multiplied by the angle χ, the later being AB's central angle expressed in radians. To further simplify the trigonometry, define a unit sphere by setting the radius of the sphere to one; thus the length of any arc in a spherical triangle equals the magnitude of its central angle.

So how do spherical triangles relate to equatorial coordinates? When you look at the night sky, you are looking at the celestial sphere from an angle equal to your latitude. Figure 2.6 shows how this affects your view of the stars; the equations of spherical triangles, plus the trigonometric equations developed earlier, allow you to convert an equatorial position in right ascension and declination to horizon coordinates of altitude and azimuth.

Figure 2.6 - Relationship of Equatorial and Horizon Coordinates

Equation Equation 2.4 describes the simple relationship of the sidereal time, right ascension, and hour angle:

Equation 2.4

To calculate an object's hour angle, subtract the right ascension from the sidereal time. Note that you measure both azimuth and hour angle in a clockwise direction from the south meridian.

Your latitude defines the angle of rotation between the celestial equator and your horizon. Figure 2.6 shows how you can form a spherical triangle, at the top of the diagram, from the star's altitude, declination, and the celestial pole. That triangle is a tool for relating various quantities to angles in a rotated coordinate system. These relationships include:

Equation 2.5

In cases where an angle is being subtracted from 90°, we can use these trigonometric relationships to simplify equations:

Equation 2.6

Substituting identities in Equation 2.5 into the equations derived in Equation 2.3 gives us a set of expressions for converting equatorial to horizon coordinates.

Equation 2.7

The inverse set of substitutions provides equations that convert horizon coordinates to the equatorial system.

Equation 2.8

Since the vernal equinox is specified in terms of time, angles of right ascension (RA) measure counterclockwise angles of between 0 to 24 hours. Each hour of right ascension equals 15 degrees. Declination, like latitude, has a value between +90° and -90°.

Catalogs

For millennia, astronomers have compiled lists of objects in the heavens, tabulating statistics and coordinates. The Alexandrian astronomer Ptolemy created first formal star catalog in the second century B.C., based on the work of Hipparchus. Ptolemy's Almagest (a Latin corruption of the Arabic title Al-mijisti) contained position data for more than a thousand stars, and was in wide use until the 17th century.

Between 1859 and 1932, a modern compilation of star coordinates was developed as a series of catalogs begun by the German astronomer Friedrich Wilhelm August Argelander. He began with the Bonner Durchmusterung, which contained positions for 320,000 stars visible from the northern hemisphere. Other astronomers added extensions for the southern hemisphere, and the joint effort includes more than a million stars. Most modern catalogs include cross-references to the Durchmusterung catalogs, and they were the largest catalogs until the creation of the Hubble Guide Star Catalog, which contains more then 14 million objects used by the space telescope.

Today's standard references include The Fourth General Catalog of Variable Stars (GCVS4), the Yale Bright Star Catalog (YBSC or HR), the Smithsonian Astronomical Observatory Catalog (SAO), the Third General Reference Catalog of Bright Galaxies (3C), and the New General Catalogue of Nebulae and Clusters of Stars (NGC). The Fifth Fundamental Katalog (FK5) is the definitive, high-precision reference for the positions of 1535 reference stars; an extension to FK5 adds another 3117 stars. Other modern catalogs include specialized listings of spectrums, X-ray sources, quasars, nebulae, pulsars, and double stars.

You can find these catalogs on the Internet at the Astronomical Data Center (ADC) maintained by the Goddard Space Flight Center in Maryland. The ADC stores hundreds of catalogs, ranging from the Almagest to the most recent compilations. Access the ADC's extensive collection of electronic catalogs at the address http://adc.gsfc.nasa.gov/.

An excellent European resource is the Strasbourg Astronomical Data Center (CDS) in France, which stores more than 1500 astronomical catalogs. The CDS Internet address is http://cdsweb.u-strabg.fr/Cats.html

Examples

Were I going out on 8 February 1998 for an observing session at 20:00 Mountain Standard Time (MST), I could calculate the locations of objects from their right ascensions and declinations. For this example, I'll use Castor -- a lovely and bright binary star -- and M101, a face-on spiral galaxy in Ursa Major. The coordinates for M101 are RA 14^h 03^m, Dec 54° 27'; Castor will be found at RA 7^h 33^m, Dec 31° 58'. Assume my observing location is at 37° 48' North latitude and 107° 40' West longitude.

From the formulas in Time and Place, we can calculate the Julian Date as 2450853.5 (at 0UT) and the Local Mean Sidereal Time as 05:05. Also in Time and Place, I calculated my geocentric latitude as 37° 37'. Here's how I plug the coordinates for M101 into the equations that calculate its location. First, I compute the hour angle from the LMST and sidereal time, according to equation 3.4:

Then, I calculate the altitude a using the third equation in Equation 2.7):

I rearrange the last equation in Equation 2.8) to determine M101's azimuth:

And now for Castor:

So, M101 will be at altitude 10�, azimuth 24� 54', while Castor can be found at altitude 59� 17', azimuth 89� 26'. This tells me that M101 will be a bit too close to the horizon for viewing, but Castor will be nicely located, nearly 60° high to the east.

In calculating the azimuth, be aware that you'll sometimes need to "adjust" your result so that the answer lies within the correct quadrant. If sin(h) is negative, then subtract the azimuth you've calculated from 360° to obtain a correct value.

You can use this chapter's formulas to identify objects from their horizon coordinates. If, during the same observing session, I see a very bright star with an altitude of 30° 50' and an azimuth of 152°, I could solve for the star's right ascension and declination using the equations above. In this case, the unknown star has an RA of 6^h 45^m and a declination of -16° 43'. Looking in a star catalog, I find that the observed star was Sirius, the Dog Star in Canis Major.

Thus far, I've described computations that should be more than adequate for the vast majority of observers. Given a sky atlas with coordinates, you should be able to plot a reasonably accurate map of any section of the sky. But "reasonably accurate" isn't always good enough for astrophotographers and others who require great precision from their observations. Obtaining great precision requires the understanding and application of several factors.

Epochs

The universe is an unstable place; the gravity that holds the universe together also causes periodic changes in the coordinates of celestial objects. Planets tug at each other, slightly altering orbits; stars move around the center of the galaxy and relative to each other. With universe in constant flux, we must reference coordinates to a specific point in time called an epoch. We can the adjust coordinates, from the base epoch, to a time of observation, by including the periodic affects of relative motion and perturbations in the Earth's orbit.

The IAU specified in 1984 that epochs would be calculated using the following formula, with the standard epoch being set at noon on 1 January 2000.

Equation 2.9

For instance, the epoch for 2 July 1996 at 4:00UT would be J1996.5 -- the middle of the year, and 3� years prior to the standard epoch of J2000. Such differences between epochs provide measures for calculating coordinate changes in time. Modern references express coordinates in terms of J2000, expecting an astronomer to adjust coordinates for a specific time of observation. Most algorithms alter coordinates from J2000 to a specified destination epoch; a few calculations count years from B1900, which has a Julian date of 2415020.0.

Prior to 1984, many star catalogs based coordinates on the Besselian epoch of B1950. That system was named in honor of Friedrich Wilhelm Bessel, a nineteenth century Prussian astronomer with a penchant for precise calculations and observations. The Besselian epoch formulation accounted for the exact length of the year:

Equation 2.10

The Besselian epoch B1950 is the Julian date 2433282.423. Some algorithms require a number of Julian centuries between epochs, calculated by dividing the difference in Julian dates by 36525 days.

Precession

We owe much of our astronomical science to Hipparchus, a Greek scientist of the second century B.C. A meticulous researcher, his estimate of the tropical year differed from modern calculation by only 6.5 minutes. Hipparchus invented trigonometry and devised a system of latitude and longitude; he also created the first compilation of star positions, based on careful observation. And in creating his list of stars, Hipparchus discovered the precession of the equinoxes.

Precession is the steady change in the direction of the celestial pole. While the angle of the Earth's axis is kept steady by the gyroscope effect, the direction in which the axis points is changing due to the gravitational influence of other bodies in the solar system. Precession causes the celestial pole to circle 23.5° away from a point in the constellation Draco. While the north celestial pole is now located near Polaris, the pole will be close to Errai (g Cepheus) in twenty-five hundred years. Four thousand years ago, when the Egyptians built the pyramids, Thuban (a Draconis) marked the celestial pole. In 26,000 years, the pole will return to the vicinity of Polaris, completing it's cycle.

Precession is a primary cause of horological changes in celestial coordinates; to accurately locate an object, it's coordinates must be adjusted from the epoch of calculation to the epoch of observation. Such a conversion is accomplished via a rotation matrix that adjusts the rectangular coordinates of an object from one epoch to another. Three angles, ζA, zA, and θA, reflect the movement of the pole, providing values for the matrix when converting to and from the epoch J2000.

Equation 2.11

Use the equations in Equation 2.1 to calculate an xyz-frame coordinate for an object on a unit sphere, storing the result in a three element vector r. To convert coordinates from J2000 to a new epoch, multiply the coordinate vector r by a matrix P containing these values:

Equation 2.12

To convert the coordinate vector to J2000 from another epoch, transpose the matrix before performing the multiplication. In terms of matrix math, the formulas are:

Equation 2.13

The final step is to convert the adjusted xyz-frame coordinates back to right ascension and declination values:

Equation 2.14

If the sine of the right ascension is negative, subtract a from 360°. Finally, divide the right ascension, which is in degrees, by 15°, converting it to hours.

Changes in the Earth's axis aren't alone in altering the coordinates of objects in time. We also need to account for proper motion, the movement of stars relative to our solar system.

Proper Motion

Everything in the universe is in motion, changing the relative positions of celestial objects. Proper motion measures changes in right ascension and declination, usually providing yearly values. In calculating the future coordinates of objects, we must note both changes in the Earth's orbit and movements in objects themselves.

Figure 2.7 - Proper Motion in Auriga

Figure 2.7 shows the changes wrought by proper motion on the stars of the constellation Auriga, over the next 100,000 years. While several stars move only slightly, Menkalinan and h travel about two degrees -- and Capella sweeps completely across the constellation! Capella is one of the "fastest" stars in the heavens; Arcturus and Sirius also have larger proper motions. Barnard's Star, a red dwarf less than 6 light years away, moves an astonishing 10.3 arc seconds annually.

Proper motion, m, has two components for movement in right ascension and declination, ma and md. Star catalogs usually define these values in arc seconds per century; for example, in the FK5, Capella has a Δα of 0.728 and a Δδ of -42.47. These equations calculate the change in position during a given period of time:

Equation 2.15

In general, before plotting the position of a star, you'll want to adjust it's coordinates to the epoch (Julian date) of the time of observation.

Aberration

Since the speed of light is finite, motion toward or away from an object changes its apparent position. Known as aberration, this phenomenon occurs in astronomical observations due to the orbit of the Earth around the sun. Hurtling through space at 30 km/sec, our view of the cosmos is skewed by up to 21 seconds of arc. To include the consequences of aberration in right ascension and declination, use these formulas.

Equation 2.16

X, Y, and Z are derivatives of the Earth's rectangular coordinates in reference to the sun. The value λ is the apparent longitude of the sun for Julian date JD. Their values can be approximated by these formulas:

Equation 2.17

Aberration changes with the position of the Earth in its orbit, and it does not alter the positions of stars relative to each other or to the epoch. Apply aberration to the RA and declination coordinates before computing an azimuth and altitude for a time of observation, and after including changes due to precession and proper motion.

The formulas above calculate annual aberration, which stems from the Earth's movement around the sun. Another form of aberration is diurnal, caused by the rotation of the Earth on its axis. At most, diurnal aberration only changes our perspective by a third of an arc second -- and it can be practically ignored.

Refraction

The Earth's atmosphere causes significant changes in horizon coordinates by refracting the light of celestial objects. Directly overhead, refraction is virtually nonexistent -- but at the horizon, the atmosphere bends light by as much as a half-degree; the sun has physically traveled below the horizon when we see it starting to "set". In essence, refraction raises the altitude of an object with respect to the horizon, and we need to account for it in our observations.

The precise amount of refraction depends on temperature, atmospheric turbulence, and barometric pressure. By nature, these factors are inconstant, making it nearly impossible to determine the precise amount of refraction. For practical purposes, an approximation is adequate. In Equation 2.16), a is an object's non-refracted altitude, P is the barometric pressure in millibars and T is the temperature (°C).

Equation 2.18

Apply refraction after computing horizon coordinates and before displaying or plotting an object's position.

In adjusting coordinates for precession and proper motion, the program differs by no more than a few tenths of an arc second from the values given in the FK5. Such accuracy is more than sufficient, since variations in refraction and other factors would obscure further precision.

Onward

The motions of the Sun and Moon define our calendars and clocks. In the next article, I'll look into techniques for calculating the rising and setting of the sun and moon; I'll also discuss the phases of the moon, librations, nutation, and demonstrate the prediction of solar and lunar eclipses.