Author: Star Life
Reviewer: 阿白特尔

  When we look up at the night sky under good conditions, we can see a sky full of stars. To understand these celestial objects, people observe and measure them, then use the resulting data in theoretical calculations to reach the conclusions they seek. The branch of astronomy concerned with observing stars, planets, deep-sky objects, and other bodies in the universe to obtain the quantities needed for such calculations is called astrometry. Today, let us look at how astronomers measure the distances to stars.

  We know that stars lie at different distances from the Sun, but all of them are extremely far away. Ordinary units of length used on Earth are therefore impractical for stellar distances, so we use other units instead. One is the light-year, defined as the distance light travels through a vacuum in one year—about 9.4605 trillion kilometers. Stellar distances are immense: even Proxima Centauri, the nearest star, is 4.2 light-years from the Sun, while other stars are hundreds, thousands, or even hundreds of millions of light-years away. The light-year is not our only unit. When measuring very distant stars and galaxies beyond the Milky Way, even a light-year can seem small, so astronomers established another, larger unit from observations: the parsec.

  The positions of stars that we see in the night sky are apparent positions—their projections onto the celestial sphere. As Earth travels around the Sun, the same star will appear in slightly different positions when Earth reaches opposite sides of its orbit. If we measure that difference, geometry allows us to calculate the star’s distance.

One arcsecond of parallax

  This change in a star’s apparent position caused by Earth’s revolution is called annual stellar parallax. Annual parallax also gives us a unit of length: if a star’s annual parallax is 1 arcsecond—an arcsecond is an angular unit equal to one three-thousand-six-hundredth of a degree—then the star is 1 parsec from Earth, or about 3.26 light-years or 30.8 trillion kilometers.

  There is a formula based on annual stellar parallax: the distance r from Earth to a star is 1/π, where π is half the star’s total annual parallax.

  Next, consider this question:

  An astronomical unit is the distance from Earth to the Sun, about 150 million kilometers—far less than one light-year. One parsec equals 3.26 light-years, so the answer is C. We generally use astronomical units for planets within the Solar System, and light-years or parsecs for stars and more distant galaxies.

  Once we know the units used for stellar distances, we can begin measuring those distances. With sufficiently precise observations, the method above lets us determine annual stellar parallax and then use the formula to calculate distance. This is called the trigonometric-parallax method. It played a major role in the early measurement of stars, but the farther a star is from us, the smaller its annual parallax becomes. The measurement grows more difficult and the error larger, until it is almost beyond our ability to measure. Trigonometric parallax is therefore unsuitable for distant stars, and other methods are needed.

  Another useful astronomical technique is spectroscopic parallax. It uses the empirical linear relationship between the intensity ratios of certain lines in a star’s spectrum and its absolute magnitude. Measuring the ratios of selected pairs of spectral lines yields the absolute magnitude, which can then be used to find the distance. Absolute magnitude represents a star’s intrinsic brightness, while observations tell us how bright it appears from our location. Because brightness decreases with distance, those two brightness values allow us to calculate the distance. Spectroscopic parallax is fairly accurate and has its own formula. After the absolute magnitude has been found from the spectral-line ratios, the distance d follows from the distance-modulus equation mv − Mv = 5lgd − 5.

  For example, suppose an observed star has an apparent magnitude of +15 and its spectrum identifies it as a G2 V star. The H–R diagram then gives it an absolute magnitude of +5. The distance-modulus formula gives d=1000 PC = 3260 l.y. (light-years).

  Note that l.y. means light-year, while the symbol for parsec is p.c.

  Spectroscopic parallax can accurately measure the distances of many stars. The Cepheid-variable method is another common technique and is specifically suited to Cepheid variables. A Cepheid’s pulsation period—the time required for one cycle of brightness variation—is proportional to its luminosity, so these stars can be used to measure interstellar and intergalactic distances. Most Cepheids are F-type stars at maximum luminosity, meaning moderately hot stars, and G-type stars at minimum luminosity, meaning cooler stars like the Sun. The archetype is Delta Cephei. John Goodricke discovered its variability in 1784, and in 1912 Henrietta Leavitt of the Harvard College Observatory discovered the period–luminosity relationship described above. Cepheid variables are therefore known as “cosmic yardsticks.” These are several of the methods used to measure stellar distances, and each has its own formulas and calculations. Astronomers have many other methods as well, each suited to a different range.

  For this question, the properties of trigonometric parallax tell us that the farther a star is from us, the smaller its annual parallax will be. This is one reason the method cannot measure very distant stars. The image shows that star B has the smaller annual parallax, so the answer is B.

  It is easier to measure the annual parallax of a nearby star, so the answer is A.

  These are several ways of measuring stellar distances. The vast universe is waiting to be explored—so what are we waiting for? Let us get started!