## Distances and sizes Distance ladder

The relative sizes of some astrophysical objects and the distances to them are listed in Table 2 and illustrated on logarithmic scales in Fig. 2. The large ranges mentioned above are very evident here. For example, the ratio of the size of the observable

266 9 Properties and distances of celestial objects Table 9.2. Size (radius) and distance examples

Object

Neutron star

Moon

Earth

Solar system Nearest star (Prox. Cen) Crab nebula Center of the Galaxy Galaxy (Milky Way) Andromeda galaxy, M31 Virgo cluster of galaxies 3C273 quasar Observable universe

Radius of object (m)a

3 x 103 (3 km) 1 x104 (10 km) 1.7 x 106 (1738 km) 6.4 x106 (6371km) 6.96 x 108

6 x 1012 (Pluto orbit)

-5 x 1020 (50 kLY) -7 x 1020 (70 kLY) 5 x 1022 (5 MLY)

Distance to object (m)a

4.1 x 1016 (4.3 LY) 6 x 1019 (6 kLY) 2.4 x 1020 (25 kLY)d

2.4 x 1022 (2.5 MLY)e 4.9 x 1023 (52 ± 5 MLY) f 2.3 x 1025 (2.4 GLY)« ~1.4x 1026 (15 GLY)h a Values are from C. W. Allen, Astrophysics Quantities, 3rd Ed., Athlone, 1976 and the 4th Ed., ed. A. N. Cox, Springer Verlag/AIP, 2000 or Zombeck, Handbook of Space Astronomy & Astrophysics, 2nd Ed., Cambridge University Press, 1990, unless otherwise referenced.

b The moon distance varies by 11% due to its elliptical orbit.

c Mean value = semimajor axis of Earth orbit; the earth-sun distance varies by 3% due to the earth's elliptical orbit. d Nominal value used for this text; result from Fig. 5 data is 23 ± 5 kLY. e Freedman and Madore, ApJ 365, 186 (1990). f Jacoby et al, Publ. Astron. Soc. Pacific 104, 599 (1992).

« Extragalactic distances are referenced to a Hubble constant of 20 km s-1 MLY-1 (65 kms-1 Mpc-1) and q0 = 1 with relation D = cz/H0. A recent "best fit" value to available data is 71 ± 3 km s-1 Mpc-1; Spergel et al., ApJ (2003); astroph/0302209. h Hubble distance R = c/H0; approx. distance to the observable limit of the universe.

universe (1026 m) to that of a neutron star (104 m) is equal to the ratio of the sizes of the earth (107 m) and the proton (10-15 m), a factor of 1022 in each case. Distances can be given in meters, light years, or parsecs.

The sizes of the earth, the sun, the Galaxy, and the observable universe serve as excellent benchmarks as do the distances to the moon, the sun, the nearest star, and the center of the Galaxy. You would do well to memorize them. Try to picture the relation between the sizes and spacings. For example, if the sun were the size of a soccer ball, how big would the similarly scaled earth be, and how distant would it be from the soccer ball? I bet the answers surprise you!

The distance to a celestial object is difficult to obtain, in principle, because the sky appears two-dimensional to us. The angular position of a star on the celestial sphere requires only the two angular coordinates. Indirect means must be used to obtain the

Figure 9.2. Sizes (actually radii) and distances of astronomical objects on logarithmic scales. The "parsec" is often used as a distance measure in astronomical literature; in this text we choose to use light years (LY). The logarithmic range of sizes is compared to human and subatomic sizes.

Figure 9.2. Sizes (actually radii) and distances of astronomical objects on logarithmic scales. The "parsec" is often used as a distance measure in astronomical literature; in this text we choose to use light years (LY). The logarithmic range of sizes is compared to human and subatomic sizes.

third coordinate, the distance. There is a so-called distance ladder whereby different techniques and objects are employed for different regimes of distance.

The ladder begins with the closer objects. Based on these distances, another technique or type of object (a standard candle) is used to obtain distances of more distant objects. For example, if the distance to a cluster of stars in the Galaxy is determined, the luminosity of any star in it is easily obtained from the measured flux. If one of the stars is highly luminous and recognizable due to its spectrum or oscillations, that type of star could be used as a standard candle to obtain the distances of galaxies in which the star is found. This is done by comparing the observed flux and luminosity of the standard candle; see (8.9). As one works out to greater and greater distances, the accuracy of the later steps depends on the accuracy of the previous steps. Uncertainties can thus be large.

Moon, earth, and sun

The distance to the moon from the earth center was determined with poor accuracy by the Greek astronomer Aristarchus (~270 BCE) from timing of lunar eclipses.

This method was improved upon by a later Greek astronomer, Hipparchus (~140 BCE). Ptolemy (second century CE) used a different method; he measured the parallax (Section 4.3) of the moon relative to the distant stars. He noted that the apparent position of the moon in the sky is different for two well separated observers on the earth. A single observer can perform the experiment by letting the earth carry her to a different position as it rotates, but she must subtract the effect of the moon's orbital motion during the interval. The closer the moon is to the earth, the more it will appear to be displaced from its expected track through the stars as the observer rotates with the earth.

Ptolemy obtained a lunar distance of 59 times the earth radius; the actual mean value is 60.3. This is quite good agreement considering that the moon's distance varies by ~ 12% due to its elliptical orbit. The absolute distance of the moon follows if the radius of the earth is known.

Indeed, the radius of the earth had been found several centuries earlier by Eratosthenes. He assumed that the rays of the sun arrive as a plane wave (as if from a distant point source). Since the earth is indeed spherical, observers at two locations would (in most cases) observe the sun to be at different elevations (angle above the horizon) at the same time. If the observers are directly north-south of one another by a known distance D, the difference between the two measured elevations Ad of the sun (at the same time) will yield the radius R of the earth according to R = D/A0.

The distance to the sun is found most directly by the measurement of the speed of the earth in its orbit. This is found by spectroscopic observations of stars in the forward and backward directions. Stars exhibit spectral lines (Section 11.4) that can be measured to obtain the frequency v of the radiation. If, at one point in its orbit, the earth is moving directly away from a star with speed vr, the spectral line from that star will be Doppler shifted to lower frequency, or toward the red. The Doppler frequency shift Av is related to vr for non-relativistic speeds as vr v — v0 Av

c v0 v where vr is the earth speed, c is the speed of light, v is the observed frequency, and v0 is the rest frequency that would be observed by an observer at rest relative to the emitting or absorbing atom. A measurement of Av and v thus yields vr.

The studies of stars in many directions surrounding the earth show red and blue shifts that oscillate on an annual basis due to the earth's orbital motion. (There are also constant shifts due to the sun's motion relative to the other stars; see below.) From the oscillations of stars located in a number of different directions, one can deduce the orbital velocity vE of the earth as well as the period PE of the earth's orbit. Of course, the period is also known from observations of the yearly apparent motion of the sun through the constellations and from the annual north-south motion of the sun relative to the earth.

If the earth's orbit is perfectly circular (an approximation) at a distance rAU from the sun, the distance around the orbit is

The two factors ve and PE together yield the radius of the earth's orbit about the sun. In fact, though, the elliptical orbit must be taken into account. Measurement of the changing speed ve and changing angular size of the sun reveal the parameters of the orbit including its semimajor axis a. The semimajor axis of the orbit is known as the astronomical unit (AU). As quoted previously (4.7), a = 1.496 x 1011 m (9.18)

This is the desired distance to the sun.

### Trigonometric parallax

The technique of trigonometric parallax yields the distances of the closer stars. As described in Section 4.3, nearby stars appear to move relative to the more distant stars as the earth moves about the sun (see Fig. 4.2). This motion appears as a circle, ellipse, or line depending on the star's location relative to the plane of the earth's orbit about the sun. As noted, the distance r to a star in the direction normal to the ecliptic may be expressed in terms of the semimajor axis of the nearly circular track on the sky, 0par, and the semimajor axis of the earth's orbit. From (4.5), a 1.5 x 1011m r =-~--(m; parallax distance) (9.19)

tan 0par 0par (rad)

where tan 0par ~ 0par for the small angles involved. The distances may be expressed in meters or AU.

The distance r may be expressed in parsecs. Rewrite (19) with a = 1.0 AU and the angle in arcseconds to obtain the definition

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