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 ...... 5

2 Age and Size 6 2.1 Measuring Age ...... 6 2.2 Measuring Size ...... 7

3 Variable 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 ...... 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) ...... 12 3.3.6 (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 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, , 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 . 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 exists where all of the stars can be assumed to be traveling in the same direction, but appear from 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 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( )

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 ’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 . With the , 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- 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 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. Tighter fits can be obtained by introducing the effective radius (the radius by which half of the galaxy’s light is emitted), which depends on the surface brightness at the effective radius I. 2.65 .65 −.82 1.24 L α σ re and re α Ie σ

The Globular Cluster Luminosity Function The brightness of globular clusters appears to be N(−7.2 mag, 1.1 mag). If more data support this assertion, globular clusters can be used as standard candles.

2 Age and Size

2.1 Measuring Age

9 F e For stars younger than age of the universe − 10 (when Fe was first formed in stars), use the He ratio, also known as the  F e  N  N  = log F e star − log F e sun He NH star NH sun Where N is the amount of the specified element per unit volume. Old stars have values around - O 5.4 and young stars have values closer to .6. H ratios can also be used, but only for stars of age age of the universe − 107 years. Fe is typically used since it’s easier to measure. As a reference, hydrogen represents 91.2% of the number of atoms and 71.0% of the total of the sun, while iron represents .0030% of the number of atoms and .14% of the total mass.

Note that metallicity is sometimes alternatively defined as the ratio of metals in a star to the ratio of nonmetals.

6 2.2 Measuring Size

The fractional change λ for large redshifts is equal to the fractional change in the universe’s size from the time when the radiation was emitted. radius obs = 1 + z radiusemitted Therefore, to scale space (so that the density of objects can be compared), divide the number of objects in the past by (1 + z)3. For example, if there is a redshift of 3, the universe is 1 + 3 = 4 times as large now.

3 Variable Stars

3.1 Pulsating Variable Stars

3.1.1 Cepheid Variables

Population I Cepheids (classical Cepheids) are young, metal rich stars. They are found in Magellanic clouds, and disks. Population II Cepheids (W Virginis Cepheids) are old, non-massive, metal poor stars. They are found in globular clusters and halos.

These populations follow slightly different period-luminosity relationships. In general, the luminosity of population II cepheids is 1.5 magnitudes less (meaning +1.5 magnitudes) than that of population I cepheids for a given period.

Historical Background: Before 1942, astronomers failed to realize the difference between cepheids in different populations and between cepheids and RR lyrae variables. Walter Baade realized that the cepheids in Andromeda (population I) upon which distance measures were based were not of the same brightness as cepheids which the period-luminosity relation was then calibrated on (population II). A difference of 1.5 magnitudes corresponds to a factor of four change in luminosity, or, in other words, a factor of 2 difference in distance (essentially, m − M was decreased by 1.5 to eliminate the error. Hence, the distance estimate of Andromeda and every other extragalactic object was doubled.

Mechanisms of Pulsation: As heat leaves the center fo the star, He+ is ionized to He+2. This results in more opacity, reducing the rate of heat transfer from the center, resulting in an increased pressure. This results in an increase in size. Of course, the increase in size leads to decreased temperatures, and some He+2 gains an electron to become He+. The opacity decreases, as does pressure as the star shrinks. Because the temperature is greater when the star is smaller, cepheids are brighter when they are smaller.

Trivia: The precision of the period-luminosity relation is much greater in IR due to interstellar absorption disrupting blue wavelengths.

7 3.1.2 RR Lyrae Variables

These are horizontal branch population II stars of spectral class F or A burning He in their cores. Originated from stars of about .8 to 1 Msun but have lost mass so that they have only about .5 Msun when they are experiencing variability. They are more common than cepheids. Most known examples are in globular clusters. Three types exist: RRab-the majority, with steep rise in brightness (about 91%, normal pulsation, enter from the red side of the HR diagram); RRc-shorter periods, more sinusoidal variation (about 9%, first overtone pulsation-a shell of material expands and then contracts inside the star, enter from the blue side of the HR diagram); RRd-rare double-mode pulsators (both pulsation modes). The period of pulsation is .2 to 2 days with an amplitude of .3 to 2 magnitudes. Some RR Lyraes experience the Blazhko Effect: A secondary pulsation resulting in, essentially, 2 periods (a common, short one and also some long term variation 60 days). It is explained by either a double mode pulsator (2 frequencies of pulsation) or by magnetic field distortions resulting from oblique rotation.

3.1.3 RV Tauri Variables

Post-AGB supergiants which experience radial pulsation. RVa variables do not change in mean bright- ness; RVb variables do experience mean brightness changes over 600 to 1500 timescales. The period of these variables is 30 to 150 days; the variation in magnitude is about 4. Spectral class ranges from F to M. These stars emit strongly in IR and can be either population I or population II, with the latter being more common. In the future, RV Tauri stars will eject their outer layers to form planetary nebulae and the cores will become white dwarfs.

3.1.4 Long Period/Semiregular Variables

These stars are red giants or AGB stars of a red or orange coloration experiencing pulsation (note: not necessarily radial pulsations-some examples are more egg-shaped). Miras (1sun) are classic LPVs, and have periods of 80 to 1000 days and vary in magnitude by 2.5 to 11 mag. Semiregular variables (such as Betelgeuse) have similar characteristics, but these are much less defined, so it is more difficult to determine these characteristics accurately. The sun will likely experience pulsation as a LPV when it is a red giant before becoming a .

3.2 Binary Variables

Binary systems are classified as follows: Visual Binaries: the angular seperation between the two stars is great enough for both stars to be resolved visually. Eclipsing Binaries: If the orbital plane of the system is parallel to the observer’s line of sight, there will be periodic changes in the amount of light received by observers on earth from the system. Spectroscopic Binaries: The binary system is detected because the visible star’s absorption spectrum is seen to be red shifted and then blue shifted on regular intervals, thus indicating its around a

8 center of mass. Spectrum Binary: Similar to spectroscopic binaries, except each star’s spectrum can be individually resolved; this occurs when neither component is significantly more massive than the other Eclipsing Binaries: The stars lie in a plane close enough to the observer’s line of sight that the stars eclipse each other thus periodically diminishing the observed luminosity of the system. Astrometric Binaries: When the parallax of a star can be measured to determine its position, it can occur that the star appears to be orbiting an empty space. This is the result of a much dimmer companion star.

Possible system configurations are as follows: Detached Binaries: Each star is completely contained within its own and evolves indepen- dently to its companion star. Semi-Detached Binaries: One of the stars in the system completely fills its Roche Lobe. This causes mass to be accreted onto the other star and creates an around the mass-acceptor. Contact Binaries: Both stars in the binary system completely fill their Roche Lobe and share a gas envelope. Most contact binaries are eclipsing binaries and often the stars end up merging.

Mass determinations Because the center of mass of the system is not accelerating due to the com- ponents of the system, the following relation holds: m r a α v 1 = 2 = 2 = 2 = 2 m2 r1 a1 α1 v1 The second to last equality is based on the small angle approximation and the preceding ratio of semi- major axes, and the last assumes circular orbits.

But we also have 4π2 P 2 = a3 G (m1 + m2)

So for Visual Binaries, solve the preceding two equations simultaneously for m1 and m2.

If we only have velocity curves, note that

 2 2 v3P 1 + m1 Gm1m2 m1v1 1 m2 2 = → m2 = (a1 + a2) a1 2πG

If the (defined as the angle between the plane of the sky and the plane of the orbit; ideally 0 in the following formula) is nonzero, then observed values will be the projection of the true values onto the plane of the sky. But this is irrelevant as far as mass ratios are concerned, as both observed angular separations will be off by the same factor of cos i, which cancels out of the equation. A larger problem occurs in Kepler’s law, which can be addressed as follows, using the fact

projskyα α˜ that α = cos i = cos i : 4π2 d !3 α˜3 m + m = 1 2 G cos i P 2

9 And for Spectroscopic Binaries, we can use the ratio of velocities1 (for the ratio, observed are fine, as any inclination cancels out) and the equation P (v + v )3 m + m = 1 2 1 2 2πG sin3 i to find the .

If we only know one velocity (if one star is so luminous as to block the other’s light), the mass ratio can be substituted for one of the velocities to get a mass function, which cannot give the masses but will allow one to set lower bounds: 3 m2 3 P 3 2 sin i = v1 (m1 + m2) 2πG

We can also derive much information from light curves alone. Let rs be the radius of the smaller star, rl be the radius of the larger star, ta be the time of first contact, tb be the time of minimum light, and tc the final time at minimum light (so if the minimum is only a point, tb = tc). Then v r = (t − t ) s 2 b a v v r = (t − t ) = r + (t − t ) l 2 c a s 2 c b where v = vs + vl is the relative velocity of the two stars.

And the temperature ratio is given by

B B − B 1 − p F 0 p = B0 = r s Bs B0 − Bs 1 − Fr l B0 where F = σT 4 and  2 2  B0 = k πrl Fr l + πrs Fr s 2 Bp = kπrl Fr l  2 2 2 Bs = k πrl − πrs Fr l + kπrs Fr s k is a constant depending on observing conditions, distance, etc. Note that these equations were written with a smaller, hotter, and brighter primary star being totally eclipsed by a larger, cooler, dimmer secondary, and that they may need to be tweaked in different circumstances. Also recognize that, when reading light curves, sharp points at minimums indicate departures from ideal inclinations or objects whose separation is comparable with their radii.

−Gm m Orbital Energy = 1 2 2a

2 P2  2 1  To find the relative velocity between objects in a system, use v = G 1 mi r − a where a is the semi-major axis and r is the current separation.

1When we write “v”, it is implied that this is the observed velocity if i is being taken into account, and if i is missing, then it represents the true velocity (if i = 90◦, then both are the same)

10 3.3 Cataclysmic Variables

3.3.1 Classical Nova

This is a class of nova where produces only one very bright outburst from 6-19 magnitudes that is caused by thermonuclear fusion of hydrogen-material accreted onto the surface of a white dwarf.

3.3.2 Recurrent Novae

Recurrent novae have multiple outbursts. The outbursts typically occur every 10 to 80 years with magnitudes ranging from 4 to 9. There are two main classes of recurrent novae: 1. An evolved secondary star which has lost most of its outer layers is transferring matter onto a hot, massive, white dwarf. Examples include U Sco, V394 CrA, or LMC 2990#2. 2. A red and a massive white dwarf where thermonuclear runaway on the white dwarf causes an outburst in the outer layers of the red giant.

3.3.3 Dwarf Novae (U Geminorum)

A class of cataclysmic variables where repeating outbursts of 2-5 magnitudes are observed with intervals between outbursts ranging from days to decades. Outbursts usually last from 2 -20 days. Dwarf Novae are thought to be caused by instabilities or sudden mass transfers in the accretion disk. There are three main subclasses: 1. Z Camelopardalis stars - exhibit standstills about 0.7 magnitudes below the maximum brightness. Outbursts cease during these standstills for tens of days to years. 2. SU Ursae Majoris stars - exhibit occasional superoutbursts which are typically 0.7 magnitudes brighter than normal outbursts. The outburst lifetime in these cases is on the order of 5 times the lifetime of a normal outburst. 3. SS Cygni - Undergo well-defined outbursts from 2-6 magnitudes. The outbursts typically reach peak brightness after 1-2 days and decline in brightness more slowly afterwards. The interval between outbursts ranges from 10 days to several years. SS Cygni stars are distinguished by having distinct long and short outburst.

3.3.4 X-Ray Binary

Consists of a neutron star or a black hole (and occasionally a white dwarf) as well as another younger star which is losing mass to its compact, massive companion. Based on the mass of the companion, X-ray binaries are categorized as high mass, intermediate mass, or low mass. As the mass of the main-sequence star decreases, the proportion of the system’s emissions in visible wavelengths decreases as well, from almost all to almost none. Microquasars are very low mass quasars which are only accreting mass from a single star and emit significant amounts in radio as well as in x-ray. Sometimes, if the magnetic poles a the neutron star in an X-ray binary system are not well aligned with the poles of rotation, the system

11 can be detected because the jets of X-rays, originating at the magnetic poles where accreted matter falls to, vary in strength as seen from earth.

3.3.5 Polar (AM Herculis) star

Consists of a closely orbiting M or K star with a white dwarf with an extremely strong magnetic field that it prevents the formation of an accretion disk and synchronizes its rotational period with its . The system becomes highly linearly and circularly polarized. Matter from the star is funneled onto the white dwarf at velocities at or above 3000km/s. This causes the matter’s kinetic energy to be converted into X-rays. Changes in brightness can be due to a change in mass transfer rate or an eclipse of the mass transfer stream.

3.3.6 Intermediate Polar (DQ Herculis) star

Similar to a polar but the white dwarf generates a weaker field on the order of 106 to 107 gauss. This means that a partial accretion disk can form around the white dwarf. Also, the orbit and rotational periods are not synchronized.

3.3.7 Super Soft Source (SSS)

38 ergs Objects with temperatures between 200,000 and 800,000 K and around 10 s . More than 90% of their observed X-ray emission is below 0.5 keV. SSSs are thought to be white dwarfs with classic hydrogen fusion occurring from material accreting onto their surfaces. This makes SSSs the progenitor for Type Ia supernovae.

3.3.8 VY Sculptoris stars

These are stars which occasionally drop in brightness by more than one magnitude, with very occasional dwarf-nova-type outbursts during the dim state. They may be a subclass of polars. The characteristic fading of the light curve for long periods is now believed to be caused by variations in mass transfer from the secondary to the white dwarf.Ex. TT Ari

3.3.9 AM Canum Venaticorum stars

Consists of two white-dwarfs orbiting extremely close together in an accreting semi-detached system. AM CVn stars are helium-rich with no detectable trace of hydrogen, have very short orbital periods - that of the prototype is about 18 minutes - and show permanent super-humps. They are strong producers of gravitational radiation.

12 3.3.10 SW Sextantis stars

These are like dwarf novae but have the accretion disc in a steady state, so don’t show outbursts; the disc emits non-uniformly. They have a very high mass transfer rate and single-peaked emission lines rather than the double-peaked lines characteristic of disk-accreting CVs.

3.3.11 Symbiotic Stars

A system with a red giant and a white dwarf that are both surround by nebulous matter. Both individual stars and the nebulous region emit radiation. This nebulous region is thought to originate from mass lost by the red giant. The symbiotic phase represents a late stage in and a brief span in the life of the binary. Because of the short timescale involved, symbiotic stars are rare objects. Less than 200 are known. Ex. CH Cygni

3.3.12 Pulsating White Dwarfs

DAV (ZZ Ceti): Pulsation arises from non-radial gravity wave pulsations. Periods range from 30 seconds to 25 minutes. Occurs in some white dwarfs that have hydrogen atmospheres. Temperatures range from 12,500 to 11,100 K.

DBV: White dwarfs with helium atmospheres that pulsate. Temperatures are about 25,000K. Ex: V77 Herr.

GW VIR (often subdivided into DOV and PNNV): Pre-white dwarfs with surface temperatures from 70,000 to 200,000 K and helium, carbon, and oxygen atmospheres. Periods range from 300 to 5,000 seconds.

DQV: White dwarfs that pulsate with carbon atmospheres. None have ever been observed.

The following is a table of white dwarf classification by spectral type Primary and Secondary Features A H lines present, no He I or metal lines B He I lines, no H or metal lines C Continuous spectrum, no lines O He II lies accomponied by He I or H lines Z Metal lines, no H or He I lines Q Carbon lines present X Unclear or unclassifiable spectrum Secondary Features Only P Magnetic White Dwarf with detectable polarization H Magnetic White Dwarf without detectable polarization E Emission lines present V Variable

13 4 Galaxy Classification

4.1 Elliptical Galaxies

 b  b Elliptical galaxies range from E0 to E7, where the number after E equals 10 1 − a , where a is the axial ratio of the galaxy in question (which happens to depend on the orientation of the galaxy in relation to the MWG, so it’s a flawed system). If a = b, it’s an oblate spheroid, if b = c, then it’s a prolate spheroid. There is little active star formation. Irregular galaxies have much star formation, as do spirals, with S0 galaxies having intermediate amounts. The surface brightness for an elliptical galaxy or the bulge of a spiral galaxy is given by " 1 # I(r)  r  4 log = −3.3307 − 1 in pc and Lsun Ie re 1 re is the radius at which 2 of the galaxy’s light is emitted, Ie is the corresponding luminosity. Should Ie and re not be available, but another scale factor a and the central brightness I0 are, use  r −2 I = I + 1 0 a

The D − σ correlation between the diameter D at which the surface brightness is 20.75 µB and the velocity dispersion is given by D σ !1.33 n = 2.05 kpc 100 km/s Also, V s  rot = σ 1 −  1−b Where σ is the velocity dispersion,  is the ellipticity (equal to a , and v is the rotational velocity).

vrot  ∗ ( σ )observed V If vrot = σ > .7, then the galaxy is said to be rotationally supported. If it’s not (say, is ( σ )predicted around .4 instead), then the galaxy is pressure supported. Generally, the broader the galaxy, the more it’s pressure supported.

Summary of Elliptical cD E S0 or dE dSph BCd Galaxies SB0 MB -22 -15 -17 -13 -8 to -14 to to to to -15 to -25 -23 -22 -19 -17 13 8 10 7 7 9 Msun 10 10 10 10 10 10 to to to to to 1014 1013 1012 109 108 Diameter (kpc) 30- 1-200 10- 1-10 .1-.5 ¡3 1,000 100 M L ¿100 10- 10 10 5-100 .1-10 100

14 There are several types of elliptical galaxies. cD galaxies are immense centers of galaxy clusters M with high L ratios (¿750, lots of dark matter). Normal ellipticals include giants (gE) and compact (cE) and sometimes lenticular galaxies as well. Dwarph spheroidal galaxies (dSph)have low L, low surface brightness with very little gas. Blue compact dwarfs (BCDs) are small and blue wiht many A stars and M current star formation, low L ratio, 15 to 20% gas by mass.

If the temperature of an elliptical galaxy can be measured and if the distance from the center to the outermost gas clouds can be calculated, then the mass required to hold such distant objects in place can be found as well.

4.2 Spirals

log(radius in kpc) = −.249MB − 4

Most spirals have trailing structures, but some anomalous galaxies have leading structures.

The Equation for spiral surface brightness profiles is

− R I(R) = I(0)e Rh

2 2 where I(0) is usually about 140L /pc or 21.7 mag/arcsec (for the MWG, Rh = .7 kpc). It is related to the equation for disk (excluding bulge) luminosity, which is

2 LD = 2 ∗ π ∗ I(0) ∗ hR

Recall that, for rotation curves to be used, the assumption of a symmetric mass distribution must be made. This causes problems for barred spirals, and rotation curves are often created using 21cm lines.

LSa < LSb < LSc

Spiral Comparison Sa or SBa Sb or SBb Sc or SBc MB -17 to -23 -17 to -23 -16 to -22 9 12 9 12 9 12 Msun 10 to 10 10 to 10 10 to 10 Lbulge .3 .13 .05 Ltotal Diameter 5 to 100 kpc 5 to 100 kpc 5 to 100 kpc M L 6.2 4.5 2.6 Vmax 299 km/s 222 km/s 175 km/s B-V .75 .64 .52 Mgas .04 .08 .16 Mtot MHII 2.2 1.8 .73 MHI Pitch Angle2 (degrees) 6 12 18 Generally, Sc and SBc galaxies are bluer and have more massive young stars (low mass to light ratios).

15 4.3 Classification

Galaxy classification systems are based primarily on visual wavelengths (blue light), which is justified by the (untrue) assumption that dark objects are unimportant. Also, it was once falsely hypothesized that galaxies evolved along the tuning fork. Therefore, elliptical galaxies are sometimes known as ”early” and spiral galaxies are known as ”late.”

The Hubble Tuning fork is the main classification scheme. Note that Hubble did not know of any S0 (lenticular) galaxies; these were only observed later.

Hubble -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Stage T de Vau- cE E E+ S0−1S00 S0+ S0/aSa Sab Sb Sbc Sc Scd Sd SdmSm Im coulers class approximate E E E S0 S0 S0 S0/aSa Sa- Sb Sb- Sc Sc Sc Sc- Irr Irr Hubble b c Irr I I class de Vaucouleurs classification divides spirals into a third category based on the presence or absence of rings (in addition to SA for non-barred spirals and SB for barred; SA0 and SB0 are classifications of lenticulars based on bars as well). If ringed, add r, if not ringed, add s, if transitional, add rs. Also, some galaxies are classified as irregular in the Hubble scheme are now Sd (SBd), implying a diffuse and broken-up nature, or Sm (SBm), implying irregularity without the presence of a bulge. These are known as Magellanic spirals and irregulars, respectively. For example, the large Magellanic cloud is SBm. Subscripts indicate increasing amounts of dust (from 1 to 3).

C implies a supergiant system, D implies a diffuse envelope.

Van Den Bergh Luminosity Class For numbers I to V, increasing value implies decreasing luminosity and decreasing order in spirals (this is because massive, luminous galaxies ahve well-defined spirals as gas clouds build up more rapidly in areas of higher concentration and form more stars more quickly).

Sometimes, r indicates and inner ring and R indicates an outer ring. Some improvements have been made by adding + or − to the designation depending on the strength of the disk and the proportion of dust. Still, the system is reasonably accurate considering its origins.

Grand Design Spirals Two symmetric, well-defined arms. Flocculent Spirals Not well defined

The most frequent galaxy types are dwarf ellipticals and irregulars. Spirals produce the largest propor- tion of light (not in Virgo, though). For example, in Virgo, 12% elliptical, 26% S0, 62%S or Irr.

The Yerkes (Morgan) scheme divides galaxies based on the spectral type of their most prominent stars, as seen in the following table:

16 Spectral Type Explanation Inclination Explanation a Prominent A stars 1 Galaxy is “face on” af Prominent A-F stars 2 f Prominent F stars 3 fg Prominent F-G stars 4 g Prominent G stars 5 gk Prominent G-K stars 6 k Prominent K stars 7 Galaxy is “edge on”

B Barred Spiral D Rotational symmetry without pronounced spiral or elliptical structure E Elliptical Ep Elliptical with dust absorption I Irregular L Low surface brightness N Small, bright nucleus S Spiral

Example: The Andromeda galaxy is classified as kS5

Galaxy Color-Magnitude Diagram This diagram shows the relationship between absolute magnitude, luminosity, and mass of galaxies. The red sequence includes most red galaxies such as ellipticals. The blue cloud includes most of the spiral galaxies. The green valley contains red spiral galaxies. The diagram also shows considerable evolution through time: the red sequence earlier in evolution of the universe was more constant in color across magnitudes and the blue cloud was not as uniformly distributed but showed sequence progression.

Normal galaxies Are the opposite of active galaxies, and have a constant luminosity. They tend to have a high proportion of non-stellar radiation, and make up 98% of galaxies.

Starburst Galaxies show a high rate of star formation, such as M82. There are several common definitions (from Wikipedia): 1. Continued star formation within the SFR would exhaust the available gas reservoir in much less time than the age of the universe (Hubble time). This is sometimes referred to as a ”true” starburst. 2. Continued star formation within the SFR would exhaust the available gas reservoir in much less than the dynamical timescale of the galaxy (perhaps one rotation of a spiral galaxy). 3. The current SFR, normalized by the past averaged SFR is much greater than unity. This ratio is referred to as the birthrate parameter. Identify starburst galaxies by noting that they have strong far IR emissions caused by the interactions of shortwave light from new stars with surrounding clouds of dust and gas. They also contain supermassive (> 100Msun) stars.

Many starbursts are caused by the interaction of various galaxies causing disturbances in structure (example: cartwheel galaxy). They may cause superwinds, which sweep up surrounding materials and are found in HII regions and other areas of active star formation, and they distribute them over a wide area.

17 Types of Starburst Galaxies 1. Blue compact galaxies have low mass, metal poor, young stars (as well as a few older ones) and do not contain dust (the have much gas, of course) 2. Ultra Luminous Infared Galaxies have much dust (see above discussion on emission lines) and are often caused by the collision of galaxies. 3. Wolf-Rayet galaxies contain and are forming many massive Wolf-Rayet3 stars. Nomenclature:The galaxies with the highest rates of star formation, even for starburst galaxies, are baby boom galaxies.

Look for spectra from HII regions to identify starbursts.

Population Synthesis refers to the investigation of the stellar contents of galaxies, done by guessing and checking hypothesized distributions.

Mass is usually measured by the strength of 21 cm lines and the amount of x-ray emission from HII regions.

Open Clusters are groups of objects which formed simultaneously in the disk. These live for less than 109 years, which is less than the lifetimes of their component objects.

Evaporation Gravitational interactions between stars in an open cluster give individual stars enough energy to escape, dissipating the cluster. Open clusters are also torn apart by clouds and differential rotation.

OB associations Regions with high proportions of O and B stars.

3Recall: Wolf-Rayet stars are supermassive stars in an evolutionary state where they emit strong He and N lines

18 4.4 The Milky Way Galaxy (MWG

Dimensions Component Shape Dimensions Stellar Gaseous Mass Motion Baryonic Baryonic (Msun) Matter Matter Dark Mat- Oblate > 50 kpc ? ? 1012 ? ter Halo Spheroid Disk Flat Disk r = 15 kpc Thin Disk Spiral Arms Thickness = Pop Dense and Stars 1011; Circular 1kpc I;z=.005- diffuse Gas 1010; differential .04; O, B clouds; Dust 108 rotation; stars in intercloud confined spiral arms medium; to plane of HII regions disk Think Disk Thickness = Old; in- Little to no Stars 1010 Almost 2kpc termediate gas circulars, pop; z=.004 scale height = 1kpc Spheroid Pop II Stellar Halo Oblate radius> z < .002 Very little Stars 109; Elliptical spheroid; 20 kpc gas, high Gas negligi- Orbits of- c/a=.8 velocity ble ten highly clouds inclined due ot galactic plane Nuclear Triaxial el- radius=3 z=.02; total 1010 6 Bulge lipsoid (bar) kpc 2.6·10 Msun black hole at center Hot Corona Tenuous hot gas

The local group is collapsing and the MWG will collide with Andromeda in 6.3 · 109 years (it’s approaching at 119 km/s).

4.4.1 Scale Height

The distance from the galactic plane over which the density of disk stars decreases to 1/e of the disk plane density. Is about 300 to 400 pc.

Recall that 95% of the MWG’s stars are in the thin disk and that the density at height R is given −R −3 −3 by n(R) = n0e 2 to 4 kpc where n0 .05 Msun pc , or about .02 stars pc .

19 The sun has a value R=30 pc. New O and B stars have R as large as 50 to 60 pc. Note: they symbol “h” is often used instead of “r”.

Types of stars in the disk: 1. Pop 1, spiral arm. Youngest (age ¡108 years). In open clusters, include classical cepheids, T Tauri, massive O and B stars. Metallicity (using definition mass metal/mass total) z=.02 to .04. Associated with HII regions. 2. Pop 1, thin disk. Ages 1 to 10· 109 years, z=.005 to .04. Survived long enough to move from birthplace, scattered throughout disk. Found within 500 pc of the galactic plane. 3. Intermediate population think disk stars. z=.002 to .01, age about 1010 years. Orbit out of plane.

Warped Disk: Gas moves 1 to 2 kpc out of the plane of the galaxy. Occurs in about 25% of spirals.

Rate of star formation α pn, n > 1.

Isophotes are curves connecting regions of equal surface brightness on a galaxy.

4.4.2 Magellanic Clouds

Irregular Dwarf galaxies which orbit the milky way. The Magellanic clouds are the distorted remains of barred spiral galaxies. They have a higher density of gas than the Milky Way and are more metal-poor (max z for LMC is .5zsun, max for SMG is .25zsun.

The distance of the LMC is 48 kpc and it is undergoing vigorous star formation. For instance, the Tarantula is the most active star-forming region of the local group.

Distance of SMG is 200 kly.

5 Galaxy Interactions

q tcMG For objects spiraling into the center of a galaxy, r = 2πv where r is the largest distance at which an object could have been pulled into the galaxy during its lifetime, t is the galaxy’s age, c is 23 for LMC or 76 for a globular cluster or 160 for an elliptical, M is the mass of the galaxy, G is the gravitational constant, and v is the cluster’s speed. Therefore, massive clusters spiral into the nucleus of a galaxy before less massive ones.

Trivia As a consequence, the Magellanic clouds will merge with the MWG in 14 · 109 years.

Dynamical Friction is defined as a net gravitational force on an object passing through space which slows it down. It is caused by a high density object’s wake accumulating other objects and also by a transfer of KE from the moving object to become PE as the separation increases. G2M 2 F = c ρ v2

20 where c is defined as in the previous formula and M, v refer to the mass and velocity of the object, respectively.

Rapid Encounters If two galaxies collide quickly, objects will not have enough time to change positions. Therefore, PE is converted into KE for stars in a random way, with E α minimum separation−4. But since 2KE = −PE, PE must increase, so gas is expelled. Example: cartwheel galaxy.

Tidal Stripping The dissipation of orbital energy by the removal of dust. Ex. Magellanic streams. The tidal radius of a galaxy size of the galaxy’s Roche lobe.

Polar Ring Galaxies are galaxies orbited by gas, dust, and stars stripped from other galaxies. Also 9 known as dust lane ellipticals. Rings have about 10 Msun and form around S0 and ellipticals only. g-Dwarf Problem Only about 2% of F, G stars have metallicity z > .02zsun, but theory predicts that 25% should. Therefore, either gas and dust are entering the MWG or there were metals in what became the disk when the MWG was formed.

Trivia: Collapsing nebulae radiate energy at a high enough rate for trapped energy to be so insignificant that it does not slow the collapse.

6 Interstellar Medium

One must apply a correction factor A to account for interstellar extinction:

m = M + 5 log d − 5 + A

Where AV = 3.2E(B − V ) And E(B-V) represents the reddening of the object due to extinction.

For a source of apparent intensity I and intrinsic intensity I0,

−τ I = I0e

Where τ is the optical depth of the penetrated medium. Optical depth is the average number of paths through which a ray can travel to the surface of a medium. Also,

A = 1.085τ

The Extinction Coefficient is given by Q = σλ , where σ is the scattering cross section (area which would σg λ be hit by scattered particles) and σg is the area of the dust particles causing the scattering (assumed to by πr2). Red light can more easily pass through dust clouds, as large λ allow waves to bypass more dust particles. Therefore, each wavelength results in different values of A.

21 Reflection Nebulae -The blue cloud seen to surround a dust cloud with a star behind it. The star appears red, as red light predominantly passes through the cloud, but the surrounding space appears blue, as some of the blue light reflected from the cloud is again reflected back in the direction of earth. This is analogous to Rayleigh scattering which explains the blue sky on earth.

Some compounds such as graphite, silicon, or polycyclic aromatic hydrocarbons (PAHs) absorb unusual amounts of short-wavelength light. Generally, molecular clouds are surrounded by HI shells as well as H2.[H2] α [CO] in dust clouds.

Types of Clouds 1. Diffuse/translucent molecular clouds: T = 15 to 50K, 3 − 10Msun, d = a few pc 5 2. Giant molecular clouds: T = 15K, 10 Msun, d = 50pc 4 3. Dark cloud complexes: AV = 5, 10 Msun, 10K, d = 10pc. These contain dense cores with O and B stars. 3 4. Bok Globules: AV = 10,T = 10K, 1 − 10 Msun, d < 1pc. These are regions of active star formation.

Generally, the ISM is heated by the interactions between photons and H and H2. The photons separate e− from the atoms or molecules and some of their newfound KE raises the temperature. Clouds cool through the emission of photons in IR.

7 8 10 The interstellar medium contains hot H gas (10 K) which is an x-ray emitter (about 10 to 10 Msun for a S0 or E galaxy). Cold HI regions (100 K), warm Hα is also present at 104 K in HII regions.

7 Active Galactic Nuclei

Active Galactic Nuclei are types of objects formed by the accretion of matter onto supermassive, rotating black holes. There are several types, and classification depends on how each object is viewed from earth. However, there is a division between radio loud sources in elliptical galaxies with jets and radio soft sources in spirals without jets. Galaxies which host active galactic nuclei are called active galaxies. Due to the mass and other characteristics of the black hole/accretion disk system, AGNs have high Eddington Luminosities, which are defined as the point at which the gravitational force in equals the force from radiation out. According to the Unified AGN Model, 42.3% of a particle’s rest mass can be converted to energy and released in the accretion disk. About 1 to 10 Msun per year are accredited. In fact, the radiation is so powerful as to create a background glow of x-rays from AGNs.

Synchrotron Radiation is radiation which is produced by electrons as they spin around their magnetic field lines. This is the type of radiation emitted in large quantities by AGNs. Sometimes, synchrotron self-absorption occurs and plasma of e− becomes opaque to its own radiation, causing a decrease in flux.

22 7.1 AGN Equations

2 Ldisk = ηMc where M is the mass accretion rate and η is the efficiency (.057 < η < .423) for the accretion disk.

F α ν−α where F is flux from AGN (measured in W m−2Hz−1), α is the spectral index (about 1, but varies), and ν is the frequency of the radiation (Hz). So, given a distance, α, and a F0, ν0 pair, evaluate the following integral to find the total power output (W) over the frequencies where emissions are significant: Z ν2  ν −α F0 dν ν1 ν0 Also, keep in mind the essential equation s 2λ 2kT ∆λ = c m This equation gives the (total) width of spectral lines based on the temperature of the gas which is producing them, the mass of one molecule m, the temperature in T, and the Boltzmann constant 1.38 × 10−23JK−1 (if broadening is given as a speed caused by temperature, it is the most probable speed). Also, the factor of 2 gives the total width, as width usually means full width at half maximum, or total width/2. If vrot is substituted for the most probable speed (keeping the factor of 2), the rotational velocity can be determined from the width of the spectral lines.

31 The Eddington Limit (Eddington Luminosity) is given by LEd = 1.5 × 10 M ( in Msun, W) , or Led = 4 3.8 × 10 M ( in Lsun, Msun) . An object cannot radiate more energy than this limit, as its outer layers would be expelled (the pressure from the radiation would exceed the gravitational force). 4πGc Alternative Definition: The Eddington limit is given by LEd = κ¯ M whereκ ¯ is the mean flux-mean opacity coefficient.

Dust cannot exist above 2000K. We can then calculate the smallest radius r at which spherical dust of radius a can exist from an object of luminosity L by equating flux at distance r over an area of πa2 q r (incoming radiation) with the luminosity of the dust (outgoing radiation). Solving for r gives r = 16πσT 4 (for dust, T = 2,000 K). Evidence for the small size of AGNs is provided by the observation that, if a source gives off varying amounts of radiation, the radius of material which is observable from earth must be small enough so that not all the variability is lost in scatter where subsequent radiation reaches earth before the furthest previous radiation does. The time delay is related to the radius of the object r by ∆t = c . Therefore, if meaningful ∆t are observed, the radius must be given by that equation (to within about a factor of 10, this is a pretty rough approximation; still, it’s better than l = d · θ).

Principal: Adopt a Shwartzchild radius 10 times smaller than the size of the AGN for calculations. The brightest part of an AGN is at five times the Shwartzchild radius.

r A rough size approximation may be made using ∆t = c .

23 Summary of Broad and Narrow Lines Broad line regions are dense, fast moving clouds with r < rdust torus from the AGN. They may be obscured by said torus. Narrow line regions are of low density outside of the torus which move more slowly.

The following is a detailed AGN table: Class Sub Class Description Seyferts-tend to reside Type 1 Broad and narrow emission lines, weak in spirals radio, x-ray emission, spiral galaxies, variable Type 2 Narrow emisssion lines, weak radio, weak x-ray, spiral galaxies, not variable Quasars Radio Loud (10%) (QSR) Broad and narrow lines, strong radio, some polarization, FR II, variable Radio Quiet (QSO) Broad and narrow emission lines, weak radio emission, weak polarization, vari- able Radio Galaxies -tend BLRG (broad-line radio Broad and narrow emission lines, to reside in giant ellip- galaxies) strong radio, FR II, weak polarization, ticals elliptical galaxies, variable. Bright nu- clei surrounded by faint envelopes NLRG (narrow-line radio Narrow emission lines, strong radio, FR galaxies) I and FR II, no polarization, elliptical galaxies, not variable. Found at the center of giant ellipticals (cD galaxies), since only large ellipticals have experi- enced the mergers of large black holes required to produce AGN which rotate quickly enough to generate these emis- sions Blazars BL Lacs (named after pro- Devoid of emission lines, strong radio, totype BL lacerate) strong polarization, rapid variability, 90% in ellipticals OVV quasars (optically vio- broad and narrow emission lines, strong lent variable) radio, strong polarization, rapid vari- ability, more luminous than BL Lacs and higher redshifts ULIRSs (ultra luminous infrared Dust enshrouded quasar or starburst galaxies) phenomenon LINERs (low ionization nuclear Low luminosity Seyfert 2, low ioniza- emission line region tion emission lines, mainly in spirals, may be starburst phenomena or HII re- gions

AGN can be identified by strong OIII (at 495.9 and 500.7 nm), Hβ, Hγ, lines which are wider than expected. These wide lines cannot be produced solely by temperature due to hte types of emissions, so rotation from AGNs must be the source.

24 Seyferts These contain “forbidden” spectral lines which are formed with low frequencies such as OIII, indicating low gas densities. These lines are narrow, at about 400 km/s. Seyferts are found in Sb and SBb spiral galaxies. The spectra of most AGN contain big blue bumps and IR bumps. The former are caused yb the optics of the accretion disk and the latter by emission from surrounding dust. Doppler broadening (see equation above) shows that Seyfert 1 emissions (non-forbidden, broader lines) come from sources moving at 1,000 to 5,000 km/s and Seyfert 2 emissions come from sources moving at 500 km/s. Seyferts are 100 times more abundant than radio galaxies.

Quasars are very luminous. A single quasar can be 105 times more luminous than the MWG. Hard to identify initially due to large redshifts (v = .15c). Very blue, with B − V = .38 and U − B < .4. Despite the name (quasi-stellar radio sources), 90% are radio-quiet. Quasars have decreased in average luminosiy over time. Generally, they have lifespans similar to those of galaxies and are associated with large ellipticals. The largest redshifts observed from quasars have been around 6, very strong Lyman (n=2 to 1) alpha lines at 121.6 nm.

Fanaroff-Reily Luminosity Classes Class I-ratio of distance between brightest spots of emission on either side of the center (not including the center) to the full extent of the source ¡ .5, Class II ¿ .5. Class I have 2 jets and less luminosity, class II have one straight jet and one very dim jet and are more luminous.

8 Spectra

Forbidden Lines: A spectral type that can only be produced by a very low density gas and cannot be produced in a laboratory. Example: OIII at 495.9 and 500.7 nm.

Balmer lines: HI lines

Balmer, Paschen Jump: Term for a sharp decrease in flux at 380 nm (Balmer) or 855 nm (Paschen).

Stark Effect: For a given class O, B, A,..., smaller stars have wider spectral lines.

Bolometric Luminosity: L over all wavelengths

Bolometric correction: The magnitude which must be subtracted from the visual magnitude to get the bolometric magnitude.

Wilson-Bappu Effect: Some spectral features imply spectral class even if T is unknown. For example, the k absorption line of Ca is broadest in spectral type K0 stars and, for G, K, and M stars, a narrow emission line is present in the middle of an absorption region. Its width can be used to find MV .

Masing: The 1.7 GHz line of OH and the 22 GHz line of H2O can be abnormally bright if these molecules are near stars which allow additional energy (either directly or indirectly due to collisions with H2 molecules) to excite OH or H2O to unusual levels.

Free-free radiation: Radiation produced when electrical forces from ions and e− deflect other electrons

25 onto curved paths and cause them to radiate (for example, x-ray emissions around the center of the galaxy and HII regions emit free-free radiation in radio wavelengths).

synchrotron radiation: Radiation given off by electrons moving close to the speed of light under a strong magnetic field. This type of radiation is often associated with SN events and powers their (and the center of the MWG’s) radio emissions.

Types of Emissions 1. Recombinant Radiation: Electrons move to a lower energy level 2. Fine Structure Transitions: Coupling between an electron’s angular momentum and spin. Exam- ple: 100 to 300 µm lines from C, O, and N (this radiation does not penetrate earth’s atmosphere). 3. Hyperfine transitions: Coupling between nuclear spin and orbiting electron spin. See 21 cm line.

8.1 21 cm line

HI (neutral H) emits at 21 cm as long as occasional collisions of H atoms provide enough energy (some low T is required in space for this to occur). The proton and electron after such a collision have parallel spins, but this is a higher energy state than would exist if they had different spins. By releasing a photon with 9.5 · 10−25 J of energy (relevant T used in collision: .046 K) at 21 cm, the spins become opposite. Note that these emissions are generally blocked by gas or dust. In conditions where H2 can form, no such flips can occur, so one must examine other molecular lines such as those of CO, which emits at 1.3 and 2.6 mm. While on average this only occurs once every 10 Myr, there are enough H atoms to make this emissions frequent.

9 Black Holes

9.1 Stellar Black Holes

Formed from the collapse of a massive star with a mass greater than about 10Msun. Only 3.6 Msun are needed to form a lack hole, but a star loses much of its mass during its stage. Stellar black holes range in mass form 3 to 15 Msun, and often have an accretion disk. As with all black holes, only three properties can describe them: mass, charge, and spin (angular momentum). While a black hole of any mass could exist, no black holes less than the order of a are known. Many stellar black holes are visible due to the presence of a hot accretion disk radiating in x rays from a companion star (see x-ray binary). It is difficult to determine the difference between a massive neutron star and a small black hole in such systems.

26 9.2 Super-massive Black Holes

These are often located at the center of galaxies. Supermassive black holes can have masses of 106 to 109 or more Msun. They form huge accretion disks and expel massive jets of superheated gas. However, their densities (defined as mass/volume inside the Schwarzschild radius) are low (less than air, 1.2kg/m3), because the radius is proportional to mass but volume is proportional to raidus3, so it grows faster. Similarly, tidal forces are weak outside of the event horizon. The method by which supermassive black holes are formed is uncertain, but one hypothesis is that stars in the center of a galaxy combine together to form an enormous (by mass) neutron star that forms a massive black hole through a non-understood process involving instability in the pressure in the center of the star, bypassing a supernova event which would blow off most mass. Another states that a dense cloud of gas in the center of a galaxy could collapse into a block hole. Finally, some astronomers think that when galaxies are formed, their cores naturally collapse into a supermassive black hole. Alternatively, accretion of matter by a black hole could eventually result in a supermassive black hole, but current hypothesis indicate that very little angular momentum can be present for so much mass to fall onto the black hole. For AGN, a black hole of about 108 Msun is required (see AGN section for details).

Method of supermassive Black Hole mergers: 2 black holes in merging galaxies migrate to the center of mass. They enter orbits. As other stars pass nearby, they are ejected and take angular momentum and energy with them, thereby decreasing the separation of the black hole system. Eventually, gravitational radiation takes enough energy away from the system, now with a small separation, for the black holes to merge.

9.3 Mid-Sized black holes

: An object on the outskirts of the ESO 243-49 galaxy named HLX-1 has been identified as having a mass of 500 Msun. This is a possible candidate for the theorized “mid-sized” black holes that are said to combine together to form supermassive black holes. Evidence for the existence of such black holes comes from the existence of ultraluminous x-ray sources, which are more luminous than stars but less luminous than AGN. The luminosity exceeds the Eddington limit for stellar mass black holes and neutron stars, leading astronomers to suspect a more massive object. Also, the power law flux αfrequency−α (see AGN section) shows similarities to that of a scaled up x-ray binary system with alternating α.

9.4 Micro-Black Holes

A Hypothesized but not observed miniature black hole of a mass above the plank mass (found by h −8 rearranging c, G, and 2π to get a unit mass of 2.17 · 10 kg). These would emit Hawking radiation and “evaporate”-lose elementary particles and disappear almost instantaneously (in about the age of the universe).

27 10 Fates of Massive Stars and Supernovae

6 Luminous Blue Variables (LBVs): High T (15,000 to 30,000 K), L (> 10 Lsun). These are near the upper limit of the main sequence (the most massive stars known, near their Eddington limits). Variability is gradual, but punctuated by periods of intense mass loss. There is some debate as to the precise Eddington limit, and how much of massive stars’ mass loss can be attributed to it. Some believe a lower limit should be set, which would explain the tendency of massive stars to lose mass.

Wolf-Rayet Stars (WR): Exhibit strong, broad emission lines rather than traditional absorption lines. −5 May be evolved LBVs. Mass loss of 10 Msun/year. 1. WN (He, N lines) 2. WC (He, C, no H, no N lines) 3. WO (O lines, rarest) Essentially, the loss of varyous layers of material is responsible for these variants (i.e. WN→WC→WO).

Humphreys-Davidson luminosity limit: Diagonal cutoff on the HR diagram running from highest T, L, down. Prevents supergiants from becoming red supergiants (that is, as T decreases, the maximum luminosity decreases as well so supergiants cannot travel at the top of the HR diagram all the way to the right).

1 The Collision Mean Free Path is given by l = nσ where n is the number density and σ is the cross section.

10.1 Supernovae Classification

SN (year) (letter A→ 1st,B→ 2nd, etc.) No H lines→ Type I Strong SiII line at 615 nm→ Ia Strong He lines→ Ib Weak/no He lines→ Ic H lines→ Type II Max MV for a type Ia is -18.4 Types Ib, Ic are fainter by 1.5 to 2 magnitudes. Type II are fainter by 1.5 magnitudes.

Supernovae Taxonomy Type I No Hydrogen lines (Balmer series) Type Ia Lacks hydrogen and presents singly ionized silicon (Si II) line at 615.0 nm near peak height Type Ib Non-ionized helium (He I) line at 587.6 nm and no strong silicon absorbtion feature near 615 nm Type Ic Weak or no helium lines and no strong silicon absorption feature near 615 nm Type II Hydrogen lines (Balmer series) Type IIP Reaches a “plateau” in its light curve Type IIL Displays a “linear” decrease in its light curve (linear magnitude versus time)

The rate of dimming for type I is about .065 magnitudes per day before dropping to .015 magnitudes

28 per day (Ia) or .01 magnitudes per day (Ib, Ic) after 50 days. Type II curves: IIP indicates the presence of of a plateau 30 to 80 days after max MV , IIL indicates a linear decrease. IIP are more common Elements heavier than iron are formed via endothermic processes and therefore do not contribute to the luminosity of the star. Less energy per unit mass is generated as elements become heavier.

10.2 Process of core collapse photodisintegration: energetic photons can destroy heavy nuclei. Ex:

56 4 4 + 26F e + γ → 132He + 4n 2He + γ → 2p + 2n

Once sufficient T, Fe core masses have been reached, this process occurs and is highly endothermic. The thermal energy drops and the core collapses (the core has about 1.3 Msun in a 10 Msun star and is 2.5 + − Msun in a 50 Msun star). Now excess protons react with free electrons: p + e → n + ve. This removes electrons so there is less core degeneracy pressure. The speed of collapse increases, and is roughly proportional to the distance from the center of the core, unless the local sped of sound is exceeded, in which case the parts of the core separate and can reach speeds of up to 70,000 km/s. Once ρ = 8 · 1017 kg/m3, the strong force becomes repulsive (Pauli exclusion principle) and a pressure wave builds up, with in falling material accreting onto it. With the addition of energy from trapped neutrinos, the star begins to explode, with light appearing once the material becomes optically thin at 1013 m (100 au). Note: the more metal rich the star is, the more massive it can be to satisify degeneracy limits. Most of the energy which leaves the SN is carried out by neutrinos ( 3 · 1046 J). 56 56 The plateau observed in most type II SN is cause by the radioactive decay Ni → Co + β + ve + γ, which results in additional energy emissions for a time.

The following are decay reactions for radioactive isotopes such as 56Ni: d .434 ln 2 d 1.086 ln 2 log L = − or M = dt t 1 dt t 1 2 2 The denser the progentor star, the longer the SN event will take to reach max L, as more energy will initially need to be spent blowing off the other layers of the star.

Trivia: The fact that neutrinos arrived before photons from SN 1987A (they left the star a bit earlier) gives an upper mass limit of 16 eV for neutrinos, calculated from their extremely fast speeds (labs give an upper limit of 2.2 eV).

Solar lithium Problem: The sun has less lithium than meteorites of the same age which should have had the same initial composition, and Li decay models consistently overestimate how much should remain. It is thought that Li is transported deeper into the sun than anticipated, where the high temperatures will cause it to decay.

S and R Processes: Consider the reactions

a a+1 a+1 a+a − z x + n →z x + γ (1) z x →z+1 x + e +v ¯e + γ (2)

29 If (1) occurs on a shorter timescale than (2), then rapid (r) process. If (2) occurs on a shorter timescale than (1), then slow (s) process. The R process occurs when there are a large number of neutrons in the core of a supernova. The nuclei of atoms quickly gain large number of neutrons. Later on, they decay to stable nuclei.

The S process occurs in AGB stars and describes a process where nuclei capture neutrons, and undergo β decay to produce stable heavy elements. This process produces elements up to bismuth.

10.3 Gamma Ray bursts

L Are isotropic-arrive with equal frequency from all directions. Recall that F = 4πr2 F: observed flux, L: luminosity, r: distance. If we assume that L is fixed and that the events are distributed homogeneously throughout space, then, if we choose F ∗, all events such that r < r∗ must have F > F ∗. Let there be n 3/2 ∗ 4 3 4  L  bursts per unit volume, therefore, the number of sources N with F > F is n 3 πr = n 3 π F ∗4π , or, simply, the number of bursts with F > F ∗ α (F ∗)−3/2. This relationship does not hold. Therefore, the distribution is not homogeneous.

Classification Long-soft have supernovae origin Gamma rays are formed as jets from SN producs’ (black holes or neutron stars) pass through accretion −1/2 1  u2  disks of dust. For a jet with a solid angle of θ = γ , γ = 1 − c2 Short-hard have neutron star origin Note: L is usually “pointed” in one direction, so the total luminosity is not what one would predict from a spherical propagation model.

10.4 Cosmic Rays and Background Radiation

A wide range of particles has been observed: e−, e+, p+, , µ, C, , O, , F e, etc. 1 particle 1 particle Fluxes also vary: m2·s (low energy) to km2· year (high energy) F lux α energy−a, a: constant. Most cosmic rays come from the sun but are of low energy. High energy particles (approaching 1021 eV) are thought to originate from events such as AGN, with intermediate energy particles produced by SN.

Background radiation was produced when R(t) = 1 , time is 1 Myr since universe’s start, T = R(t0) 100 3000K. The formation process has to do withe the recombination of H atoms, and the resulting escape of photons. It peaks at 10−6 KeV, λ 1mm. The escape of these photons allowed matter to collapse into stars. Now we can see light from when the universe was 1Gyr old. Background radiation has T=2.728 K now, and it has a density of 420 photons per cm3. Its energy is starlight in outer regions of MWG, −6 W i.e. 10 m2 .

30 11 Red Giants

Post main sequence evolutionary phase for .5 to 5 Msun stars. Once fusion in the core stops, the star’s core contracts, resulting in increaced temperatures around the core. Fusion of H to He starts in a shell, causing a great increase in size. The star then follows the following paths: red giant branch Hydrogen is fused to helium in a shell surrounding the helium core. Depending on the position on the HR diagram, different methods are used to fuse He in the H burning shell. On the main sequence, less massive stars use the p-p process H + H → D; D + H → He3; He3 +He3 → He4 +2H and more massive stars (which contain some metals already) use the CNO 12 1 13 13 13 + 13 1 14 14 1 15 process C + H → N + photon; N → C + e + nue; C + H → N + photon; N + H → 15 15 + 15 1 12 4 O + photon; O → N + e + nue; and then, most likely, N + H → C + He.(nue is an electron neutrino, mass ¡ 2.2 eV).

Horizontal Branch Stars produce carbon from helium (triple α process: He + H → Be; Be + HE → C or He + He + He → C. This branch begins with a “helium flash” where the He core begins to fuse C throughout at one time for stars of M < 2.5Msun. More massive stars, whose cores are not solely supported by degeneracy pressure, do not experience this flash due toe temperature increasing in the core before degeneracy effects predominate. Note that the horizontal branch crosses both the main sequence and the instability strip.

Asymptotic red giant branch This is analogous to the initial red giant branch, except the core of the stars entering is C, and there is a H → C shell surrounding it. Depending on mass, this cycle may or may not continue. 1Msun stars do not fuse C to O, but many others do, as carbon oxygen white dwarfs are common (these would re-enter the horizontal branch next). The CNO cycle takes place in stars whose centers can reach 15 · 106 K. For many stars, the disruption caused by the ignition of the helium shell at the tip of the red giant branch precipitates the formation of winds and ejects much matter into space. No elements heavier than iron are produced in known stars.

12 Star Formation

For a cloud to collapse, 2KE < U (Virial Theorem). It can be shown that a cloud which satisfies the Jeans Mass (M > Mj) or the Jeans Length (R > Rj) will collapse. 5kT !3/2 s 3 s 15kT Mj = Rj = GµmH 4πρ0 4πGµmH ρ0 where k is the Boltzmann constant (1.38 · 10−23J/K), T is the temperature, G is the gravitational constant, ρ0 is the initial density, and µmH is the mean molecular weight.

As long as the cloud remains optically thin, the collapse is isothermal. The free fall time is independent of the initial mass of the cloud. The density increases everywhere at the same rate (homogeneous collapse).

Inside-Out collapse: A collapse where there is a higher density at the center initially. Fragmentation

31 of clouds occurs if the collapse produces regions of different densities, and these individual satisfy the Jeans criteria and collapse. A general trend is that magnetic fields help prevent collapses. This is seen most vividly in neutron stars but applies to clouds as well.

Ambipolar diffusion: The slow migration of neutral particles across space even if charged particles are held in place by a magnetic field.

Steps of Protostar collapse: 1. The dust cloud reaches a mass necessary to collapse 2. A shock wave builds up and takes away much of the energy from the falling particles (increases L) 3. Once T=1000K, dust vaporizes, T increases more, and L increases more as well as well. 4. At 2000K, H2 → 2H, which takes energy. The core collapses before stabilizing at r = 1.3rsun. 5. Deuterium begins to burn and T drops. 6. H− builds up in the outer layers, increasing opacity. T increases, L drops. 7. Hayashi Track: refers to the drop in L immediately before the mains sequence (i.e. is the path followed onto the main sequence). To the right of the track, instability is caused by insufficient heat transfer mechanisms. 8. Eventually, once hydrostatic equilibrium is achieved, the PP1 chain starts. Notes: The upward branch is missing for M < .5Msun stars because these stars don’t tend to have a high enough temperature (ρ α M, density can’t get large enough) for stable H burning phase.

Initial Mass (Msun) Pre-Main sequency lifetime (Myr) 60 .0282 25 .0708 15 .117 9 .228 5 1.15 3 7.24 2 23.4 1.5 35.4 1 38.9 .5 68.4

 .7 Once the stars are on the main sequence, the following equations may be used: RR M and sun Msun  α LL M where α=5 for M < M , α=3.9 for 10M > M > 1M , and α=2.2 for M > 10M . sun Msun sun sun sun sun

32 13 Classification of other DSOs

13.1 Globular Clusters

Class I, II, III : Visible high stellar density at their core. With a halo around decreasing in luminosity as a function of the distance from the core. Example: M75 is a globular cluster of class I in Sagittarius

Class IV, V, VI : The core stellar density is still visible, but is more spread out and not as dense. Example: M62 is a globular cluster of class IV.

Class VII, VIII, IX : The cluster stellar density is more homogeneous and less contrasted. Example: M22 is a globular cluster of class VII in Sagittarius.

Class X, XI, and XII : The cluster surface luminosity is completely homogeneous with no increase in stellar density visible at the core. Example:M55 is a globular cluster of class XI in Sagittarius.

13.2 Open Clusters

Concentration and detachment from the surrounding star field Class I: The cluster is strongly detached from the stellar background with a strong core stellar density. Class II: The cluster is detached from the stellar background with a light core stellar density. Class III: The cluster is detached from the stellar background without a denser core. Class IV: The cluster is weakly detached from the stellar background, the area having a higher stellar density but no visible core. Range in Brightness Class 1: All stars present have about the same brightness Class 2: The stars present have a range of brightnesses Class 3: Besides some very brights stars, many weaker stars with a wide magnitude range Number of Stars p: The cluster is poor in stars (less than 50) m: The cluster has a medium number of stars (from 50 to 100) r: The cluster is rich in stars (more than 100) The letter “n” at the end of the classification indicates a nebula linked to the cluster.

13.3 Planetary Nebulae

I: Stellar image (like a star) II: Regular disk a) A shinier core b) Uniform brightness c) Presence of an annular structure

33 III: Irregular disk a) Irregular brightness b) Presence of an annular structure IV: Annular structure V: Irregular form between a and a diffuse nebula VI: Abnormal form without a regular structure (like an S or an 8, etc.).

14 Special Relativity

Postulates: 1. Physical laws remain the same regardless of their frames of reference 2. The speed of light is always constant regardless of one’s frame of reference

Lorentz Transformations, where the intertial reference frame S’ is moving in the positive x direction with velocity ~u relative to the frame S. x − ut x0 = q u2 1 − c2 y0 = y z0 = z t − ux t0 = c2 q u2 1 − c2 To find the inverse transformation, swithc primed and unprimed quantities and replace u with −u.

True Redshift v u λobs − λemit u1 + vr/c = z = t − 1 λemit 1 − vr/c

Time Dilation: For an observer observing a clock traveling quickly w.r.t. the observer’s frame of reference, the clock is running slow. The equation which describes this is ∆t0 = ∆t where ∆t is the q 2 1− v c2 time an observer in the same reference frame measures and ∆t0 is the time an observer in a different frame of reference measures (note ∆t0 > ∆t).

Length Contraction: For an observer observing an object traveling quickly w.r.t. the observer’s frame of reference, the object is smaller than it appears to be to an observer in the same frame of reference as 0 L q v2 0 the object. The equation which describes this is L = γ(v) = L 1 − c2 , where L is the length measured by an observer in a different reference frame and L is the length measured by an observer in the same reference frame as the object in question. Note: γ(v) is the Lorentz Factor.

Other relativistic effects can be considered to yield an accurate mass-energy relationship E = mc2 q 2 1− v c2 and definition for momentum ~p = m~v . q 2 1− v c2

34 In an attempt to simplify some formulae, the values of certain fundamental constants are often normal- ized to 1. The best known system for achieving this is the Plank System, where c = G = h¯ = 1 = 4π0 √ −3 kB = 1 and e = α, where α is the dimensionless constant 7.297 · 10 and e is the elementary charge carried by one proton.

15 Miscellaneous

M30 (NGC 7099) is a globular cluster which contains 12 variable stars.

Class V Capricornus 21:40.4 (h:m) Distance 26.1 kly Visual Brightness 7.2 mag Apparent Dimension 12.0 arcmin Radial Velocity 181.9 km/s Age 13 · 109 years

Has undergone core collapse, so half of the mass is concentrated within 8.7 light years. Also contains blue stragglers4, blue stars whose origins are unknown bur appear on the main sequence above the turnoff point for the cluster they reside in.

Chandra multiwavelength plane survey (ChaMPlane) is a project designed to locate various x-ray sources in or near the plane of the MWG, including x-ray binaries, neutron stars, and black holes.

WIYN Open Cluster Study (WOCS) is a project designed to determine precise measurements of the position of various open clusters. It has observed NGC 188, a 7 Gyr classical globular cluster in the disk, NGC 2451, a 60 Myr old cluster which is among the 12 closest open clusters; WOCS’s measurements of distance confirm the parallax values, NGC 6882/5, the former a 100 Myr, 1350 pc cluster and the latter a part of a larger structure, and a Car, a suspected cluster which does not appear to actually be very coherent; instead, it’s part of a larger young star complex in Carnia.

The brightness of the night sky is 22 Bmag/arcsec2.

For the reddening of stars with high metal content, metal → more opacity (e−/atom) → expansion of star → temperature decrease → red color. so generally, the redder the region, the more metal rich it is. Metal content α L. 4Blue stragglers are main sequence stars in clusters which are bluer than the main sequence turn off point for said cluster. They may be caused by the collapse of binary systems to form more massive stars, or mass transfer could lead to unexpected massive stars without combination

35 15.1 The Moon

The angle of inclination of the moon’s orbit is 5.1◦. The phases of the moon are, in order, new, waxing crescent, first quarter, waxing gibbons, full, waning gibbons, third (last) quarter, and waning crescent. A day on the moon is 29.53 days.

15.2 Solar System, Orbits, and the Sky

Bernard’s star has the largest of any object. Its proper motion is 10.3 arcsec/year The solar apex is the direction that the sun travels w.r.t. the local standard of rest. It is in the direction of the constellation southwest of the star Vega (RA 18h 03m 50.2s and Dec 30◦ 000 16.800). Radar is not effective at long ranges since the strength of the signal follows an inverse fourth power law. The Greeks named seven “wanderers” which move relative to the background. They are the Sun, Moon, Venus, Jupiter, Saturn, Mercury, and Mars. Zenith:= The vertical direction opposite the net gravitational force at a location. Nadir:= The vertical direction coinciding with the net gravitational force at a location. The Solar system is divided into two parts: The inner solar system (up to the asteroid belt) and the outer solar system (beyond the asteroid belt). Apsides:= Points where the distance between two orbiting objects is at max or min Periapsis:= Point at which the secondary is closest to the primary and is moving at its fastest point (perihelion) Apoapsis:= Point at which the secondary is farthest away and slowest (aphelion) Nodes of an orbit are points at which the secondary passes through a horizontal reference plane a−b Oblate spheroid: a shape whose flattening f = a . For the earth, the aspect ratio is .9966 and the flattening is .00335286. The only satellite orbits which are not inclined are geostationary orbits over the equator. Precession:= The gradual change of the inclination of the axis of rotation of an orbiting body (that is, how tilted earth is on its orbit). The earth goes through one cycle in 26,000 years. Precession of the ecliptic refers to changes in the inclination of the plane of an orbit, and the period of earth is about 10,000 years. Hour angle:= local sidereal time - right ascension. Note: One hour of right ascension equals 15 degrees. A sidereal day is defined as the time earth takes to complete one rotation relative to the vernal equinox and equals 23.93447 hours. The age of the sun is 4.57 · 109 years, and it will produce 3 · 1044 J of energy during its main sequence lifetime.

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