Astronomy - Galaxies

(View Cosmos DVD 6, episode 10, Milky Way simulation.) Also see this site at the University of Hawaii.

We can learn much about the populations of stars in a galaxy like ours by looking statistically at the stars we know best: those closest to us. Using a recent survey (source), we have culled spectral class, apparent magnitude and parallax data for 3633 stars near the Sun. This data is available in an Excel spreadsheet. Using the parallax data, you can compute each star's distance from Earth as

distance_pc = 1000 / parallax_mas

From the relation

absolute magnitude = apparent magnitude + 5 - 5 * log₁₀ distance_pc

you can then compute each star's absolute magnitude. The absolute magnitude can be used to determine the luminosity from the relation

absolute magnitude = 4.83 - 2.5 * log₁₀ luminosity_solar

luminosity = 10^{(4.83 - absolute magnitude) / 2.5}

Assuming that luminosity is approximately equal to mass^3.5, you can then compute each star's mass as

mass_solar = luminosity_solar^(1/3.5)

Portfolio Exercise 1: Create a histogram of the spectral classes of this population of stars. Omit stars with spectral classes other than O, B, A, F, G, K or M, and stars with a range of possible classes. Include stars of all subclasses in the main class (ie., count M and M1 through M9.5 all as class M). Note the percentage of stars for which there is no subclass; what does this say about the difficulty of stellar classification?
Create a histogram of the known luminosity classes of this population of stars. Omit stars with luminosity classes other than I-V and D/SD (white dwarf and sub-dwarf, sometimes designated VII and VI, respectively). Note the percentage of stars for which there is no luminosity class; what does this say about the difficulty of luminosity classification?
Create a histogram of the masses of this population of stars. What is the average mass to luminosity ratio (in solar units)?

Galaxy Morphology

Note that the classification of a given galaxy is not always universally agreed upon. The classifications in parentheses below follow the morphological types quoted in SIMBAD:

Spiral (Sa) - M 104 (source)
Spiral (Sb) - M 64 (source)
Spiral (Sb) - NGC 4622 (source)
Spiral (Sc) - M 51 (source)

(central bulge gets smaller, arms more separated ~ increasing age)

Spiral - ESO 510-G13 (source)
Barred Spiral (SBa) - NGC 4314 (source)
Barred Spiral (SBb) - NGC 1300 (source)
Barred Spiral (SBc) - M 83, M 83 (source), and M 83 (source)
Elliptical (E) - NGC 1132 (source)
Ellipticals are often subclassed from E0 (nearly spherical) to E7 (very elongated); the stellar populations of most ellipticals are very old. Also, clusters of galaxies in the early universe seem to have a larger percentage of spirals than nearer clusters. Elliptical galaxies have almost no interstellar gas or dust.
Lenticular (S0: "S-zero") - NGC 5866 (source)
Lenticular (SB0) - NGC 2787 (source)
Irr I - NGC 4449 (source)
Irr II - M 82 (source)
Some references (including SIMBAD) do not distinguish between Irregular types, or use distinctions other than I and II.
Ring - Hoag's Object (source)
Interacting, or Merging - NGC 4038, 4039 (source), Arp 147 (source) Arp 148 (source)
Also note the NGC 4013 tidal stream. The Hercules Cluster indicates that galaxy interactions may be common. Interactions and mergers probably contribute to differences in morphology.

Some references also consider dwarf galaxies as a separate morphological type. Dwarf irregulars like I Zwicky 18 are thought to combine into larger galaxies over time (source).

Since 2001 it has been known that there is a correlation between the mass of supermassive central black holes in galaxies and the mass of the central bulge: the median black hole mass is .0013 times the bulge mass. This correlation hints that the events which lead to the formation of supermassive central black holes are the same as the events which lead to the formation of central bulges (in spiral or elliptical galaxies). (source)
More recently, it has been discovered that this ratio is 20 to 30 times larger for galaxies at high red shift (Z > 6). (source) This seems to indicate that the black holes form before the central bulges. Interesting problem!

Note that many galaxies (for example, Centaurus A) when viewed in multiple wavelengths challenge our notions of galactic morphology gleaned from visible light observations. Also, see NGC 2915: a "hidden spiral".

(View Cosmos DVD 6, episode 10, galaxy interaction simulations.)

Portfolio Exercise 2: Find an additional example of each morphological type (Sa, Sb, Sc, SBa, SBb, SBc, E, S0, SB0, I, Ring and Interacting) and verify the type using SIMBAD. Include an image and as much information as you can (including all URLs used) for each example.

Doppler Shifts and the Hubble Relation

(View Cosmos DVD 6, episode 10, dramatization of red shift discoveries at Mt. Wilson.)

In 1998 and 1999, the GHASP survey of spiral and irregular galaxies (source) measured the central wavelength and width of the Hydrogen-a line at 6562.78 Angstroms for the following galaxies in the UGC catalog (values were derived from the paper for the purposes of this course):

UGC # l_central (A) width (A) diameter_angular (am) M m

2034 6575.29 2.43286 2.818 -17 12.9

2080 6582.36 13.1647 5.754 -19.5 11.2

3574 6594.17 6.98982 3.467 -17.6 14.1

4325 6573.78 4.62356 3.388 -17.7 12.1

4499 6577.66 3.3546 2.344 -16.2 14.1

5253 6591.66 11.3157 4.168 -20.8 10.8

5316 6585.16 5.77299 4.073 -19.7 11.5

5721 6574.37 4.2076 1.819 -16.9 12.6

5789 6578.75 5.30686 5.888 -19.2 11.2

5829 6576.52 2.10449 5.128 -17.1 12.9

5931 6597.89 8.18138 1.949 -20.1 11.7

5935 6599.49 2.66179 3.715 -20.3 11.5

5982 6597.1 9.7857 4.073 -20.3 11.5

6778 6583.74 13.0248 4.073 -20.6 10.4

7524 6569.74 3.10968 13.8 -17.3 10.6

8490 6566.96 3.83073 4.677 -17.7 11.1

9969 6617.91 14.6256 5.128 -21.7 11.2

10310 6578.2 3.70572 2.884 -17.4 13.1

12060 6582.29 4.21267 1.318 -16 14.8

12754 6579.01 5.92111 4.073 -18.9 11.5

(The angular diameter is taken to be the major axis length reported by SIMBAD.)

UGC #	l_central (A)	width (A)	diameter_angular (am)	M	m
2034	6575.29	2.43286	2.818	-17	12.9
2080	6582.36	13.1647	5.754	-19.5	11.2
3574	6594.17	6.98982	3.467	-17.6	14.1
4325	6573.78	4.62356	3.388	-17.7	12.1
4499	6577.66	3.3546	2.344	-16.2	14.1
5253	6591.66	11.3157	4.168	-20.8	10.8
5316	6585.16	5.77299	4.073	-19.7	11.5
5721	6574.37	4.2076	1.819	-16.9	12.6
5789	6578.75	5.30686	5.888	-19.2	11.2
5829	6576.52	2.10449	5.128	-17.1	12.9
5931	6597.89	8.18138	1.949	-20.1	11.7
5935	6599.49	2.66179	3.715	-20.3	11.5
5982	6597.1	9.7857	4.073	-20.3	11.5
6778	6583.74	13.0248	4.073	-20.6	10.4
7524	6569.74	3.10968	13.8	-17.3	10.6
8490	6566.96	3.83073	4.677	-17.7	11.1
9969	6617.91	14.6256	5.128	-21.7	11.2
10310	6578.2	3.70572	2.884	-17.4	13.1
12060	6582.29	4.21267	1.318	-16	14.8
12754	6579.01	5.92111	4.073	-18.9	11.5

Let's take a closer look at UGC 2034. Using the relation for Doppler Shift:

apparent wavelength / actual wavelength = 1 + recession velocity / c

we can solve for the recession velocity

v = c * (l_central / 6562.78 - 1)
= c * (6575.29 / 6562.78 - 1)
= 572000 m / s

Using our relation between absolute and apparent magnitude, we can solve for the distance

distance_pc = 10^{(5 + apparent magnitude - absolute magnitude) / 5}
= 10^{(5 + 12.9 - (-17)) / 5}
= 9.55 * 10⁶ pc

If we do this for all of these galaxies, and plot the recession velocities vs the distances, we get the following plot:

The red line is the best linear fit to the data points, and its slope is called the Hubble Parameter:

H₀ = 70.1 km/s / Mpc

(this value is the current "best" value from all known observations; the fit for our 20 galaxies had a slope of 70.783 km/s/Mpc).

It is common when dealing with distances on this scale to use the Hubble Relation

recessional velocity = H₀ * distance

to compute the distance.

Portfolio Exercise 3: Recreate the Hubble Graph above for our sample of 20 galaxies.

Astronomical Distances

The Hubble Relation is one of several techniques astronomers use to determine astronomical distances. We have discussed a number of these, and place them here in relation to one another:

technique approximate maximum effective distance

radar ranging (distance = c * echo time / 2) Solar System

stellar parallax (distance_pc = 1000 / parallax_mas) 200 pc

H-R Diagram (main sequence spectral class gives us absolute magnitude; distance_pc = 10^{(apparent magnitude - absolute magnitude + 5)/5} 5000 pc

RR Lyrae / Cepheid Variables: Period/Luminosity relation (Absolute Magnitude ~ 1.608 period_days^0.275), calibrated to known distance to Cepheids in the Large Magellanic Cloud 20 Mpc (~ Virgo Cluster)

Tully-Fisher Relation: absolute magnitude of a spiral galaxy = a - b Log (width of spectral line), a and b depend on morphological type and wavelength 100 Mpc

Type Ia Supernovae: C-O White Dwarfs explode with characteristic luminosity as a function of their light curves 1000 Mpc

Hubble Relation (recessional velocity = H₀ * distance) z = 1

technique	approximate maximum effective distance
radar ranging (distance = c * echo time / 2)	Solar System
stellar parallax (distance_pc = 1000 / parallax_mas)	200 pc
H-R Diagram (main sequence spectral class gives us absolute magnitude; distance_pc = 10^{(apparent magnitude - absolute magnitude + 5)/5}	5000 pc
RR Lyrae / Cepheid Variables: Period/Luminosity relation (Absolute Magnitude ~ 1.608 period_days^0.275), calibrated to known distance to Cepheids in the Large Magellanic Cloud	20 Mpc (~ Virgo Cluster)
Tully-Fisher Relation: absolute magnitude of a spiral galaxy = a - b Log (width of spectral line), a and b depend on morphological type and wavelength	100 Mpc
Type Ia Supernovae: C-O White Dwarfs explode with characteristic luminosity as a function of their light curves	1000 Mpc
Hubble Relation (recessional velocity = H₀ * distance)	z = 1

There are numerous variations on these themes, some involving very clever uses of statistics which are beyond the scope of this course.

All of the techniques based on luminosity measurement depend on our knowledge of the extinction.

Note that these techniques must be calibrated to each other: radar ranging is necessary to accurately determine 1 AU, which is needed for stellar parallax, which is used to calibrate the magnitude-distance relation, etc.

Each technique has limits to its accuracy, for some approaching 20%, and they should in principle all agree where they overlap. At present that agreement is better than 10%. That it is not zero is a testament to the difficulty of consistently measuring many of these quantities.

The maximum distance for the Hubble Relation is listed as "z=1". This refers to the redshift, about which more later.

Galactic Rotation Curves and Dark Matter (I)

Returning now to the case of UGC 2034, the Hubble Relation gives us a distance of

d = v / H₀
= 572 km / s / (70.1 km/s / Mpc)
= 8.16 Mpc

We should view this as a correction to the absolute magnitude, but for the sake of simplicity we will continue to use the value of M from the table above.

We can compute the maximum speed of UGC 2034's rotation from the relation

line width = 2 * speed * central wavelength / c

by solving for the rotational speed (and including the Doppler Shift!) as

v = ((c / 2) * width / l_central) / (1 + v_recession / c)
= ((c / 2) * 2.43286 / 6562.78) / (1 + 572000 / c)
= 55500 m / s

The absolute magnitude can be used (as above) to determine the luminosity as

L = 10^{(4.83 - (-17)) / 2.5}
= 5.4 * 10⁸ L_solar

From our statistical discussion above, the average mass to luminosity ratio in solar units is around 1/2. Hence the mass of UGC 2034 should be in the neighborhood of 2.7 * 10⁸ M_solar. But we can use Newton's Laws to check this. First, the radius of UGC 2034 can be computed from the angular diameter (converting from arc minutes to radians) as

r = d * diameter_angular * p / (60 * 180) / 2
= 8.16 Mpc * 2.818 * p / (60 * 180) / 2
= 3344 pc

Then we can solve the relation for orbital velocity

orbital velocity = (G * mass / radius)^1/2

for mass, obtaining

m = v² * r / G
= 55500² * (3344 * 3.086 * 10¹⁶) / G
= 4.764 * 10³⁹ kg
= 2.38 * 10⁹ M_solar

This is almost 9 times our estimate from the luminosity. We have probably over-estimated the mass because our rotational velocity represents the maximum possible consistent with the observations. But even if our velocity is twice the actual (which is unlikely), there is still more than twice as much mass in UGC 2034 as we can see.

While there is significant variation in the mass to luminosity ratio depending on both size and morphology, the general conclusions are the same: assuming that Newton's Laws hold throughout the universe, it is clear that galaxies have more matter than is visible to our telescopes!

Here is a plot of the measured rotation curves for a sample of Sb and Sc galaxies (source):

This issue raises an important question about the Tully-Fisher Relation between rotational velocities and luminosity: how is the dark matter, which contributes to the rotational velocities but not to the luminosity, linked to the luminous matter?

Portfolio Exercise 4: Create a table with the Hubble Distance, Doppler rotational speed, luminosity, luminous mass, angular diameter and total mass (from the orbital velocty relation) for all 20 galaxies in this sample.

Clusters of Galaxies

Here is a segment of the Coma Cluster, in visual with labels on the larger galaxies in the cluster, and in x-rays (the foreground star in the upper right is HD 112887):

(source)

Note that much of the x-rays emanate from a region which is visually devoid of matter; although this radiation comes from hot gases which are found in most clusters, it is not dark matter.

Galactic Rotation Curves and Dark Matter (II)

Since galactic rotation curves first indicated the presence of dark matter, gravitational lensing has enabled astronomers to image dark matter in the Bullet Cluster, here showing gas glowing red (in x-rays) from colliding clusters, and their dark matter in blue passing through relatively unimpeded:

(source); to compare distributions of visible and dark matter:

(source); and to describe the evolution of dark matter over some of the "recent" history of the universe:

(source). It is instructive to compare the dark matter in cluster Cl 0024+17 (in light blue) with the gravitationally lensed images of galaxies on the far side:

(source).

This applet lets you "design" a galaxy and view the corresponding rotation curve. It assumes that within the galactic disc, the mass is evenly distributed by volume. The same assumptions are made for the central bulge and the dark matter halo, which is drawn in violet. The disc thickness is assumed to be 1% of its radius, and the bulge radius is limited to 50% of the disc radius. The width of the graph is twice the largest chosen radius, and the color intensity is scaled so that the largest mass selected corresponds to the brightest colors.

You need a Java-capable browser to be able to use the applets. If they do not work with your Windows system, download the Java VM (Virtual Machine) for your version of Windows at the download section at java.sun.com.

Portfolio Exercise 5: In the sample galaxies we have been working with, the paper lists all as spirals except 2034, 5829, 5935 and 12060. For three of those spiral galaxies, use the applet above to create dark matter profiles. Start with the values you have calculated for the luminous mass, disc radius and dark matter mass. Use SIMBAD and the Aladin viewer to obtain an image of each galaxy, and estimate the bulge radius and the percentage of luminous matter in the bulge using the image. If a blue ("O" filter) JPEG is available, it is often the clearest image. If you cannot decide from the image, you may assume that the bulge radius is 20 to 30 % of the disc radius, and that the percentage of luminous mass in the bulge is about 30.
Begin with a dark matter radius of about 3 times the disc radius. Then vary the dark matter mass and radius until the rotational velocity curve (across the top) looks approximately like the ones in the graph above (or like the curve shown when the applet starts, which is approximately what is expected for the Milky Way). In most cases you will have to lower the dark mass from your calculated value to reproduce the expected velocity curve. Include images of your final profiles in your portfolio.

Please send comments or suggestions to the author.