Long-Term Stellar Motions, part 2: Shortcuts

In the previous page on this site, you've seen how to calculate how stars move in space over long times. Unfortunately, the calculations are often very detailed and lengthy, and sometimes you don't need all that information.

If all you're interested in is the distance or brightness of a star in the distant past or future -- that is, you don't need its detailed coordinates -- then there are some short cuts you can take to get that information with relatively few calculations.

Figuring out distances to stars at different times

Recall that a star moving through space has a velocity, v, that can be broken down into transverse velocity vT and radial velocity vR. You can see the relationship between the two velocities (as well as an angle, \theta, which we'll use later) in this diagram:

Transverse and Radial Velocities

To figure out distances to a star, as a function of time, you will need both velocity components.

Step 1: Get the radial velocity

This is usually given in catalogs, along with the proper motion data. It will usually be given in kilometers per second. Advance warning: the Hipparcos Catalog, though an excellent resource for parallaxes and proper motions, has no radial velocities.

If the radial velocity is missing, then you can't do any more with the data. Either try another catalog or ignore that star.

Step 2: Get the transverse
velocity

You've already encountered the components of proper motion \mu\subalpha and \mu\subdelta. You'll need them again now. As with the other long-term calculations, you'll want \mu\subalpha in arcseconds per year, rather than seconds of right ascension.

Once you've gotten both proper motion components in arcseconds per year, use the relationship between them and transverse velocities:

vTA(km/s) = \mu\subalpha(arcsec/yr) * d * 4.740;
vTD(km/s) = \mu\subdelta(arcsec/yr) * d * 4.740

This is the same calculation you did to get these velocity components in the long-term motion calculation section. The same cautions for data (e.g., watch out for bad parallaxes and distance information) apply.

You'll also need the total transverse velocity vT. Get that by combining the components:

  • vT = sqrt (vTA2 + vTD2)

Finally get the total velocity v by combining its components, the radial velocity and the total transverse velocity:

  • v= sqrt (vR2 + vT2)

Step 3: Find the distance to the star, for any time

The star's total space velocity, v, is oriented at an angle to the line of sight to the star. This angle is given by \theta, and is defined from the relative sizes and orientations of the transverse and radial velocities:

Velocities Diagram Velocity Diagram 2

You can see that if the star is approaching the Earth (left diagram), \theta is between 0 and 90 degrees. It's between 90 and 180 if the star is receding (right diagram). Calculate the angle \theta by one of the two formulas below:

  • Negative vR (approaching): \theta = -tan-1 (vT / vR)
  • Positive vR (receding): \theta = 180 - tan-1 (vT / vR)

For this calculation, use the regular ATAN function, rather than ATAN2.

Now you can figure out how far away the star is at any given time. During a certain amount of time, t, the star will travel a distance l, in the direction of the velocity v. Also during this time, the star's distance from Earth will change from its original value, d, to some new value, d'. The diagram below shows an approaching star, but the same relations hold for a receding one as well:

Velocity-Angle Relations

For a given amount of time, t, the distance l is given by:

  • l = v * t

where l and v are expressed in comparable units. In general, you will use times in years, and distances in parsecs, so you must convert v to parsecs per year:

  • v (pc/yr) = v (km/s) / 977,780

Then get l from this converted velocity. Having gotten l, the most interesting quantity is d', the new distance to the star. You can get d' by the following formula:

  • d' = sqrt(d2 + l2 - 2dl cos \theta)

This calculation works equally well for an approaching star or a receding one.

Step 4: Find the brightness at the new distance

Recall from the section on calculating stellar positions from a distant reference point that changing the distance to a star changes its apparent magnitude. The version of the magnitude-change formula to use here is:

  • V' = V - 5 log10 (d/d')

Step 5: A special case: distance and brightness at closest approach

Suppose you want to know the minimum distance between the star and the Sun. When the star is at its closest, it has a unique feature: its line of sight distance d' to the Earth is perpendicular to its velocity vector:

Star at Closest Approach

Under these circumstances, you can write several of the formulas more simply:

  • d' = d sin \theta
  • l = d cos \theta

Note that you will get a negative l if the star is receding. Imagine that the star would have to back up to get to its closest position, since closest approach occurred in the past.

Since you know v, the star's total velocity, you can figure out how long it will take the star to travel the distance l. Assuming you've already converted v into pc/yr:

  • t = l / v (pc/yr)

The "sign" on t (past or future) is determined by the radial velocity. If the star's radial velocity is negative, the star's closest approach will occur t years from now. If the radial velocity is positive, its closest approach was t years ago, and it will never get any closer.

An example: the motions of 61 Cygni

Let's look at the star 61 Cygni -- the first star to have its distance measured, in the 1830s. 61 Cygni is a double star; let's use the data for the brighter star of the two, also known as 61 Cygni A. All these data were taken from the Hipparcos catalog, except the radial velocity, which I obtained from the Gliese Catalogue of Nearby Stars, 3rd edition.

  • V (magnitude) = 5.20
  • d = 3.48 pc
  • \mu\subalpha = 4.156 arcsec/yr
  • \mu\subdelta = 3.259 arcsec/yr
  • vR = -64.8 km/sec.

Now calculate various quantities from the data:

  • vTA = 3.48 pc * 4.146 arcsec/yr * 4.740 = +68.4 km/s
  • vTD = 3.48 pc * 3.259 arcsec/yr * 4.740 = +53.8 km/s
  • vT = (68.42 + 53.82) = 87.0 km/s
  • vR = -64.8 km/sec
  • v = ( (-64.8)2 + (87.0)2) = 108.5 km/s

The radial velocity is negative, so use the appropriate formula for calculating \theta:

  • \theta = -tan-1 (87.0/ -64.8 ) = 53.3 degrees

Let's find out when 61 Cygni is at its closest, and how long it will take to get there. For this, use the shortcut formulas from Step 5:

  • d' = 3.48 sin (53.3 degrees) = 2.79 pc.
  • l = 3.48 cos (53.3 degrees) = 2.08 pc.

Also get the magnitude at closest approach:

  • V' = 5.20 - 5 log10 (3.48/2.79) = 4.72

61 Cygni is approaching the Earth (vR < 0), and so at some time in the future, it will reach its minimum distance -- 2.79 pc as opposed to its current 3.48 pc -- and will thus be somewhat brighter (magnitude 4.72 instead of 5.20).

Since 61 Cygni is approaching, the time of closest approach is in the future:

  • v (pc/yr) = (108.5 km/s) / 977,780 = 1.11 x 10-4 pc/yr.
  • t' = 2.12 pc / (1.11 x 10-4 pc/yr) = 18700 yr.

61 Cygni is at its closest about twenty thousand years from now. Thereafter, it will recede -- until, at some point in the distant future, it will no longer be visible without a telescope.