Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.
Parallax is the apparent shift in position of a nearby object when it is viewed from different locations. Those might be two different observatories on the Earth's surface (as was the case in the famous transits of Venus in the eighteenth and nineteenth centuries), but, in most astronomical applications, are two spots in Earth's orbit around the Sun:
If one knows the value of the baseline distance b, and can measure the parallax angle π
Yes, I know this is confusing. Yes, astronomers back in the old days should have picked a better letter. Sometimes you see the "curly pi" ϖ used to denote this angle, but few font families include it, so the parallax angle is usually written with the ordinary Greek letter π.
then one can use simple geometry to compute the distance to the target object d.
Q: Write a simple formula for d.
Of course, that simple formula ignores many complications. For example,
But one can work around these issues, or at least place constraints on their consequences.
The importance of parallax lies in its fundamental simplicity: we humans understand geometry, and trust it. As long as we can make measurements of the angular displacements over time, we can use the resulting distances with confidence.
There may be minor issues involving corrections for the small shifts of the background reference stars themselves; but if we choose to use galaxies or quasars as the reference sources, we can avoid those completely.
Most important of all, parallax is a direct method of measuring distances. It relies on a single assumed quantity: the astronomical unit (AU) = the distance from Earth to Sun. Back in the 1800s and early 1900s, this was indeed a major issue, and astronomers made great efforts to pin down the value of the AU. But these days, thanks to radar and spacecraft flying through the solar system, we do have a very, very good idea for the size of the AU. For example, the JPL Horizons system uses a value taken from the Planetary and Lunar Ephemerides DE430 and DE431, which list
1 AU = 149597870.700 km
That's ... quite a few signficant figures.
And that's it. Heliocentric parallax does NOT require any assumptions about stellar luminosity or evolution, or the distribution of sizes and colors of stars, or the spectral energy distribution of glowing hydrogen gas.
So, if astronomers can measure a distance via parallax, they will rely on it much more than distances determined by other methods.
In 2002, the asteroid 2002 NY40 flew past the Earth at a distance of less than one million kilometers. I used a telescope at the RIT Observatory in upstate New York to take its picture ...
... at exactly the same time that one of my colleagues at the Naval Academy in Annapolis, MD, did the same with her telescope:
If you compare the two images carefully, you should see that the background stars line up in the two images -- but the nearby asteroid clearly "jumps" from one image to the next due to parallax. (Click on the image below to blink the images).
We knew the distance between Rochester and Annapolis, and we could measure the size of the shift in the asteroid's position as seen from the two sites. With a little bit of geometry -- well, no, with quite a lot, actually -- we were able to calculate the distance to the asteroid.
It sounds as if using this technique to measure the distances to other stars should be pretty easy. In practice, though, it isn't quite so simple. There are a number of complications.
First, what exactly do we expect to see? As the Earth orbits the Sun, our point of view shifts back and forth, and so we ought to see nearby stars also shift back and forth, back and forth, relative to the distant ones. A simple, back-and-forth linear motion -- right?
Well, not quite. There's a complication due to the motion of our solar system and other stars in the Milky Way Galaxy. Even though most stars are circling the center of the Galaxy in the same general direction, each moves with a slightly different speed and direction. The result is that, over periods of decades and centuries and millenia, nearby stars appear to drift slowly across the sky. We call this proper motion.
The stars in the Big Dipper, which are relatively close to the Sun (only about 20-30 parsecs away), provide a dramatic illustration. Click on the picture below to watch a movie showing 200,000 years of proper motion.
Movie courtesy of
Tony Dunn's proper motion simulator -- thanks, Tony!
So, if we make careful measurements of the position of a nearby stars over a period of several years, we expect to see it move in a path which combines
The result is something like a helix or corkscrew. You can see examples from the Hipparcos Intermediate Data archive by downloading and running their Java applet. The example below belongs to one of the brightest stars in the summer sky.
Thanks to the
European Space Agency
See also alternate version
You should be able to look at the diagram above and use the side-to-side shift to estimate the distance to this star. Can you do it?
Q: What is the amplitude of the side-to-side motion around
the diagonal straight line? The units on both
axes are milli-arcseconds.
Q: What is the amplitude of the parallax motion of this star,
expressed in arcseconds?
Q: What is the distance to this star, expressed in parsecs?
By the way, do you know which star this is? The figure above provides its Right Ascension and Declination. If you were to type those coordinates into
perhaps you could identify it.
This seems relatively straightforward: take pictures of a nearby star 20 or 40 times over the course of several years, measure its position relative to distant stars carefully, fit a model consisting of some linear motion plus an elliptical component, and *poof* out pops the star's distance. What's so hard about that?
Well, I've neglected just one thing in this discussion so far: scale. Just how large are these motions? And how precisely can we measure the position of a star from a typical image?
Consider the image below, which shows the nearby star Ross 248. This is one of many CCD images I have acquired of this star with the 12-inch telescope at the RIT Observatory.
The target has a proper motion to the South, so if we were to measure it over a three-year period, we ought to see it move in a pattern like this:
Simple, right? Measure the size of the "wiggles" in its motion, and use them to determine the parallax angle, and thence the distance to the star. Done!
There just one little complication. In the picture above, I've drawn the motion of Ross 248 in a slightly exaggerated way so that it's easier for you to see. The real motion is a bit smaller ...
Because the motion has a small amplitude, it's not quite so easy to measure accurately. Still, with some care, this is still not a big deal, right?
Nope, I'm lying. I still haven't drawn the motion at its proper size. Let's zoom in a bit more.
Okay, NOW I've drawn the motion of Ross 248 at its proper scale. Over this 3-year period, the star will move about 4.8 arcseconds to the South. Since the pixels on the camera at the RIT Observatory subtend 1.25 arcseconds each, this corresponds to a motion of about 3.8 pixels.
Oh, but wait -- it's not the long axis of this motion that we want to measure -- it's the short axis, running roughly East-West, representing the shift due to the Earth's motion around the Sun. What we really want to measure is THIS motion:
Of course, the full side-to-side motion is TWICE the parallax angle, so we really need to figure out half of that shift. As you can read elsewhere, in order to use parallax to determine the distance to a star accurately, one must measure the parallax angle with an uncertainty of at most ten percent. In the figure below, the red square illustrates the precision to which we must find that angle.
Oh, and one more thing: this particular example, Ross 248, is one of the closest stars to our solar system: the ninth-closest stellar system, in fact. A more typical parallax measurement requires precision which is 10 times, or even 100 times, higher than this.
NOW you may start to understand why measuring the distances to stars is not so easy ....
(If you wish to see an example of the analysis required to turn measurements of position into parallax angles and distances, check out my analysis of RIT Observations of GX And.)
Astronomers have known about the possibility of parallax for thousands of years. The ancient Greeks did their best, without telescopes, to detect its effect on the positions of stars, but failed to do so. That failure led some of them to argue -- quite reasonably -- that the heliocentric theory of Aristachus and others must be incorrect: if the Earth really did orbit around the Sun, then nearby stars SHOULD make small shifts over the course of a year.
The first successful measurement didn't take place unti several centuries after the invention of the telescope. As described briefly here, Friedrich Bessel published his determination of a parallax angle π = 0.314 arcseconds
No, I'm not making up that value; check out a recent historical overview or read Bessel's original paper (in English translation)
to the star 61 Cygni in 1838. Several other astronomers soon followed with measurements of a few other stars, and soon observers were working to create catalogs of tens and hundreds of measurements.
Let's examine our estimates of the distance to just one star to see how it has changed with time. Which star?
Below are three estimates of the parallax angle to this star.
Year Technique Parallax angle
(arcseconds)
------------------------------------------------------------------
1918 photographic plates 0.0054
1997 Hipparcos satellite 0.00176 +/- 0.00082
2022 Gaia satellite (DR3) 0.00227 +/- 0.00011
------------------------------------ ------------------------------
For each of these estimates, calculate the distance in parsecs. Use the uncertainty in the parallax angle to calculate an uncertainty in the distance by using the following simple method:
1
best distance = -----------
angle
1
max distance = ---------------
angle - uncert
1
min distance = ---------------
angle + uncert
What are the distances implied by each of these three measurements? How well (if at all) do they agree with each other?
If our measurements of parallax are incorrect, then our estimates of many other stellar properties will be wrong, too. For example, the absolute magnitude M depends on the apparent magnitude m and the distance d according to the formula
The apparent V-band magnitude of VV Ori is mV = 5.34.
Q: What is the absolute magnitude MV if we use the
parallax measurement from 1918?
Q: What if we use the measurement from Hipparcos?
Q: What if we use the measurement from Gaia?
As you can see, if one can't measure parallax angles accurately, one can't determine the absolute magnitudes -- or luminosities -- of stars. Any HR diagrams based on inaccurate parallax measurements will have a lot of errors.
Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.