The Memphis Astronomical Society Presents A SHORT COURSE in ASTRONOMY

CHAPTER 8 MEASURING DISTANCES in the MILKY WAY Dr. William J. Busler Astrophysical Chemistry 439

MEASURING DISTANCES in the MILKY WAY Overview: In this chapter, we will explore quantitatively some of the concepts covered in a more narrative fashion in the previous chapter (“The Early History of Astronomy”).

A. A. Measuring the Distances in the Sun-Earth-Moon system Recall from Chapter 7 that Aristarchus ( BC) made the first attempt at measuring the scale of the Sun-Earth-Moon system, using units based on Earth’s diameter. He figured (correctly) that when the Moon is at its First Quarter phase, it is not at right angles to the Sun in the sky, unless the Sun were an infinite distance away. (Refer to the diagram on the next slide.)

A. A. Measuring the Distances in the Sun-Earth-Moon system Aristarchus figured that when the Moon is at its First Quarter phase, it is not at right angles to the Sun in the sky, unless the Sun were an infinite distance away. (Refer to the diagram below.) Last Q. 1st Q. Sun Moon Earth

A. A. Measuring the Distances in the Sun-Earth-Moon system Aristarchus tried to measure the Sun-Earth-Moon angle at First Quarter, but found it was impossible to measure any difference from 90  using the instruments available to him. He then surmised (correctly) that it should take longer for the Moon to move from its First Quarter position to its Last Quarter position than to move from the Last Quarter back to First Quarter.

A. A. Measuring the Distances in the Sun-Earth-Moon system After years of observations, which were made difficult by inaccuracies in clocks and the ellipticity of the Moon’s orbit, he detected an average time difference of about an hour in the two halves of the lunar month. (The actual difference is 1 hour, 4 minutes, and 38 seconds.) Using his unreliable data, he calculated the various distances and sizes in the table on the next slide:

A. A. Measuring the Distances in the Sun-Earth-Moon system Aristarchus’ various lunar distances and sizes: Moon’s Distance Moon’s Diameter Sun’s Distance Sun’s Diameter Aristarchus 10 Earth Diameters Modern ,

A. A. Measuring the Distances in the Sun-Earth-Moon system Of course, we would now consider these results to be seriously inaccurate. However, these were probably the first astronomical measurements ever made. As such, they were quite an accomplishment for the time of Aristarchus.

B. B. Measuring the Diameter of the Earth Eratosthenes ( BC) was a Greek astronomer who made the first attempt to measure the diameter of the Earth. He noticed that the Sun would shine directly down a well on the first day of summer at Syene (modern Aswân), while on the same day and time, the Sun was 7.2  south of the zenith at Alexandria, about 500 miles north of Aswân. Eratosthenes believed that the Earth was spherical and that the Sun was so far away that all of its rays which strike the Earth are essentially parallel.

B. B. Measuring the Diameter of the Earth Eratosthenes made the following calculation: 7.2  / 360  = 500 miles / x; x = 25,000 miles = the circumference of the Earth. 25,000 miles /  = 7960 miles = the diameter of the Earth. (Accepted value = 7930 miles.) Center of EarthSyene From Sun 7.2° Alexandria

C. C. Finding the Distance to the Moon by Parallax This method was developed by Ptolemy, about 140 AD. Parallax is the phenomenon by which a nearby object appears to shift its position against remote background objects when the observer changes position. (Try holding one finger about 12 inches in front of your nose; look at it first with one eye, then the other; notice the shift in your finger's apparent position against a distant background.)

C. C. Finding the Distance to the Moon by Parallax In this method, two observers simultaneously observe the Moon against the starry background from positions 6000 miles apart. The Moon’s position against the stars differs by 1.4 . To calculate the Earth-Moon distance, X: sin (0.7  ) = 3000 miles / X; = 3000 miles / X; X = 3000 miles / = 246,000 miles miles X Earth Moon 0.7  **********

C. C. Finding the Distance to the Moon by Parallax In actual practice, two simultaneous observations are not necessary. The rotation of the Earth during the night will carry a single observer from one point to the other. During this time, however, the Moon will have moved a short distance in its orbit. This motion must be subtracted before the true parallax can be determined miles X Earth Moon 0.7  **********

D. D. The Scale Model of the Solar System In his book De Revolutionibus Orbium Cœlestium, published in 1543, Copernicus calculated and tabulated the distances of the planets from the Sun in terms of the Earth-Sun distance (AU). To do this, he used the time it took for each planet to move from opposition (or conjunction) to quadrature – right angles to the Sun. Since he knew the sidereal periods of the planet and of the Earth, he could calculate the fraction of a complete orbit which had been traversed, and thus the angle with the Sun.

D. D. The Scale Model of the Solar System From there, it was a matter of simple geometry to calculate the planet’s distance from the Sun compared to the Earth’s. Mercury Venus Earth Saturn Mars Jupiter Planet Copernicus Modern

E. E. Using Kepler’s 3rd Law to Calculate Sun-Planet Distances Recall from Chapter 7 that Kepler had used Tycho’s data to formulate his three laws of planetary motion. Once these mathematical laws had been obtained, it was possible to make exact calculations which were not dependent on any uncertainties in the original data. For example, once a planet’s sidereal period (year) is known, its exact distance from the Sun can be calculated using the third (harmonic) law:

E. E. Using Kepler’s 3rd Law to Calculate Sun-Planet Distances For Mars, the orbital period (P) is 1.88 years. P 2 = D = D = D 3. D = = 1.52 AU. This remarkable equation allows the distance of any planet to be calculated as accurately as its sidereal period is known.

E. E. Using Kepler’s 3rd Law to Calculate Sun-Planet Distances However, an error (of less than 1%) is introduced for the most massive planets, Jupiter and Saturn, because Kepler’s equation neglects the mass of the planet orbiting the Sun. Newton made this correction years later, but for small bodies orbiting the massive Sun, it is not really necessary.

F. F. Newton’s More General Form of Kepler’s 3rd Law In the Principia (1687), Newton explained and derived Kepler’s laws from fundamental principles of celestial mechanics, not observational data. The general (i.e., good for all cases) version of the third (harmonic) law is as follows: (m S + m P )  P 2 = (D S + D P ) 3, where m S and m P are the masses of the Sun and the planet (in units of solar masses), and D S and D P are the distances of the Sun and planet from their common center of mass (in AU).

F. F. Newton’s More General Form of Kepler’s 3rd Law (m S + m P )  P 2 = (D S + D P ) 3 When the planet is much smaller than the Sun (e.g., Mars), m P and D S may be dropped, and the equation becomes identical to Kepler’s third law. For a very massive planet (e.g., Jupiter or Saturn), this equation removes the discrepancies observed between observational data and Kepler’s third law.

G. G. Measuring Distances to Stars by the Parallax Method The parallax method used for the Moon won’t work for stars; the baseline (the diameter of the Earth) is too short. Instead, the baseline must be the diameter of the Earth’s orbit. Even with a baseline of 186 million miles, the parallax is very small – generally less than 1 arc- second! Recall that many pioneer astronomers ruled out a moving Earth because no stellar parallax could be seen as a result of the Earth’s revolution.

G. G. Measuring Distances to Stars by the Parallax Method There is a detailed example with an explanatory drawing of the parallax method in Chapter 9, pages 3 and 4. It is strongly suggested that you study that example at this time.

DISTANCE DETERMINATION by PARALLAX Recall that as the Earth revolves around the Sun, it changes its position in space by a distance of 2 astronomical units million miles. This causes the nearer stars to exhibit parallax, i.e., to appear to move back and forth slightly against the background of the more distant stars. (Ch. 8.) Through the use of trigonometry, parallax measurements allow us to calculate the distances to some of the “nearer” stars -- those within a few hundred light-years of our Solar System.

The annual parallax of a star depends upon its distance from the Solar System: The farther away it is, the smaller the angle of parallax.

Usually, the star is observed from two points in the Earth’s orbit 6 months apart. This provides a baseline of 2 astronomical units (186 million miles) for triangulation. This drawing is not to scale! The parallax angle for even the nearest stars is less than 1 second of arc ( 1 / 3600 of one degree).

Suppose a star is at a distance such that its parallax is exactly 1" either side of its average position (i.e., 2" total) when viewed from diametrically opposite points in the Earth’s orbit. This distance is known as one parsec (from parallax-second). We can easily calculate the distance of the star, using trigonometry:

tan  = ; tan 0  0'1" = tan   =  = x = = 1.92  miles = 19.2 trillion miles = light-years = 1 parsec. Opposite  Adjacent

G. G. Measuring Distances to Stars by the Parallax Method An additional example: Sirius exhibits a total annual parallax of 0.77  ; in other words, its true parallax, based on the radius of the Earth’s orbit of 93 million miles, is . What is its distance in light-years and in parsecs? Easy Solution: 1 / = 2.60 parsecs; 2.60 psc  l-y / psc = 8.5 light-years.

G. G. Measuring Distances to Stars by the Parallax Method Same problem; direct trigonometric solution: Sirius exhibits a total annual parallax of 0.77  ; in other words, its true parallax, based on the radius of the Earth’s orbit of 93 million miles, is . What is its distance in light-years and in parsecs? tan (0.385  ) = 9.3  10 7 miles / X   1  /3600  =  . X = 9.3  10 7 miles / tan (   ) = 9.3  10 7 miles /  =  miles.

G. G. Measuring Distances to Stars by the Parallax Method  miles  1 l-y / 5.87  miles = 8.5 light-years; 8.5 light-years  1 psc / l-y = 2.60 parsecs.

G. G. Measuring Distances to Stars by the Parallax Method Although Bessel (1838) used visual observations to measure the parallax of 61 Cygni, it is preferable to use photographic methods now that they are available. It is possible to measure stellar distances up to about 200 light-years with about 20% accuracy using the parallax method. Computer measurement of photographic plates and the Hubble Space Telescope have extended the reach of the parallax method.

H. H. Other Methods of Determining Stellar Distances 1. Radial Velocity / Proper Motion: Most stars have some intrinsic motion (space velocity); they wander about through the Galaxy. If we assume that all stars have about the same space velocity (risky!), then those with the most apparent motion against the background of very remote stars are probably the closest. This motion may be across the sky (proper motion); it may be detected by comparing a series of photographs taken over long periods of time.

H. H. Other Methods of Determining Stellar Distances 1. Radial Velocity / Proper Motion: If the motion is along our line of sight (radial velocity), it may be detected by looking for Doppler shifts in the star’s spectral lines. The spectral lines will be shifted towards the blue if the star is approaching; towards the red if it is receding. (There will be further discussion of Doppler shifts in Chapter 12.)

H. H. Other Methods of Determining Stellar Distances 2. Moving star clusters: If a star cluster is moving through space, it may be assumed that all of its stars have the same space velocity and direction. However, if the paths of the stars appear to be converging or diverging, this is a perspective effect; the greater the degree of convergence or divergence, the nearer the cluster. This method was useful for determining the distances to the Hyades and the five middle stars of the Big Dipper.

H. H. Other Methods of Determining Stellar Distances 3. Spectral Class / Inverse Square Law: In Chapter 9, we will see that if a star’s spectral class (essentially its color) is known, the Hertzsprung-Russell diagram may be used to provide a good estimate of its intrinsic luminosity. Comparing the star’s luminosity with its apparent brightness leads to the determination of its distance.

H. H. Other Methods of Determining Stellar Distances 4. Binary Star Systems: (Refer to Newton’s version of Kepler’s third law, in section F of this chapter.) We can measure the period of revolution (P) of the system from its light curve (eclipsing binaries) or from visual observations of the stars’ changing positions and separation. For stars whose distance from us is already known, we can easily obtain the true distance between them, and then calculate the sum of the masses.

H. H. Other Methods of Determining Stellar Distances 4. Binary Star Systems: If the distance is not known, we can estimate the masses from the spectral classes (Chapter 9), and then calculate the separation distance. By comparing this known separation with the angular separation in the sky, we can compute the distance of the system.

H. H. Other Methods of Determining Stellar Distances 5. Interstellar absorption: There is a slight amount of gas in interstellar space through which a star’s light must travel to reach us. The degree of absorption of a star’s light (as evidenced by the intensity of the absorption lines in its spectrum) is a function of its distance from us.

H. H. Other Methods of Determining Stellar Distances 6. Nova / Supernova light-shell front: After a nova or a supernova explosion, a “shell” of light expands away from the residual star at the speed of light. By measuring the angular expansion of the shell over a period of time, and knowing that the true velocity of expansion is the speed of light, the object’s distance from us can easily be calculated.

H. H. Other Methods of Determining Stellar Distances 7. Nova / Supernova Remnant Expansion: This is similar to the previous method, but uses the much slower expansion of the physical material in the remnant, rather than the light front. The true rate of expansion is nowhere near the speed of light; it may be estimated from the Doppler shift of material near the center of the remnant.

MEASURING THE S P A C E V E L O C I T Y OF A STAR

A star moves through space from point a to point b. What is its true “space velocity”? 1. Measure its “proper motion”: the angular speed across the sky (arc-sec/year). Measuring the “space velocity” of a star  Solar System a b 1

A star moves through space from point a to point b. What is its true “space velocity”? 1. Measure its “proper motion”: the angular speed across the sky (arc-sec/year). 2. Measure the distance to the star, using the parallax method. Measuring the “space velocity” of a star  Solar System a b 1 2

A star moves through space from point a to point b. What is its true “space velocity”? 1. Measure its “proper motion”: the angular speed across the sky (arc-sec/year). 2. Measure the distance to the star, using the parallax method. 3. This allows us to calculate its “transverse velocity”, the tangential component of its true “space velocity”: tan (or sin)  = Measuring the “space velocity” of a star  Solar System a b 1 2 3

A star moves through space from point a to point b. What is its true “space velocity”? 1. Measure its “proper motion”: the angular speed across the sky (arc-sec/year). 2. Measure the distance to the star, using the parallax method. 3. This allows us to calculate its “transverse velocity”, the tangential component of its true “space velocity”. 4. Using the Doppler shift, calculate the “radial (line of sight) velocity” of the star: v / c =   Measuring the “space velocity” of a star  Solar System a b 1 2 4

A star moves through space from point a to point b. What is its true “space velocity”? 1. Measure its “proper motion”: the angular speed across the sky (arc-sec/year). 2. Measure the distance to the star, using the parallax method. 3. This allows us to calculate its “transverse velocity”, the tangential component of its true “space velocity”. 4. Using the Doppler shift, calculate the “radial (line of sight) velocity” of the star. 5. Knowing the transverse and radial velocities, use the Pythagorean theorem to calculate the hypotenuse -- the “space velocity” of the star. Measuring the “space velocity” of a star  Solar System a b

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