How Astronomers Measure Distances
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PART 2 Further Detail: How Astronomers Measure Distances by Lesa Moore
Astronomers use some clever techniques to measure distances to objects beyond our Solar System. Different methods work over different distance ranges. The whole range of methods is known as the “distance ladder”. It starts with the most accurate, which is annual stellar parallax. Next is the period-luminosity relationship of cepheid variable stars, which are known as reliable “standard candles”. Beyond that, other celestial objects are used as standard candles, including planetary nebulae, whole galaxies and, at the greatest distances, type 1a supernovae.
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- You may or may not know how surveyors measure distances to or heights of landmarks that they can’t walk to, e.g. working out the distance to a tree on the other side of a river. The answer is trigonometry. Perhaps you did some of this in school – sin, cos, tan, opposite over hypotenuse, 30/60 triangles and so on. Trigonometry allows us to know the relationships between angles and sides of right-angled triangles in such a way that if you know one of the smaller angles and the length of just one of the sides in the triangle, you can compute everything else about that triangle. You may not think there are many triangles in space, but the most important one (Figure 1 below) is described in detail on the Wikipedia page here: https://en.wikipedia.org/wiki/Stellar_parallax.
Figure 1 below - Geometry for measuring annual stellar parallax.
Image Credit:https://en.wikipedia.org/wiki/Stellar_parallax
- The radius of the Earth’s orbit around the Sun is a side of known length (parallax was also used to make this measurement: https://phys.org/news/2015-01-distance-sun.html). The angle subtended at the star by the Sun-star line and the Earth-star line is measured from the field of background stars using photography (Figure 2). When you make an image of the nearby star against the background stars, its position relative to the background stars will change throughout the year. The angular change in position is taken from the positions of the nearby star on the photographs, with the fraction of arcsecond change in position calculated from the known positions of the background stars and/or the scale of the image, e.g., the nearby star may be measured to have changed position relative to the background stars by 0.3 of an arcsecond over a six-month period. Remember, separation between stars in an image is a measure of angle, not distance.
- Using the known and measured values, the distance to the star (the long side and the hypotenuse are virtually the same length) can be calculated. In fact, the angle that is measured is the double angle formed by Earth at the two extremities of its orbit. One of the diagrams on the Wikipedia page (reproduced below) shows multiple measurements of the star (S) throughout the year, with the full ellipse extrapolated to include the observed points.
Figure 2 below - Measuring position of the star: Simulated image of background stars and positions of foreground star, S, whose distance is being measured.
Image Credit:https://en.wikipedia.org/wiki/Stellar_parallax
- From trigonometry, the length we are trying to calculate is the adjacent side where the tan of the angle (0.3 arcseconds in the example above) is equal to the opposite side (radius of Earth’s orbit) over adjacent (distance from the Sun to the star). This is simplified by using the “small-angle approximation”, but we don’t need to know the maths.
We end up with the formula:
d (pc) ≈ 1/p (arcsec).
where d is distance in parsecs and p is the parallactic (measured) angle. Astronomers’ use of parsecs for measuring distances derives from this technique, where an object at a distance of 1 parsec would subtend an angle of one second of arc. No stars are this close. As an example, Proxima Centauri (the nearest star to Earth other than the Sun), whose parallax is 0.7685, is 1 / 0.7685 parsecs = 1.301 parsecs (4.24 ly) distant ( https://en.wikipedia.org/wiki/Stellar_parallax). - This method works with reasonable accuracy out to a few hundred parsecs (about 1000 light years, 1 pc = 3.26156 ly). Space-based parallax measurements have been made by the ESA’s Hipparcos and Gaia missions.
- Henrietta Swan Leavitt worked at Harvard College Observatory from 1903 identifying variable stars in the Magellanic Clouds from photographic plates. She published papers in 1908 and 1912, the earlier noting that the brighter variables had longer periods and the later plotting data for 25 variable stars in the Small Magellanic Cloud (Periods of 25 Variable Stars in the Small Magellanic Cloud submitted by Edward C. Pickering, but acknowledging that the statement was “prepared by Miss Leavitt”). Refer Figure 3 below.
- What Leavitt had found was that there was a direct relationship between the intrinsic luminosity of Cepheid-type variable stars and their period of variability. This relationship, refined since its first discovery (separating type-1 and type-2 Cepheids), is known as the period-luminosity relationship for Cepheid variable stars.
Figure 3 below - The Period-Luminosity Relation for Magellanic Cloud Cepheid variable stars: Data from Storm et al. 2011, A&A, 534.
Image Credit: Dbenford, https://commons.wikimedia.org/wiki/File:Storm2011_Cepheid_Data.svg, licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
- This relationship means that astronomers may measure the period of a Cepheid variable (distinguishable from other types of variable star by its light curve) to determine its intrinsic brightness, or absolute magnitude (M) from the relationship. The difference between the absolute magnitude and the apparent magnitude (m) is known as the Distance Modulus (m-M). The distance to the star can be calculated from the Distance Modulus using a formula that takes into account the fact that brightness decreases as the inverse square of distance (refer Distance Modulus and Distance https://www.asnsw.com/i2ao2b#dm).
- The most distant observable standard candle is the Type 1a Supernova. These end-of-life explosions are all instigated in the same way.
- A star in a binary system has evolved off the main sequence to become a white dwarf. The companion star swells up as it goes through its red-giant stage. This allows material from the red giant to be gravitationally captured by the white dwarf. The white dwarf’s mass increases until it reaches the critical mass for a supernova explosion, the Chandrasekhar limit of 1.44 solar masses.
- This regularity in the formation process produces a supernova with a known peak absolute magnitude (M) of -19.5.
- The distance to the supernova may be determined, again, by measuring apparent magnitude and using the Distance Modulus formula (refer Distance Modulus and Distance https://www.asnsw.com/i2ao2b#dm).
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Author: Lesa Moore, 15th April 2024