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How far are these stars from each other?
Jul. 26th, 2010 11:19 amTaking a break from the star survey for today… If you’re going to write a story involving Sol and more than one other star, you’re going to want to know not only how far those other stars are from us, but also from each other. How can you figure this out?
Let’s take a specific example and work it through. Suppose your story involves a little interstellar empire comprised of Sol and the two closest stars that possess habitable planets. Let’s say those stars happen to be Lalande 21185 and Sigma Draconis, which are 8.3 and 18.8 light years away, respectively. And let’s call their colonized planets “Lal” and “Sig”.
How far apart are Lal and Sig from each other? If the Empress is making a goodwill visit to Lal and a crisis demanding her presence suddenly breaks out on Sig, how far will she have to travel if she goes directly from one colony to the other? And how far in total will she have to go if she wants to make a stop on Earth first?
Well, as we plot the Empress’ journey, we need to start with the stars’ XYZ coordinates.
You probably remember plotting things out on a Cartesian plane in high school, with the X axis being the horizontal one and the Y axis being the vertical one. (If you’re saying “Whaaa?” at this point, here’s a link to some basic information: http://en.wikipedia.org/wiki/Cartesian_coordinate_system). Now imagine a third axis, the Z axis, which brings in the third dimension, is perpendicular to the first two, and has the same zero point separating negative values from positive ones.
Now imagine that Sol is at the 0,0,0 point of such a three-dimensional Cartesian area, one where the X axis points towards the centre of the galaxy; the Y axis points in the direction the galaxy is rotating; and the Z axis runs parallel to the galaxy’s axis of rotation (also known as “galactic north” and “galactic south”). Each star will have a set of “galactic XYZ” coordinates, based on how far it is from Sol along each of these three axes.
(You can also do this exercise with “celestial XYZ” coordinates, based on the same principle but with the axes pointing in other directions… but I’m trying to keep this simple.)
I’ve already mentioned the Internet Stellar Database, which has XYZ coordinates for a whole bunch of stars out to about 75 light years. It tells us that Lalande 21185’s galactic XYZ coordinates are (rounding down a bit): -3.36, -0.32 and 7.6. Sigma Draconis’ numbers are -3.45, 17.1 and 7.03. (If you do this exercise yourself using the ISDB, remember to use galactic and not celestial XYZ coordinates, and above all not to mix the two up!)
Now let’s take those numbers and plug them into an Excel spreadsheet, at cells A1 through F1. Next, let’s plug in the following formula at cell A2: =SQRT(((A1-D1)^2)+((B1-E1)^2)+((C1-F1)^2)) . With this formula, we subtract one X coordinate from the other, then square the difference; repeat this with the Y and Z coordinates; add up the three results and then find the square root of the total. There’s your distance between the two stars, which in the case of Lalande 21185 and Sigma Draconis is 17.4 light years. Since a trip from Lal to Earth and only then to Sig would be about 27 light years, the Empress may want to rethink her plan to stop off at home first.
On the other hand, if the empire’s two colony worlds were around Sirius and Ross 154 (8.6 and 9.7 light years from Sol respectively, and 17.1 light years from each other), a stopover here would only add a light year or so to her travel distance, so it might be a sensible plan.
And if they were around Epsilon Eridani and Tau Ceti (11 and 12 light years from Sol, but only 5 light years from each other), she’d have to have an awfully compelling reason for her detour.
Another source for galactic XYZ coordinates is the HabHYG database, a spreadsheet which contains information on tends of thousands of stars that could have habitable planets. It can be found at the “3 D Starmaps” website at http://www.projectrho.com/smap06.html. Distances are in parsecs. Do have a look around the entire website -- it’s chock full of fascinating info.
Let’s take a specific example and work it through. Suppose your story involves a little interstellar empire comprised of Sol and the two closest stars that possess habitable planets. Let’s say those stars happen to be Lalande 21185 and Sigma Draconis, which are 8.3 and 18.8 light years away, respectively. And let’s call their colonized planets “Lal” and “Sig”.
How far apart are Lal and Sig from each other? If the Empress is making a goodwill visit to Lal and a crisis demanding her presence suddenly breaks out on Sig, how far will she have to travel if she goes directly from one colony to the other? And how far in total will she have to go if she wants to make a stop on Earth first?
Well, as we plot the Empress’ journey, we need to start with the stars’ XYZ coordinates.
You probably remember plotting things out on a Cartesian plane in high school, with the X axis being the horizontal one and the Y axis being the vertical one. (If you’re saying “Whaaa?” at this point, here’s a link to some basic information: http://en.wikipedia.org/wiki/Cartesian_coordinate_system). Now imagine a third axis, the Z axis, which brings in the third dimension, is perpendicular to the first two, and has the same zero point separating negative values from positive ones.
Now imagine that Sol is at the 0,0,0 point of such a three-dimensional Cartesian area, one where the X axis points towards the centre of the galaxy; the Y axis points in the direction the galaxy is rotating; and the Z axis runs parallel to the galaxy’s axis of rotation (also known as “galactic north” and “galactic south”). Each star will have a set of “galactic XYZ” coordinates, based on how far it is from Sol along each of these three axes.
(You can also do this exercise with “celestial XYZ” coordinates, based on the same principle but with the axes pointing in other directions… but I’m trying to keep this simple.)
I’ve already mentioned the Internet Stellar Database, which has XYZ coordinates for a whole bunch of stars out to about 75 light years. It tells us that Lalande 21185’s galactic XYZ coordinates are (rounding down a bit): -3.36, -0.32 and 7.6. Sigma Draconis’ numbers are -3.45, 17.1 and 7.03. (If you do this exercise yourself using the ISDB, remember to use galactic and not celestial XYZ coordinates, and above all not to mix the two up!)
Now let’s take those numbers and plug them into an Excel spreadsheet, at cells A1 through F1. Next, let’s plug in the following formula at cell A2: =SQRT(((A1-D1)^2)+((B1-E1)^2)+((C1-F1)^2)) . With this formula, we subtract one X coordinate from the other, then square the difference; repeat this with the Y and Z coordinates; add up the three results and then find the square root of the total. There’s your distance between the two stars, which in the case of Lalande 21185 and Sigma Draconis is 17.4 light years. Since a trip from Lal to Earth and only then to Sig would be about 27 light years, the Empress may want to rethink her plan to stop off at home first.
On the other hand, if the empire’s two colony worlds were around Sirius and Ross 154 (8.6 and 9.7 light years from Sol respectively, and 17.1 light years from each other), a stopover here would only add a light year or so to her travel distance, so it might be a sensible plan.
And if they were around Epsilon Eridani and Tau Ceti (11 and 12 light years from Sol, but only 5 light years from each other), she’d have to have an awfully compelling reason for her detour.
Another source for galactic XYZ coordinates is the HabHYG database, a spreadsheet which contains information on tends of thousands of stars that could have habitable planets. It can be found at the “3 D Starmaps” website at http://www.projectrho.com/smap06.html. Distances are in parsecs. Do have a look around the entire website -- it’s chock full of fascinating info.