Astronomy Binder Bloomington High School South 2011 Contents 1 Astronomical Distances 2 1.1 Geometric Methods . 2 1.2 Spectroscopic Methods . 4 1.3 Standard Candle Methods . 4 1.4 Cosmological Redshift . 5 1.5 Distances to Galaxies . 5 2 Age and Size 6 2.1 Measuring Age . 6 2.2 Measuring Size . 7 3 Variable Stars 7 3.1 Pulsating Variable Stars . 7 3.1.1 Cepheid Variables . 7 3.1.2 RR Lyrae Variables . 8 3.1.3 RV Tauri Variables . 8 3.1.4 Long Period/Semiregular Variables . 8 3.2 Binary Variables . 8 3.3 Cataclysmic Variables . 11 3.3.1 Classical Nova . 11 3.3.2 Recurrent Novae . 11 3.3.3 Dwarf Novae (U Geminorum) . 11 3.3.4 X-Ray Binary . 11 3.3.5 Polar (AM Herculis) star . 12 3.3.6 Intermediate Polar (DQ Herculis) star . 12 3.3.7 Super Soft Source (SSS) . 12 3.3.8 VY Sculptoris stars . 12 3.3.9 AM Canum Venaticorum stars . 12 3.3.10 SW Sextantis stars . 13 3.3.11 Symbiotic Stars . 13 3.3.12 Pulsating White Dwarfs . 13 4 Galaxy Classification 14 4.1 Elliptical Galaxies . 14 4.2 Spirals . 15 4.3 Classification . 16 4.4 The Milky Way Galaxy (MWG . 19 4.4.1 Scale Height . 19 4.4.2 Magellanic Clouds . 20 5 Galaxy Interactions 20 6 Interstellar Medium 21 7 Active Galactic Nuclei 22 7.1 AGN Equations . 23 1 8 Spectra 25 8.1 21 cm line . 26 9 Black Holes 26 9.1 Stellar Black Holes . 26 9.2 Super-massive Black Holes . 27 9.3 Mid-Sized black holes . 27 9.4 Micro-Black Holes . 27 10 Fates of Massive Stars and Supernovae 28 10.1 Supernovae Classification . 28 10.2 Process of core collapse . 29 10.3 Gamma Ray bursts . 30 10.4 Cosmic Rays and Background Radiation . 30 11 Red Giants 31 12 Star Formation 31 13 Classification of other DSOs 33 13.1 Globular Clusters . 33 13.2 Open Clusters . 33 13.3 Planetary Nebulae . 33 14 Special Relativity 34 15 Miscellaneous 35 15.1 The Moon . 36 15.2 Solar System, Orbits, and the Sky . 36 1 Astronomical Distances 1.1 Geometric Methods Trigonometric Parallax If one compares an object to an infinitely distant background from two different viewpoints, the object appears to move less with respect to the background between the viewpoints if it is more distant. Parallax can be measured using radio waves to achieve higher accuracy. 1 µ = d , where µ is parallax in arcsec and d is distance in parsecs. Range: 0 - 100 pc Angular Diameter If one knows the actual size of an object and its angular diameter, one can find its distance. tan θ r = 2 d 2 Where θ is angular diameter, r is actual radius, and d is distance Moving Cluster Method Suppose a star cluster exists where all of the stars can be assumed to be traveling in the same direction, but appear from earth to be converging on a single point (just as parallel lines in Euclidean geometry appear to meet at infinity even though they don't). The angle θ between the cluster and this convergence dθ point can be measured, as can the rate of change dt . Also, the initial radial velocity can be found via Doppler shifts. If the tangential (orthogonal to radial) velocity can be found, it can be compared with dθ dt to give distance. This method is obsolete as its range is not significantly more than that of standard parallax, and the clusters must be very tight. It has only been successfully used once to measure distance to the clusters Hyades and Pleiades. However, it has appeared in several textbooks. −1 vt vt vt(kms ) tan θ = and d = dθ or D(pc) = 00 −1 vr dt 4:74u( year ) Specular Parallax This method attempts to expand the range of the trigonometric parallax method by increasing the distance traveled by the earth to more than 2 au. Specifically, it takes advantage of the sun's motion. au The sun moves 4.09 year with respect to the local standard of rest. If the group of stars one is measuring the distance to can be assumed to have an average motion of zero with respect to the local standard of rest, distance can be measured. There may be mathematical issues with the accuracy of this method in some circumstances. distance traveled by sun sin θ = distance to star Expansion parallax If an object is expanding uniformly, one can find the velocity of expansion using Doppler shifts. Then, if the apparent angular expansion is known, distance can be found. Useful for SN remnants but can be inaccurate if assumption of uniform expansion is not met. vexp d = dθ dt Light Echo A ring around a bright object which emits pulses will be illuminated by the object intermittently. If the time delay can be measured, the radius of the object can be found, which yields the distance when compared with angular diameter. t2 represents the time taken for light traveling to the farthest point on the ring to reach earth, t1 represents the time taken for light traveling to the nearest point on the ring to reach earth, and t0 represents the time taken for light traveling in a straight line from the object to reach earth. r (1 − cos i) r (1 + cos i) c (t − t + t − t ) r t − t = and t − t = so r = 1 0 2 0 ! d = 1 0 c 2 0 c 2 θ 3 1.2 Spectroscopic Methods Absorption Spectra One can use the absorption spectra of a star to find its spectral class and then absolute magnitude. With the apparent magnitude, one can find the distance to the star. m − M = 5 log d − 5 H-R diagram The apparent magnitudes of all stars in a cluster are plotted with surface temperature (calculated from absorption spectra). This H-R diagram is compared with the H-R diagram based on apparent magnitudes thus allowing the distance of the cluster to be found Equations: Same as above; Range 100 to 10,000 pc. Baade-Wesselink Method This applies to pulsating stars only. It involves setting up a ratio of radii at two points in a star's period. Once this has been achieved, the difference of radii can be found using Doppler shifts of the star's layers and the time taken for those shifts. The resulting equations can be solved for radius rt1 θt1 = and rt1 − rt2 = vave∆t so d = r/θ rt2 θt2 1.3 Standard Candle Methods Cepheids and RR Lyrae Cephied variable stars have a known period-luminosity relation, and RR Lyrae stars have a constant absolute magnitude. The apparent magnitudes can be used to find distances; this method is effective from 100 to 107pc. Cephieds: Mv = −2:76 (log(P ) − 1) − 4:16 RR Lyrae: Mv = :75 Type Ia Supernovae The absolute magnitude of any type Ia supernovae is -19.3, and this can be used to find distance. Range: > 106pc. 2 4 For SN events, L = 4π(vejectiont) σT Also, it is possible to match the shape of a type Ia light curve to find mv even if the peak was not itself recorded. Novae: there exists a relationship between M and the time needed to drop a given number of magnitudes: dm M max = −9:96 − 2:311 log v dt 4 dm where dt is the average rate of decline of the magnitude for the first two magnitudes. Other possibilities: Mv for the three brightest stars is -8 Mv for globular clusters is -6.6 Mv for plaetary nebulae at 5007A˚ is -4.53 1.4 Cosmological Redshift Hubble's Law The fabric of the universe is expanding. Objects that are further away have more space between them and us. Since that space is expanding, further objects appear to be receding faster. This gives them a large redshift, which can be used to calculate distances, for objects sufficiently (> 106pc) far away. Note that K corrections must be used to account for the redshift of light from distant galaxies (i.e. received IR radiation may not actually be from the IR portion of the spectrum). km v ( ) = 70d (Mpc) s 1.5 Distances to Galaxies The following methods make use of rearrangements of the following three equations, with the assumption M of a constant surface brightness for all galaxies and also the assumption of a constant L ratio: r θ = d L f = 4πd2 GM v2 = R f Note that surface brightness, defined as θ2 , is constant regardless of distance because, if multiplied by 1 2 1 x, f is multiplied by x2 , but θ is cut by a factor of x, and θ is therefore multiplied by x2 as well, maintaining a constant ratio. mag Surface Brightness S ( ) = M + 2:5 log Area (in arcsec)2 arcsec2 mag L Unit Conversion: S ( ) = 26:402 − 2:5 log S ( sun ) arcsec2 pc2 Tully-Fisher Relation The rotational velocity of spiral galaxies is related to their luminosity. This relation's range is 106 to 109pc. 4 L α vmax 5 For Sa, MB = −9:95 log Vmax + 3:15 For Sb, MB = −10:2 log Vmax + 2:17 For Sc, MB = −11:0 log Vmax + 3:31 Faber-Jackson Relation The velocity dispersion of an elliptical galaxy is proportional to the 4th root of its luminosity. This relation's range is 106 to 109pc. L α σ4 log σ = −:1MB + :2 The Fundamental Plane The Faber-Jackson relation is imprecise.