Defining a Coordinate System or Reference Frame
The first step in navigation comes in answering “Where am I?,” or determining both your position and velocity—known in interplanetary navigation as your state or state vector—at a given moment in time. When a navigator makes that confident pronouncement, “We are here at position X,” there is an implied allusion to an even more fundamental issue, “relative to position O.” A position is meaningful only when given in a specific coordinate system or reference frame (the two terms will be used synonymously).
The definition of a coordinate system starts with, well, duh, a starting point—also known as the origin. This is the central point from which all positions will be measured. For Earth-based coordinate systems, the origin is typically the center of the planet. In the well-known rectangular, or Cartesian, coordinate system, three perpendicular lines, normally called the X, Y, and Z axes, radiate from the origin (three because our universe has three spatial dimensions). Any object’s position in space can now be specified exactly by three numbers, called coordinates: the object’s distance from the origin as measured on the X axis, the Y axis, and the Z axis. At least, it can be once we take care of one more tiny detail: we must specify the orientation of the coordinate axes.
On Earth the Z axis is synonymous with the spin axis, meaning that it passes through both the north and south poles (so the X and Y axes would both be in the plane of the equator). A line drawn along the surface of Earth from pole to pole is called a meridian. By international agreement, the meridian that passes the Royal Observatory in the town of Greenwich, England, is called the Prime Meridian. Now draw a line from the center of Earth to the point where the Prime Meridian intersects the equator, and we have finally defined the location of the X axis for Earth. The good news is that we have now completely specified a three dimensional coordinate system for the surface of Earth and we understand the basics of how one might construct a three dimensional coordinate system.
Chief Galen Tyrol
Kara "Starbuck" Thrace, Sharon "Boomer" Valerii, and Lee "Apollo" Adama.
Unfortunately, in our everyday lives we do not often need three dimensions to locate a spot on Earth’s surface; we naturally assume that any place we want to go is actually on the surface. So for most of our Earthly navigation needs, we treat Earth as a two dimensional map. This way, any object’s position can then be specified by using just two angles: longitude, a value that can range from 0 degrees to 360 degrees around Earth’s equator, and latitude, a measure of the angular distance above or below the equator, ranging from -90 degrees at the South Pole, to 0 at the Equator, to +90 degrees at the North Pole.
Let’s determine the location of Universal Studios in Los Angeles, California, where Battlestar Galactica was written. If we measure along the equator and determine the angle from the Prime Meridian to the meridian passing through Universal Studios, we determine that its longitude is 118.35 degrees west. If we measure along the meridian passing through Universal Studios and determine its angle above the equator, 34.14 degrees, we have its latitude.
That works for objects on and near Earth, but not elsewhere. Since Earth is rotating, any Earth-fixed coordinate system is rotating as well. This would not be an ideal frame for navigating among the stars, since the coordinates of any star in the sky would vary dramatically over the span of 24 hours. A rotating reference frame like this is called an accelerated or non-inertial frame. For interstellar travel, we want to establish a coordinate system that has axes that are permanently fixed, or at least close to fixed, for very long time periods.
Astronomers use a similar reference frame, which also has two angles akin to latitude and longitude, to locate objects in the evening sky. Imagine that Earth is surrounded by a transparent sphere thousands of light-years in radius. Imagine also that the stars we see in the night sky are affixed to this Celestial Sphere. (Real stars in the real sky move through the Galaxy, but their movement is noticeable only over many human lifetimes; for navigational purposes, the stars are as motionless as the phosphorescent stars children affix to their ceilings.)
The coordinate system used by astronomers starts by projecting Earth’s equator onto the Celestial Sphere. This is called the Celestial Equator. If we project Earth’s north and south poles onto the Celestial Sphere, we have the north and south Celestial Poles. Like latitude and longitude, positions in the sky are measured with two angles: right ascension and declination. Right ascension is measured along the celestial equator and ranges from 0 to 360 degrees (though astronomers often use 15-degree increments called hours), similar to longi-tude on Earth. Declination is measured as an angle “north” or “south” of the celestial equator, similar to latitude, and similarly ranges in value from +90 degrees to -90 degrees. What is the equivalent of the Prime Meridian, the arbitrary place where we start counting degrees? By scientific convention, the intersection of Earth’s orbit with the Celestial Equator is a point in space near the constellation Aries (called the Vernal Equinox), and is the celestial equivalent of
The Celestial Sphere.
Greenwich, England. Such a two-angle system like latitude and longitude, or right ascension and declination, is also implied throughout the series Battlestar Galactica. How many times have we heard Lieutenant Gaeta say, “We have a Cylon der, CBDR, bearing 123 carom 45”?dh
Obviously the Colonials do not use degrees as their angular measurement, since we have heard bearings with numbers far greater than 360 on numerous occasions. This simply implies that the angular size of one of their graduations is a fraction of a degree, making them a bit more accurate than our system. As for distances in space, the Colonials have used three throughout the series. For very close distances in space near Galactica, and on the ground of a planet, they have used kilometers. Within a stellar system, the term “SU” was first used in the episode “Captain’s Hand,” and is presumably of roughly the same scale as an Astronomical Unit (or AU) within the solar system. One AU is colloquially defined as the average distance from Earth to the Sun, about 149 million kilometers, or 93 million miles. Finally, for interstellar distances the Colonials use the same term as terrestrial astronomers: the light-year. Since light is the fastest thing in the universe, it is reasonable that any spacefaring race might use it, in some way, to define vast distances. Since we think that intelligent life as we know it can exist around stars of only a very narrow range of stellar masses (say F5 to K9), and within fairly narrow zones around those, it’s reasonable to assume that a “year” defined by the Colonials ranges from 10 months to 15 months. So one Colonial light-year would range from 81 percent to 125 percent of what we colloquially call a light-year.
This coordinate system works perfectly well within the entire solar system because the size of the solar system is tiny compared to the distance of nearby stars. Presumably something like this may work even within the Twelve Colonies. What if we are traveling vast interstellar distances, however? In that case our coordinate system references will appear to shift. We have to look for something else as a basis for our coordinate system.
What about the very center of the Galaxy? We saw in chapter 15, “Our Galaxy,” that the solar system orbits the center of the Galaxy every 225 million years, give or take a few million years. Could we use the center of the Milky Way Galaxy as a reference? This is, in fact, one of many coordinate systems used by astronomers today, and is called the galactic coordinate system. If we start with our solar system as the center reference point, the Colonials would obviously start at some point within the Twelve Colonies—then one axis might start at the origin and pass through the center of the Galaxy. Another axis would be perpendicular to the plane of the Galaxy, and the third axis perpendicular to both of them. A slight improvement upon that might be if the initial axis passed either along, or perpendicular to, the long axis of the galactic bar. The position of any object is given as a distance, an angle equivalent to longitude or right ascension that ranges from 0 to 360 in the galactic plane, and an
angle like latitude or declination ranging from +90 to -90 degrees above or below the galactic plane. With FTL technology, it’s reasonable to think that the colonies may have launched FTL-capable robotic probes to map much of the Galaxy, or even on trajectories perpendicular to the galactic plane, to map our Galaxy from above/ below. This may, therefore, sound like a good choice for coordinate systems, and perhaps it would be within the colonies and for journeys to comparatively nearby star systems. But again problems arise. In the plane of our Galaxy there are not only countless billions of stars to obscure the line of sight, there are also vast dark clouds of interstellar dust that inhibit visibility over all but comparatively short galactic distances—a ship that loses track of its references may not end up “lost” per se, but may not know how to get where the crew wants it to be. Perhaps, then, we need to look outside of our Galaxy for a fixed reference frame.
The Galactic Coordinate System.
One fundamental outcome of Albert Einstein’s Theory of Special Relativity is that there is no universal standard of rest. As seen from any Galaxy, the universe is expanding uniformly in every direction—so there is no “center of the universe.” An implication of this is that we are perfectly justified in saying that our Galaxy, the Milky Way, is fixed and at the center of the universe, and all other galaxies are expanding away from ours. In looking at the universe that way, finding something that is as fixed as stars on a bedroom ceiling is impossible, but finding something that appears to be fixed is trivial. With a few local exceptions (like M31), other galaxies are so very distant that their orientations relative to one another seen from within the Milky Way will never change—not in a hundred thousand human lifetimes, and not from any vantage point. Since galaxies are not concentrated in any area of the sky, we could choose a select number of reference galaxies that are bright enough to be picked out among the stars, and out of the plane of the Milky Way, so they are less likely to be obscured by our own gas and dust. Many spacecraft today have star trackers—a system consisting of a camera that images the field of stars, and a computer that determines which stars within the field of view are those whose positions are stored in an onboard database—to determine spacecraft orientation or attitude. Similarly, it might be reasonable that after every FTL jump, there is an onboard system (or a crew member) that determines the visibility and orientations of select reference galaxies (and perhaps even celestial landmarks), so Galactica would always know its orientation with respect to its coordinate system.
In fact, throughout the show we see Galactica’s astrogators using simple astrophotographs when calculating jumps. If the astrophotographs are being used also to triangulate the position of Galactica, what might they be using as landmarks? Just having a fixed reference frame is a necessary first step, but it is insufficient to answer the question “Where are we?”
Where Are We?
As we looked for very distant objects that are fixed permanently as a basis for a coordinate system, we want to look for comparatively nearby reference points to triangulate our actual position—just as islands, lighthouses, and other landmarks were used by ancient mariners. By determining the relative bearing of three landmarks, navigators at sea “triangulated” their positions. Today’s technological variation on the notion of landmarks is the “constellation” of thirty-two satellites orbiting Earth in various orbits, all about 20,200 kilometers above its surface. These satellites constantly broadcast messages with details on their orbital positions and message transmission time. The GPS receiver, which is essentially a small single-purpose radio receiver and computer, records the time the message was received, and uses that to determine the distance to the satellite. It then figures out where the satellite is in its orbit, and calculates the many places on Earth where you would be able to see that particular satellite at that particular distance at that particular time. By using four or more satellites, those many places can be narrowed to a single place, usually to within 30 feet.
In the early 1970s two scientists found a sort of naturally occurring GPS system that triangulates Earth’s position in space. NASA was about to launch Pioneer 10, the first space probe to the planet Jupiter, and the first man-made object specifically designed to leave the solar system. Since Pioneer 10 was Earth’s first emissary to the stars, the Cornell planetary scientist Carl Sagan petitioned NASA to include a letter of introduction, from Earth to the cosmos, on the spacecraft—one that included directions to the spacecraft’s planet of origin. In only a few weeks, Sagan and Dr. Frank Drake, then a professor of astronomy at Cornell, designed a 6-x-9-inch plaque of gold anodized aluminum that carried an engraved calling card from Earth.
The Pioneer Plaque.
The plaque was unveiled at a NASA press conference shortly before the launch, and interest and uproar were instantaneous. “Nudes and Map Tell about Earth to Other Worlds,” ran the headline in the New York Daily News. Various newspapers across the country ran articles about the plaque, but displayed family-friendly versions of the plaque, selectively edited for public consumption. Yet the newspapers were not upset with the multiline “burst” in the left center of the plaque—even though for the people of planet Earth it had the potential to be the most dangerous piece of information in the entire message. That collection of lines is a map telling space aliens how to find Earth. In fact, some scientists who argued against the plaque’s inclusion on the Pioneer spacecraft in the first place claimed that it sent out a very clear message: “Here’s what’s on the menu, and here’s a map to the restaurant.”
How does the plaque reveal the position of Earth? Instead of orbiting satellites like GPS, Sagan and Drake used something more exotic and naturally occurring. Look very carefully at their map: the center of the map represents the location of Earth, as seen from above the North Galactic Pole. Each of the radial lines represents the angle and distance to fourteen different pulsars. Further, each of those lines ends with approximately 30 binary digits. The 30 binary digits are durations measured as units of time (in 1,420 million cyles per second or, more correctly, its inverse, 7 ten millionths [7/10,000,000] of a second per cycle), and they describe periods of several seconds to several milliseconds. In other words, these lines convey information on the rotation rates and rate of slowing of known pulsars at the time of Pioneer’s launch.
Sagan and Drake decided to use the rotational signatures of fourteen different pulsars, hoping that any civilization that could retrieve a small probe from deep space could also easily determine which fourteen pulsars had those particular rotation rates. By working backwards, any civilization that finds the plaque would be able to say that those fourteen pulsars had those specified rotation rates at such-and-such a period in time, and were visible at the given angles only at such-and-such a position in space. Sagan claims that if the extraterrestrials have a good pulsar database, the map will pinpoint 1971, the specific year the plaque was designed. (Without a good database, it’ll only specify the twentieth century.) Likewise, the position of Earth could be approximated to within about 60 light-years, and our exact planetary system pinpointed by the drawing of the solar system across the bottom of the plaque.
Could the Rag Tag Fleet navigate across the Galaxy using pulsars as landmarks? Perhaps while the Fleet was close to the Twelve Colonies, but certainly not much farther. To triangulate her position, Galactica navigators would need to have landmarks visible over fairly vast distances. When Sagan and Drake used pulsars as landmarks in 1971, pulsars were fairly recent discoveries. We have since learned that pulsars are only seen to “pulse” from vantage points situated on a cone centered on the pulsars’ rotational poles. Move very far away from Earth, and perhaps none of the pulsars on the Pioneer plaque may appear to “pulse.” Sagan and Drake had a good idea, but it probably would not help our brave Colonial heroes navigate across the Galaxy.
What about pattern-matching visible stars and constellations? We can rule this out for all but the shortest of journeys. Our current spacecraft determine their orientations using
star imagery, but they only operate within the solar system, where the stars look the same from Neptune as they do from Earth. We tend to forget that the stars of our well-known constellations are three-dimensional; many of the stars are at vastly different distances. Over journeys of many light-years, constellations change their apparent shapes dramatically, rendering them all but unrecognizable the further you get from home. But if we know what the star patterns look like from our destination, more and more constellations will become visible as we get closer to our target. Perhaps, as Lieutenant Gaeta did in “Revelations, Part II,” we might arrive at our destination and announce, “Visible constellations are a match!”
Can we just match observed stars to stars in a large database? It sounds possible, but again, the answer is no. Having a computer program match visible stars to a database of known stars is a type of problem that computer scientists call an NP-complete problem (where NP stands for “nondeterministic polynomial”). NP-complete problems have the nasty property that, even for small input sizes, they take the fastest computers a sizeable fraction of the age of the universe to solve. Recall that Aaron Doral said that Galactica’s computers “barely deserve the name,” and you have an unrealistic choice of navigational aids. Further, almost any star projected against the plane of the Galaxy will be difficult to detect because of the sheer number of stars visible.
The Science of Battlestar Galactica Page 20