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Motions in 3-D: Short Term Calculations

In the other pages on this site, you've seen how to calculate how a star appears in space from another star (or any other point in space). Stars don't just have positions in space -- they also move around, and over time the night sky changes dramatically. The sky that early hominids saw hundreds of thousands of years ago bore little resemblance to that we see today, because of this motion.

Approximate or short-term space motion calculations

"Short term" calculations apply when the star moves only a small distance or angle. Most stars move slowly enough that any time span less than about 10,000 years qualifies as "short term".

Step 1: Get basic data

The basic data here is the proper motion, the star's apparent angular velocity with respect to the other stars (see the Introduction to Terminology page for a more detailed description). You will need the following data, which you can get from a catalog, such as the Hipparcos catalog:

  1. \alpha, the right ascension, in decimal hours.
  2. \delta, the declination, in decimal degrees.
  3. \mu\subalpha , the proper motion in right ascension.
  4. \mu\subdelta , the proper motion in declination.

The last two will normally be in arcseconds per year, or some very similar unit. Some catalogs, like the Hipparcos catalog, have this value in milliarcseconds per year. To use that, divide it by 1000 before proceeding.

Step 2: Check units, and convert as needed

For the short term, we will simply assume the proper motion in both right ascension and declination is constant. This is a reasonable assumption so long as the star moves through a fairly small angle. We will thus calculate a change in right ascension and a change in declination by multiplying the proper motion in each by the amount of time we want to consider:

\alphat = \alpha0 + \mu\subalpha * t

\deltat= \delta0 + \mu\subdelta * t

In reality, they won't be constant and independent -- that's a subject for the next page.

As a result, we'll want to stick to the units conventionally used for right ascension (hours, minutes, seconds) and declination (degrees). So we'll use the quantity \mu\subalpha in seconds of right ascension rather than seconds of arc. In some catalogs, the \mu\subalpha is already in these units. If so, leave it alone. If not, it will almost certainly be in arcseconds, which for this calculation are an inconvenient unit. To convert arcseconds to seconds of right ascension:

  1. divide by 15,
  2. divide the result by cos \delta.

Step 3: Convert seconds into decimal quantities

Since your right ascension and declination values will be decimal, convert the proper motion quantities into decimal fractions of an hour or degree:

  • \mu\subalpha (decimal hours) = \mu\subalpha (seconds of right ascension) / 3600;
  • \mu\subdelta (decimal degrees) = \mu\subdelta (seconds of arc) / 3600;

Step 4: Calculate new right ascension and declination

For a given amount of time t (positive for future times, negative for past ones) and the converted proper motion data, calculate the new decimal right ascension and declination at time t (\alphat, \deltat) by a simple linear extrapolation:

  1. \alphat =\alpha + \mu\subalphat
  2. \deltat =\delta + \mu\subdeltat

Now you can plot the star's new position on a star chart, using these new \alpha and \delta values.

Step 5: Check your assumptions

This simple approach works well for time changes up to about 10,000 years, past or future. Some impressive online tools, e.g., the Hipparcos mission's proper motion page, use this approach. However, these calculations break down totally for longer times, since the star's distance from Earth changes appreciably, and so its proper motion changes, invalidating the simple approach we just did. As a general rule, if the star moves by more than about 30 degrees -- roughly the length of the Big Dipper, or the height of Orion -- these calculations are beginning to break down.

After a few hundred thousand years, just about any star will have moved enough to cause the values of the proper motion to change and invalidate this simple approach. The problem is worst for nearby stars like Alpha Centauri -- unfortunately, these are the stars that many 3D starmap makers are most interested in.

The next page on stellar motions addresses these issues. It shows how to add proper motion information to a third velocity, the radial velocity, to calculate the exact position and brightness of any star at any time, past or future.

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