Astronomy 1144: Introduction to Stars, Galaxies, and Cosmology

Todd Thompson
Department of Astronomy
The Ohio State University

Lecture 5: Stars I: Distances

Key Ideas

Distance is an important & difficult quantity to measure in Astronomy. It is an essential piece of information in understanding the sizes, luminosities, masses, and orbits, of essentially all things in the Universe.

Method of Trigonometric Parallaxes
Direct geometric method of finding distances

Units of Cosmic Distance:
Light Year
Parsec (Parallax second)

Key Equations


Why are Distances Important?

Distances are necessary for estimating: The problem is that distances are very hard to measure...

The problem of measuring distances

How do you measure the distance of something that is beyond the reach of your measuring instruments?

Examples of such problems:


You resort to using GEOMETRY to find the distance.

The Method of Trigonometric Parallaxes

Nearby stars appear to move with respect to more distant background stars due to the motion of the Earth around the Sun. This apparent motion (it is not "true" motion) is called Stellar Parallax.

(Click on the image to view at full scale [Size: 7Kb];(Graphic by R. Pogge) )

In the picture above, the line of sight to the star in December is different than that in June, when the Earth is on the other side of its orbit. As seen from the Earth, the nearby star appears to sweep through the angle shown. Half of this angle, is the parallax, p.

Parallax decreases with Distance

As the distance to a star increases, the its parallax decreases. This is easy to see in the following two figures:

(Graphic by R. Pogge)

In the upper figure, the star is about 2.5 times nearer than the star in the lower figure, and has a parallax angle which is 2.5 times larger.

This movie demonstrates parallaxes graphically.

This gives us a means to measure distances directly by measuring the parallaxes of nearby stars.

We call this powerful direct distance technique the Method of Trigonometric Parallaxes.

Stellar Parallaxes

Because even the nearest stars are very far away, the largest measured parallaxes are very small; less than an arcsecond.

For example, the nearest star, Proxima Centauri, has a parallax of 0.772-arcsec (the largest parallax observed for any star).

1 arcsecond (1'') is the angle subtended by a dime at a distance of 2.3 miles.

Parallax Formula:

We saw before that the smaller the parallax, the larger the distance. We can express this as a simple formula:

(Graphic by R. Pogge)


p = parallax angle in arcseconds

d = distance in "Parsecs"

Writing our parallax formula in this way allows us to define a new "natural" unit for distances in astronomy: the Parallax-Second or Parsec.

In words: The distance of a star in parsecs is one over its parallax in arc-seconds.

Parallax Second = Parsec (pc)

Fundamental unit of distance in Astronomy
"A star with a parallax of 1 arcsecond has a distance of 1 Parsec."

1 parsec (pc) is equivalent to:

206,265 AU
3.26 Light Years
3.086x1013 km

Light Years

An alternative unit of astronomical distance is the Light Year (ly).
"1 Light Year is the distance traveled by light in one year."

1 light year (ly) is equivalent to:

0.31 pc
63,270 AU

The light year is rarely used in research astronomy.

Examples of Parallax Distances:

  1. The star alpha Centauri has a parallax of p=0.742-arcsec:

  2. A star is measured to have a parallax of p=0.02-arcsec:

(Graphic by R. Pogge)

Additional information on parallax here here and here and here.


If the stars are too far away, the parallax can be too small to measure accurately.

In general, the greater the distance, the smaller the parallax, and so the less precise the distance measurement will be.

The smallest parallax measurable from the ground is about 0.01-arcsec. This means that from the ground, the method of Trigonometric Parallaxes has the following limitations.


The Hipparcos satellite (launched by the European Space Agency in 1989) measured precision parallaxes to an accuracy of about 0.001-arcsec.

Hipparcos represented a great leap in our knowledge of the distances (and motions) of nearby stars. The catalog was just released in late 1997, and is already having an impact on many areas of astronomy that rely in accurate distances.

For more information about the Hipparcos mission, visit the Hipparcos Web Site at ESA Astrophysics.

The Future

Space Interferometry Mission (SIM)

SIM is a proposed NASA mission to use the emerging technique of optical interferometry to make high-precision angular measurements. A firm launch date has not been set, but it would be no sooner than 2012....

Update: SIM is cancelled. More information here

One of SIM mission's key projects is to map the Galaxy using accurate stellar parallaxes. SIM's technical goal is to achieve a parallax precision of ~4 microacseconds, which would yield 10% distances out to 25,000 parsecs, encompassing the Galatic Center (8000pc away) and the halo of the Galaxy. SIM's science projects will select particular targets througout the Galaxy to perform focussed "pointed" studies, rather than measuring every star down to some brightness limit.

SIM is an ambitious mission with many important science goals, to learn more visit the SIM webpage at NASA's Jet Propulsion Labortory.


Gaia is a proposed European follow-up mission to Hipparcos currently slated for a late-2011 launch. It's goal is to conduct what amounts to a Galactic census. Unlike SIM, which is a pointed mission, Gaia will attempt an all-sky survey of the Galaxy that build up high-precision angular measurements through repeated measurements (whereas SIM can achieve high precision in single measurements).

Mission Goals:

To learn more about Gaia, visit the Gaia website

update 8/25/2013 by Todd Thompson
Original version by R. Pogge