StarPlot Documentation

1.1. Coordinate Systems

The first thing an astronomer needs to know about an object in the sky is where to find it. Most commonly this information comes in the form of a right ascension (RA) and declination. These are two angular coordinates, similar respectively to the longitude and latitude of a point on the Earth's surface. One can imagine a sphere centered on the Earth, whose radius is larger than that of the known universe. This fictitious sphere is called the celestial sphere. The RA and declination of an object are defined as the coordinates of the point on the celestial sphere which that object is in front of, as seen from the Earth.

For convenience, the north and south poles of the celestial sphere's coordinate system are aligned with those of the Earth. In other words, if one stands at the Earth's north pole, the north pole of the celestial sphere is directly overhead. This almost coincides with the projection of the star Alpha Ursae Minoris onto the celestial sphere; hence that star has a declination of almost 90° N, and we call it the North Star, or Polaris. (Declination, like latitude, is measured in degrees, minutes and seconds of arc.) More generally, an object in the sky with declination d will pass overhead at some point during the day or night as seen from a point on Earth with latitude d.

Right ascension is more complicated, because the Earth rotates while the rest of the universe, as a whole, does not. So astronomers can't use Earthly longitudes to determine celestial right ascensions in the same way that they do with latitudes and declinations. Instead, they define a specific direction in the sky to be the zero meridian. This point, called the first point of Aries (FPA), is the place where the Sun's projection onto the celestial sphere crosses the celestial equator at the beginning of spring. (It happens to lie in the constellation of Aries, the Ram; hence the name.)

Now, suppose the FPA happens to be directly above the Earth's southern horizon, as seen from somewhere in the northern hemisphere. As the Earth rotates, this point on the celestial sphere will appear to set in the west. After a time period called a sidereal day, the FPA will have risen again in the east and once again be directly above the southern horizon. A sidereal day is slightly shorter than a solar day, roughly 23 hours and 56 minutes. (This is because the Earth, in its orbit around the Sun, makes one additional rotation per year if considered relative to the distant stars than if considered relative to the Sun.)

A sidereal day may be divided into 24 periods called sidereal hours, each slightly shorter than a standard hour. If the current sidereal time is h (the number of sidereal hours since the FPA was last due south), then the points which are now directly over the southern horizon are said to have right ascension h. For instance, after the FPA is due south, one must wait 6 sidereal hours and 43 sidereal minutes before the star Sirius is directly over the southern horizon. Therefore the right ascension of Sirius is 6 hrs 43 min. Looking due south from a point in the northern hemisphere, right ascension increases going from right to left.

As an aside, notice that one hour of right ascension is equivalent to 360° / 24 = 15° of longitude. It is unfortunate that an hour of right ascension is divided into 60 minutes or 3600 seconds of right ascension, because these units have nothing to do with the similarly named minutes and seconds of arc into which the degrees of latitude or declination are divided.

This is probably all very confusing, so I will summarize quickly: An object in the sky located at right ascension h and declination d will pass directly overhead as seen by an observer located at Earthly latitude d when the local sidereal time is h. Local sidereal time is dependent upon both the current solar time and the day of the year. In general, the sidereal time h (in sidereal hours) equals ( t + 24f ) mod 24, where t is the local solar time (in standard hours), and f is the fraction of a year which has passed since the last vernal equinox.

The above-described coordinate system is called celestial coordinates. Given the celestial coordinates of an object in the sky, it is nearly trivial to determine where the best place on Earth is from which to observe the object, at what time the object will be highest above the horizon on a given date, and how high it will be at that time. For this pragmatic reason, the celestial coordinate system is the most often-used.

However, astronomers also frequently use a different system called galactic coordinates. This system uses the plane of our Milky Way Galaxy, projected onto the celestial sphere, as its equator. The galactic latitude of an object is therefore given by its angular distance from the plane of the Milky Way, as seen from Earth. Galactic longitude is given using meridians perpendicular to the galactic plane, with meridian zero defined as the direction to the center of the Milky Way. In this system, both latitude and longitude are measured in degrees, just like Earth-based latitude and longitude (except there are no "east" or "west" longitudes; galactic longitude starts at zero and increases in the counterclockwise direction until it reaches 360°).

Although galactic coordinates are not simply related to an object's position in the sky as celestial coordinates are, they have the advantage of being more meaningful and less Earth-centric. The knowledge that a distant star has a high galactic latitude, for example, suggests it is one of the old halo stars rather than a new star within the galactic disk. Objects outside our galaxy are typically easier to see if they have a high galactic latitude because less gas and dust is in front of them from our viewpoint.

The conversion between galactic and celestial coordinate systems involves a great deal of irritating trigonometry. StarPlot supports both coordinate systems, and allows you to switch easily between them without hassle.

1.2. Distances and Parallax

So far we have only considered two of the three variables necessary to specify a star's three-dimensional location in space. The third variable, of course, is distance from Earth. Unlike a star's apparent position on the celestial sphere, however, its distance is difficult to determine. In general two methods are available: using a star's trigonometric parallax, and estimating its distance from its spectral lines and apparent brightness.

By far the more accurate of these is the parallax. Over a period of six months, the Earth travels in its orbit from one side of the Sun to the other, a shift in position of about 300 million kilometers. Parallax is simply one-half the angle by which a distant star appears (to us) to shift position on the celestial sphere due to this motion. The more distant the star, the smaller its parallax. The nearest star excepting our Sun, Proxima Centauri, has a parallax of 0.78 seconds of arc. Recall that a second of arc is 1/3600 of a degree, which itself is 1/360 of a full circle. Stars are exceedingly far away.

Astronomers prefer to measure stellar distances with a unit called the parsec (pc). A parsec is defined as the distance at which a star would have a parallax of one second of arc, and is equal to about 31 trillion kilometers. However, another unit of similar size called the light-year (LY) is more favored in popular science and science fiction. The light-year is the distance which a beam of light would take one year to travel in a vacuum. It is roughly 9.5 trillion kilometers, or 1/3.2616 of a parsec. Because of the unit's popularity, I have chosen to use the light-year in StarPlot.

Unfortunately, we cannot accurately measure parallaxes smaller than a few hundredths of a second of arc. Hence, stars farther away than about one or two hundred light-years must have distances determined via the second method. This will be described later. It is worth noting that while stellar parallaxes may be off by an order of 10 %, distances obtained from a star's apparent brightness and spectrum might easily be wrong by a factor of two.

1.3. Stellar Magnitudes

Other than the positional data described above, the first piece of information one might want to know about a star is its brightness. This is described using a logarithmic scale called magnitude. A star has an apparent magnitude that describes its brightness as seen from Earth, and also a more objective absolute magnitude related to its intrinsic brightness. To avoid confusion, I will use the adjectives "bright" or "faint" to refer to a star's apparent brightness, and "luminous" or "dim" to refer to its objective brightness (luminosity).

Astronomers define the apparent magnitude m of a star, as seen from the Earth, to be m = -2.5 log₁₀ r, where r is the ratio of the star's brightness to that of the bright star Vega. Notice that (1) fainter stars have larger magnitudes; (2) if the difference in magnitudes between two stars is 1, then one is 10^0.4, or about 2.512, times fainter than the other. The reason for this logarithmic scale is that ophthalmologists used to believe that human night vision perceived brightnesses logarithmically.

The brightest star in the night sky is Sirius, with an apparent magnitude of -1.46. That is, Sirius is 10^{(0.4)·(1.46)} = 3.8 times brighter than Vega. (For comparison, the Sun has an apparent magnitude of -27 or so, and the full moon, -12.) The faintest stars visible to a human with good eyesight, without optical aid, in a dark area far from city lights, have apparent magnitudes between +6 and +6.5 (and thus are 1000 to 1500 times fainter than Sirius). The faintest objects detectable by any human instruments have apparent magnitudes as large as +29.

The absolute magnitude M of a star is then defined as the apparent magnitude which that star would have if it were moved to a distance of 10 parsecs (32.16 light-years) from Earth. Our Sun is a relatively dim star, having an absolute magnitude of +4.85. However, stars may be much dimmer; the vast majority are small, relatively cool, and dimmer than absolute magnitude +10 or so. On the other hand, exceptionally luminous stars may have absolute magnitudes in the range of -5 or -6.

Knowing a star's absolute and apparent magnitudes, one may use the formula D = (10 pc)·10^(m-M)/5 to calculate its distance D in parsecs. This allows for the second method of distance determination listed earlier, because often one can estimate a star's absolute magnitude from knowledge of the lines in its spectrum.

1.4. Spectral Classes

A star's spectrum can provide a great deal of useful information about it, including but not limited to its temperature, density, speed of rotation, radial velocity, possible unseen companions, and of course, composition. Most of these variables affect only the details of spectra, however. Only the composition and temperature have very obvious effects. Since most stars have approximately the same composition in their external layers, usually the only important variable is the temperature at the surface of the star.

Because temperature is a continuous scalar variable, it is possible to arrange the major types of stellar spectra in a one-dimensional sequence. Each type of spectrum has been designated by a letter, called the Harvard spectral type. From hottest to coolest, the major spectral types are O, B, A, F, G, K and M. (When this system of types was developed in the 1890's, the effects of temperature upon stellar spectra were unknown, leading to the somewhat random-seeming order of these letters.) Type O and B stars are hot and blue-white, while M stars are cool and appear red. Each spectral type is subdivided by appending a number in the range [0,10), where a subtype "0" is hotter than a subtype "9". Our Sun has spectral type G2, indicating a surface temperature of about 5800 Kelvins (10,000°F).

Using parallax data, one can make a scatter plot of star absolute magnitudes versus spectral type (or equivalently, temperature). Such a graph is known as a Hertzsprung-Russell (H-R) diagram. Traditionally the diagram is plotted with temperature increasing from right to left, and luminosity increasing (magnitude decreasing) from bottom to top. Surprisingly, the stars on such a diagram, rather than being scattered all over it, lie in several well-defined bands. The most important contains about 90% of all stars, and extends diagonally from luminous blue O stars at upper left to dim red M stars at lower right. This is the main sequence. Bands of stars lying above the main sequence (having greater luminosities) lie in the giant and supergiant regions.

The Morgan-Keenan luminosity class is an attempt at a numerical system for describing these bands. A Roman numeral between I and VI is added to the end of the spectral type, with I representing supergiants, II and III giant stars, IV the rare "subgiants," V the main sequence or "dwarfs," and VI the so-called "subdwarfs." Naturally this classification is somewhat subjective. Nonetheless, certain features of a star's spectrum, if examined closely, make it possible to guess a star's luminosity class without knowing its absolute magnitude. From the spectral type and luminosity class, one can estimate the absolute magnitude, and therefore the star's distance as well.

Of course, a small percentage of stars don't fall into these tidy categories. A class of stars called white dwarfs (not to be confused with the main sequence "dwarfs" or "subdwarfs") lies to the bottom left of a Hertzsprung-Russell diagram, combining high temperatures with surprising dimness. The Wolf-Rayet stars lie far to the left, and even more exotic objects such as pulsars and black holes don't even have a place on the chart.

1.5. Stellar Evolution

From lengthy observations and inspired hypotheses, astronomers have arrived at a reasonably good understanding of stellar evolution. Stars begin "life" via gravitational condensation from clouds of gas and dust. Thousands of them form in a cluster at once. Once nuclear fusion has begun in a star's core and it has reached an equilibrium state, it starts out in the main sequence. More massive stars start in the upper left corner of the H-R diagram, while less massive ones begin in the lower right. (The minimum mass required for a star to reach stable equilibrium with nuclear fusion is about 1/12 the mass of our Sun.)

The more massive a star is, the shorter its lifespan, for the rate of nuclear reactions is strongly dependent upon the stellar core temperature, which in turn is a function of pressure. Sunlike stars may last 10 billion years in the main sequence stage, while stars starting out at type O on the main sequence will survive for "only" a few million.

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1.6. Star Names

One large problem astronomers have with stars, since there are billions of them in our galaxy alone, is naming them. The ancient Greeks and Arabs simply named the brightest stars individually. Some of these names, like "Betelgeuse," are still well-known, while others such as "Zubeneschamali" are barely used anymore. However, this approach did not scale well, even to the only 6000 stars visible to the unaided eye.

One of the first systematic naming attempts was made by Johann Bayer in 1603. He identified the stars in each constellation with a Greek letter, starting with Alpha for the brightest star and working his way down. This was before the telescope was used in astronomy, and long before photography or electronic devices, so often he made mistakes: Alpha Orionis (a.k.a. Betelgeuse) is the second-brightest star in the constellation Orion. Rather than write out the entire constellation name all the time, astronomers will use just the standard three-letter abbreviations; Orionis becomes Ori. A list of the 88 constellations and their abbreviations can be found as an appendix in most astronomy handbooks.

As there are only 24 Greek letters, and many more stars than that in each constellation, Bayer sometimes resorted to superscripts, so we have (for instance) Pi¹ through Pi⁶ Ori. StarPlot assumes that superscripts are represented like this in its ASCII text data files: Pi(1) Ori. In the graphical display, StarPlot will convert the Greek letter names into actual Greek letters, and use superscripts where necessary: . (You do not need to have Greek fonts installed on your system for this feature to work.)

Clearly, more star names were needed. Around 1725, the astronomer John Flamsteed gave stars numbers (which, unlike Greek letters, are in inexhaustible supply) from west to east in each constellation, in order of increasing right ascension. In addition to being called Rigel and Beta Orionis, the brightest star in Orion now became 19 Orionis as well. One minor problem: Flamsteed never numbered stars which were too far south to be seen from England. Hence a lot of stars in the far southern sky which lack Bayer designations are still known by such constructs as I Carinae or L² Puppis.

One additional set of star designations uses the constellation names. Some stars vary cyclically in brightness over time; these are called variable stars. A German astronomer named F.W.A. Argelander decided that these stars needed their own system of nomenclature. He began with the letter R for the brightest variable star in each constellation, for instance, R Andromedae. (In some constellations, the letters up through Q were already taken.) After reaching Z, he started a confusing system of double letters, which finished at QZ with the 334th variable in a constellation. Rather than continue his mess, later astronomers decided to name further variables V335, V336, and so on.

More recently, constellation names have gone out of style in star naming schemes. In 1859, Argelander also began a massive star catalog called the Bonner Durchmusterung (Bonn Survey), in which stars were labeled by position. Rigel, for example, is BD -08°1063, meaning it is the 1063rd star (counting from 0 hours right ascension) which was catalogued in the strip of the celestial sphere between -8° and -9° declination. Two other surveys, which labelled stars "CD" and "CP," completed coverage of the entire celestial sphere. These surveys covered over a million stars.

In the 1910's and 1920's, the Henry Draper Catalogue (HD) was compiled as a listing of stellar spectra, containing over 200,000 stars numbered by increasing right ascension: HD 1 through HD 225300. The Smithsonian Astronomical Observatory (SAO) released its own catalog of more than 250,000 stars in 1966. Reasonably bright stars (brighter than about 10th magnitude) are almost all covered by one of the catalogs described here, usually several.

However, astronomers also use many other special-purpose catalogs. These are especially noticeable in the set of nearby stars, most of which are dim red dwarfs below the magnitude limit of any catalog already described. Commonly seen designations in the Gliese data set (available from the StarPlot web site) include the Gliese, Giclas, Wolf, Ross, Luyten and LHS catalogs. And some stars, like Barnard's Star or van Maanen's Star, are simply named after their discoverers.