Putting Your Stars in Their Places


Vision 14


Putting Your Stars in Their Places


Greg Scalise

Copyright © 2003, Greg Scalise, All Rights Reserved


As science fiction writers, we're free to range far beyond earth in our stories, from the moon to the farthest stars we can see. However, there are times when this can pose worldbuilding problems that writers in other genres don't need to worry about.

For a story I recently wrote, I had to find out how far apart Tau Ceti and BY Draconis were so I could figure out how long a message between the two systems would take for a round trip. Given that this message would be traveling at FTL speeds, I could just wave my writer's wand and make messages travel instantaneously. However, this would violate the rules I'd already set up in the story for this form of communication, and maintaining internal consistency is crucial to maintaining your reader's willing suspension of disbelief.

I already knew how far each star was from Earth, and in which direction, and I knew there had to be some way to translate this into the distance between the two stars. I found it, and it's not difficult to handle at all.

Astronomers locate the objects they study with a coordinate system comprised of three elements: Right Ascension, Declination, and Distance. Right Ascension (hereafter RA) and Declination might sound mysterious, but don't worry -- they're really not hard to understand. RA and declination are precisely analogous to longitude and latitude. The difference is that we can't specify the location of astronomical objects relative to the surface of the Earth, because the Earth rotates. Therefore, we use an imaginary reference grid in the sky to locate our objects, with the zero points defined by the point where the Sun crosses the equator on the first day of Spring for RA and the celestial equator and poles for declination. RA is measured in hours, minutes, and seconds (24 hours equals 360 degrees of "longitude"; RA increases in the eastward direction). Declination is measured in degrees (0 at the equator; 90 at the Celestial North Pole; objects below the celestial equator are assigned negative declinations).

Now that we know how astronomers locate things, we can use stellar catalog information to locate our stars and figure out the distance between them. Taking the two stars I mentioned earlier as our example, we find the location from Earth of each star is as follows: Tau Ceti: RA 1hr 44min 4.091sec Dec -15 degrees 56min 14.89sec Distance: 11.9 light years BY Draconis: RA 18h 33min 55.808sec Dec 51 degrees 43min 8.62sec Distance: 53.6 light years Armed with this information, we're ready to begin. Here's where the math kicks in. To find the distance between stars, we need to convert from the Earth-centered polar coordinate system astronomers use to a Cartesian X-Y-Z coordinate system. It sounds scary at first, but it really isn't. If you have spreadsheet software handy that can handle trig functions, it's very easy to create a spreadsheet that does all this for you.

First, we need to convert RA to degrees. We'll call this value A:

A = (hours * 15) + (minutes * 0.25) + (seconds * 0.004166)

Doing so gives us:

Tau Ceti: 26.017046

BY Draconis: 278.482530

Save these answers; we'll need them later.

Next, we need to find the value of what we'll call B. As declination is already measured in degrees, this is pretty simple.

B = ( ABS(Dec_degrees) + (Dec_minutes / 60) + (Dec_seconds / 3600)) * SIGN(Dec_Degrees)

Note that, as declination can range between 90° and negative 90°, we need to take the absolute value of the degrees term (the value as a positive number) and restore the sign after we've done our math. Otherwise, we'll get an inaccurate value for B. Using our two stars we get: Tau Ceti: -15.937469 BY Draconis: 51.719061 C is just the distance in light years (You can use parsecs if you prefer. It does you no good to use light years with one star and parsecs with the other, though!): Tau Ceti: 11.9 BY Draconis: 53.6 Now that we have all our polar coordinates in the same units, we can convert that to Cartesian X-Y-Z coordinates.

The formula for the conversions is:

X = (C * cos(B)) * cos(A)

Y = (C * cos(B)) * sin(A)

Z = C * sin(B)

Using our values for A, B, and C for each star, plugging in our numbers gives us: Tau Ceti: X1 = 10.283036 Y1 = 5.019159 Z1 = -3.267598 BY Draconis: X2 = 4.898169 Y2 = -32.842914 Z2 = 42.075062 Now we've got things located in familiar Cartesian coordinates, with the scale in light years, centered on Earth. With this, we're finally ready to find the distance between our two stars.

The formula for finding the distance between two points in a Cartesian coordinate system is:

Distance2 = (X1 - X2)2 + (Y1 - Y2)2 + (Z1 - Z2)2

Altering this to a form more suitable for pocket calculator use gives us:

Distance = SQRT((X1 - X2)2 + (Y1 - Y2)2 + (Z1 - Z2)2)

Putting our numbers in gives us the distance between two stars (and you though we'd never get here!)

Distance between Tau Ceti and BY Draconis: 60.991494 light years

Now, this may seem like a lot of work to get a number which you might not explicitly use in a story, but details like this matter when you're building a world. Even in hard SF we're allowed to break some of the laws of physics, but we need to make sure we've explained the ground rules of our world to the reader, and then rigorously stick with those rules. Otherwise, you'll risk losing your reader's willing suspension of disbelief -- and possibly that that reader's desire to keep reading your story. Sustain that willing suspension of disbelief by making your imaginary worlds vivid, detailed, and internally self-consistent, though, and you'll likely have a reader staying up past her bedtime, totally absorbed in the world you're showing her.


The Internet Stellar Database: http://www.stellar-database.com/

A searchable online database of most of the stars within 75 light years of Earth. Also gives you Earth-centered X,Y,Z coordinates, so you can save a bit of number crunching.

3-D Starmaps: http://www.projectrho.com/starmap.html

If you're interested in more information on the math involved, there's an extensive discussion here. Also includes extensive links to more information sources, online star catalogs, and software.