3-D Starmaps

Starmaps seem so crystal pure and elegant. But reality is far more messy. The night sky seem to be decorated with static stars firmly nailed to the firmament, but reality is closer to an explosion in a glitter factory.

The only reason they seem to be stationary is because

  1. space is so freaking huge
  2. stars move (relatively) slowly, and
  3. human beings think that ten thousand years is a little too long to sit and watch the action.

Some planetarium computer programs have an option to show how distorted the constellations appear millions of years in the past ( as seen by the dinosaurs ) or millions of years in the future ( as seen by all the mutant rats and cockroaches ).

And in times gone by, a star that was quite close to another could now be quite remote. This can have implications for young upstart interstellar explorers who want to loot ancient high-tech goodies in the ruins of long gone empires. What used to be a nice compact empire half a million years ago is now spread out over a large section of the spiral arm.

But again I warn you, the math is not going to be easy.

Thanks to Thomas York for the first half of these equations, visit his web site. And double thanks to Richard Powell for the other half, do check out his Atlas of the Universe.

Armchair Astrometrist.

Motion of Stars

Information about the motion of a star is generally given in terms of Radial Velocity and Proper Motion. Why? Because they are the easiest things for an astronomer to measure, that's why.

Any star's motion ( or vector ) is a complicated thing, but any vector can be split into two vectors, much in the same way that any line can be drawn using the two controls of an Etch-a-Sketch. So one knob will be the Radial Velocity knob, and the other the Proper Motion knob. The vector is called the Space Velocity.

In the diagram to the right, Vr is the Radial Velocity and Vt is the Proper Motion. Vs is the Space Velocity. Don't you just love all the weird terms astronomers invent?

Radial velocity is simply how fast the star is approaching or running away from the Sun ( the rate the RV knob is being turned ). It is easy to measure this with a spectroscope, as the star's spectrum is under the influence of the dreaded Doppler Effect. (I'm not going to attempt to explain this here, but five minutes with a web searcher should turn up a few tutorials). Most star's RV's are less than 100 kilometers per second, they are typically from 10 to 40 km/s. Positive values mean the star is receeding from the Sun, negative values are approaching.

Proper motion is how fast the star seems to move left or right in the sky, as viewed from the Sun (the rate of the PM knob is being turned). It is measured by taking photographs of stars from year to year and painstakingly measuring how much the image moves. The poor astronomer's task is complicated by the fact that all the stars are moving, so there isn't any reference point. But that's OK, that is what graduate students were invented for. What is generally given in a star catalog is the Annual Proper Motion, how many seconds of arc a star moves in one year. This is symbolized by the greek letter Mu μ. The largest known proper motion is Barnard's Star (often known as Barnard's Runaway Star) with an mu of 10.2 seconds per year.

The direction a star appears to be moving on the sky is called, surprise, surprise, the Direction of Proper Motion (sometimes called the Direction Angle of Proper Motion). It is in degrees. It is measured clockwise, and zero is north.

In the Gliese 3.0 data, Proper Motion is in the field of bytes 31-36, Radial Velocity is in bytes 44-49, and Direction Angle of Proper Motion is in bytes 38-42.

Calculating Motion

Step 1
  1. If you are provided with the Right Ascension and Declination proper motion components 'muRA' and 'muDec', then you need to multiple muRA by COS(Dec) to get the total proper motion in the Right Ascension direction(this is because lines of Right Ascension converge towards the poles).

    • pmRA = muRA COS(Dec)
    • pmDec = muDec

    where muRA, muDec, pmRA and pmDec are measured in arcsec/yr.

  2. Alternatively, if you have the proper motion (pm) and the proper motion position angle (pmRA) (the Gliese catalogue gives these), then convert them with:

    • pmRA = pm SIN(pmPA)
    • pmDec = pm COS(pmPA)

    where pm, pmRA and pmDec are measured in arcsec/yr.

  3. Alternatively, you may be given pmRA and pmDec directly. They are provided in the Hipparcos catalogue, so you can skipm steps 1a/1b.

Step 2

Obtain the components of the "space velocity" of the star in the east, north and radially outward directions by dividing the proper motions by the star's parallax (given in arcseconds), this converts the star's velocity from arcsec/year to AU/year. We also multiply by 4.7406 which converts the from AU/year to kilometres/second (1 AU/yr = 4.7406 km/sec).

  • VE = 4.7406 pmRA / parallax
  • VN = 4.7406 pmDec / parallax
  • VR = RV

where RV is the radial velocity.

Step 3

Find the space velocity in rectangular equatorial coordinates.

  • Vx = VN (-COS RA SIN Dec) + VE (-SIN RA) + VR (COS RA COS Dec)
  • Vy = VN (-SIN RA SIN Dec) + VE (-COS RA) + VR (SIN RA COS Dec)
  • Vz = VN ( COS Dec) + VR (SIN Dec)
Step 4

Rotate these velocities into galactic coordinates.

  • U = -0.0548755 Vx - 0.8734371 Vy - 0.4838350 Vz
  • V = 0.4941095 Vx - 0.4448296 Vy + 0.7469822 Vz
  • W = -0.8676661 Vx - 0.1980764 Vy + 0.4559838 Vz
Step 5

Convert these velocities from km/s to ly/year or pc/year:

  • 1 km/s = 3.3355E-6 ly/year
  • 1 km/s = 1.0226E-6 pc/year

Worked Example

Here's a worked example using Barnard's star:

Step 1
VariableSymbolValue
Proper motion (RA direction)pmRA-0.798 arcsec/yr
Proper motion (Dec direction)pmDec10.327 arcsec/yr
Radial velocityRV-111.0 km/s
Parallaxplx0.5490 arcseconds
Right Ascension (2000)RA17 57.8 = 269.45 deg
Declination (2000)Dec+04 41.6 = 4.69 deg
Step 2
  • VE = 4.7406 * -0.798 / 0.549
  • VE = -6.891 km/s
  • VN = 4.7406 * 10.327 / 0.549
  • VN = +89.171 km/s
  • VR = -111.0 km/s
Step 3
  • Vx = +89.171 (-COS 269.45 SIN 4.69) - 6.891 (-SIN 269.45) - 111.0 (COS 269.45 COS 4.69)
  • Vx = -5.76 km/s
  • Vy = +89.171 (-SIN 269.45 SIN 4.69) - 6.891 (-COS 269.45) - 111.0 (SIN 269.45 COS 4.69)
  • Vy = +117.85 km/s
  • Vz = +89.171 ( COS 4.69) - 111.0 * (SIN 4.69)
  • Vz = 79.80 km/s
Step 4
  • U = -0.0548755 (-5.76) - 0.8734371 (117.85) - 0.4838350 (79.80)
  • U = -141.2 km/s
  • V = 0.4941095 (-5.76) - 0.4448296 (117.85) + 0.7469822 (79.80)
  • V = +4.3 km/s
  • W = -0.8676661 (-5.76) - 0.1980764 (117.85) + 0.4559838 (79.80)
  • W = +18.0 km/s
Conclusion

Barnard's star is rapidly heading towards us at a velocity of 0.47 light years per millennium.

But What Will It Look Like In The Future?

I'm still working on how to calculate this backwards (i.e., how to calculate what the constellation of the Big Dipper will look like two million years from now)

It's Getting Closer!

As a matter of interest, there is a star called Gliese 710 that is apparently on a near-collision course with the Sun. The current best estimate has it passing within a mere 0.4 parsecs about 1.4 million years from now. Gliese 710 is a type M1 red dwarf currently 19 parsecs (63 light years) away.