H205 – Cosmic Origins Exploration Packet 4: Galaxies Your Name Due April 8, 2009 Part 1 – Classifying Galaxies

Part 2: The Distance to the Galaxy M100 using Cepheid Variables

(This exercise is based on an ESA/ESO Astronomy Exercise, Series 2, and has been adapted for use in our class. The original data on which the exercise is based were taken from Freedman et al. 1994, Nature, 371, 757)

From repeated observations of the galaxy Messier 100 using the Hubble Space Telescope, astronomers were able to identify more than a dozen Cepheid variables, and measure their brightness as a function of time. Cepheid variables are pulsating stars with regular periods of a few weeks. Henrietta Leavitt, working at the Harvard College Observatory early in the 20th century, discovered that the period of a Cepheid variable is related to its brightness. Brighter Cepheids pulsate with longer periods. From the Hubble measurements of Cepheid variables in M100, the periods of the variables, and hence the absolute magnitudes of the variables, could be determined. Once both the absolute magnitudes and the apparent magnitudes are known, the distance to M100 can be calculated.

The relation between a Cepheid's period and its absolute magnitude is given by the “Leavitt Law:”

M_v = -2.8 log(P) -1.4

Where M_v is the absolute magnitude and P is he period in days.

For convenience, this expression is shown in graphical form in the plot below.

Determining the distance to M100 - Use the light curves given in the ESO materials to determine the periods of each of the Cepheids, as well as the average apparent magnitude of each one.

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  The period is the time between successive maxima of the Cepheid variable.
  
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  The absolute magnitude can be determined from the period using either the chart above or the expression on the previous page.
  
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  The average apparent magnitude is half way between the maximum and minimum brightness in the light curve.
  

Work in pairs to enter the period, absolute magnitude, and average apparent magnitude for three more Cepheids in the table below. The measurements for Cepheid No. 1 have already been entered.

Cepheid Number	Period (days)	Absolute Magnitude (M)	Maximum App. Mag. (mmax)	Minimum App. Mag (mmin)	Average App. Mag. (mave)	Distance Modulus (mave-M)	Distance (Mpc)
1	53.4	-6.2	24.5	25.3	24.9	24.9-(-6.2)=31.1	16.6
2
3
4
5
6
7
8
9
10
11
12
Average distance to M100 in Megaparsecs:

In astronomy, the "absolute magnitude" of a star is the apparent magnitude that star would have if it were at a distance of 10 parsecs (a parsec is 3.26 light years). Thus the difference between the absolute magnitude and the apparent magnitude is related to the distance. The relation can be written mathematically as:

Distance in parsecs = 10^{(mave -M+5)/5}

where m_ave is the average apparent magnitude and M is the absolute magnitude.

The Cepheid variables in M100 give a wide range of distances. What are some possible reasons why the distances might vary so much from star to star?

Compare the size of M100 to its distance (M100 is similar in size to our own Milky Way, with a diameter of about 30 kiloparsecs). Could the variation in the distance to M100 from the different Cepheid period measurements result from the difference in location of the Cepheid variables within M100? Why or why not?

In the original scientific paper reporting on the distance to M100 using the Hubble observations, the distance was calculated to be 17.1 ± 1.8 Mpc (about 56 million light years). The presence of interstellar dust within M100 causes the apparent magnitudes of the Cepheids to be dimmer than they should be, so that they appear to be more distant than they really are.

Part 3: Distances from Apparent Diameters of Spiral Galaxies

We can measure the distances to spiral galaxies by measuring their angular diameters, and assuming that most spiral galaxies have about the same radius, about 22 Kpc.

Visit the website http://www.astro.washington.edu/courses/astro211/HubbleLaw/galaxies.html

and select Image for two of the following galaxies. (Please work in pairs and select galaxies so that all are measured by some group:

NGC 1357, NGC 1832, NGC 2276, HGC 2775, NGC 2903, NGC 3034, NGC 3147, NGC 3227, NGC 3368, NGC 3623, NGC 3627, NGC 4775, NGC 5248, NGC 5548, NGC 6181, NGC 6217, NGC 6643, NGC 6764.
Find the angular size of the galaxy using its image. There may be more than one galaxy in the image; the galaxy of interest is always the one closest to the center. To measure the size, click on opposite ends of the galaxy, at either end of the longest diameter. Be sure to measure all the way to the faint outer edges. Otherwise, you will dramatically underestimate the size of the galaxy, and introduce a systematic error.
Your measurements are in image pixels. Compute the diameter of the galaxy in pixels using the Pythagorean Theorem (x²+y²=diameter²). Convert the diameter from pixels to degrees as shown below.

The distance in Kpc is given by: distance (in Kpc) = 22 Kpc / tangent (diameter in degrees)

Galaxy Name ______________________ Diameter in pixels___________________________

Diameter in arc min _________________ Distance in Mpc ____________________________

Galaxy Name ______________________ Diameter in pixels___________________________

Diameter in arc min _________________ Distance in Mpc ____________________________

Part 4: Recession Velocities

The recessional velocity of a galaxy is measured using the Doppler effect and is given by:

velocity = c * (
where v is the velocity in km/sec, c is the speed of light (300,000 km/sec), is the rest wavelength of the radiation, and  is the amount the radiation has been shifted towards longer wavelengths ( = the observed wavelength minus the rest wavelength). For these spectra, wavelengths are usually measured in Angstroms (Å), rather than nanometers. An Angstrom is 1/10 of a nanometer, 10^-10 meters. The red-ward shift of a spectral line of radiation relative to its rest wavelength,  is known as the redshift, and is often denoted by the letter z.
We can determine the velocity and redshift of a galaxy from its spectrum: we measure the wavelength shift of a known absorption line and solve for v.
Example: An absorption line that is found at 5000Å in the lab is found at 5050Å when analyzing the spectrum of a particular galaxy. Therefore this galaxy is moving away from us with a velocity

v = (50/5000) * c = 3000 km/sec

Using the same website as above, click on “spectrum” for the two galaxies whose distances you measured. The optical spectrum of the galaxy is shown at the top of the spectrum page. Shown are many different spectral features, including absorption lines and emission lines, superimposed on continuum emission from the galaxy, over the entire visible-light spectrum. Below the full optical spectrum are enlarged portions of the same spectrum, in the vicinity of the spectral features of calcium and hydrogen.

The small dark bars near the lower left corner of the sub-spectrum indicate the rest wavelengths of particular lines. Measure the wavelength by clicking at the middle of the spectral line in the galaxy's spectrum. Important note: Ca H and Ca K are “absorption lines” – this means that they look like big dips in the spectrum. The spacing between these two dips is approximately the same as the spacing between the two small dark bars that show their rest wavelengths. Try to select the deepest point in each dip for your measured wavelength. In contrast, H is a strong “emission line” – the tallest sharp peak in the spectrum. Select the point at the peak for the measured wavelength of this line. Enter the positions of the two Ca features and the H-alpha feature below.
Galaxy Name __________________________
Position of Ca K ____________________ Velocity from Ca K _____________________________
Position of Ca H ____________________ Velocity from Ca H _____________________________
Position of H-alpha __________________ Velocity from H-Alpha ___________________________
Average Velocity _______________________________
Galaxy Name ______________________
Position of Ca K ____________________ Velocity from Ca K _____________________________
Position of Ca H ____________________ Velocity from Ca H _____________________________
Position of H-alpha __________________ Velocity from H-Alpha ___________________________
Average Velocity _______________________________

The velocities can be calculated from the positions of the spectral lines as follows:

For the Ca K line: km/sec

For the Ca H line: km/sec

For the H-alpha line: km/sec

Part 5: The Hubble Constant

We will collect the velocities and distances for the set of galaxies listed above to plot the recession velocity vs. distance.

What is the Hubble Constant based on our measurements?

___________________________________

Part 6 -The Age of the Universe
If the universe has been expanding at a constant speed since its beginning, the universe's age would simply be 1/H_o.
First, convert H_o to inverse-seconds (1/sec). Since 1 Mpc = 3.09 x 10¹⁹ km , this entails dividing your calculated value for the Hubble Constant by 3.09 x 10¹⁹ km.
Now calculate the "expansion age" of the universe: t = 1/H_o. This gives an approximate age of the universe in seconds. To convert this number into years, divide it by 31,536,000 (the number of seconds in a year).
Expansion age of the Universe = ___________________________________
This is a very simple model for the expansion of the universe. A better model would account for the deceleration caused by gravity. Models like this predict the age of the universe to be: t = (2/3)H_o. Re-calculate the age using this relation.
t = _______________________________________

How does this value compare to the age of the Sun, which is about 4.5 billion years old?

For Wednesday, April 8

Part 7: Supernova in M51
At least two supernovae have been detected in recent years in the nearby Whirlpool Galaxy, Messier 51. M51 is located at a distance of about 31 million light years (about 10 megaparsecs) in the direction of the constellation Canes Venatici.
Stars can explode as supernovae in different ways, and the different explosion mechanisms produce characteristically different behavior of brightness with time, as the supernova brightens and then fades again. The two dominant types of supernovae are Type Ia supernova, which are thought to be exploding white dwarfs, and Type II supernovae, which are explosions of massive stars that run out of nuclear fuel.

The light curves of supernovae of Type Ia and Type II are compared in the figure below. Type Ia supernova rise to maximum and then fall steadily in brightness, fading in just a few weeks. Type II supernovae remain bright for a longer period of time as energy is provided by the decay of radioactive products produced in the explosion.

From observations of the light curve of SN1994i in M51, determine whether the explosion is a Type Ia (exploding white dwarf) or a Type II (massive star) supernova. The observations were obtained with the Leuschner Telescope between April 2 and May 6 in 1994.


Note: Because sky conditions vary, the brightness of the supernova should be compared to the brightness of the two stars in the image, in the lower left center and lower right.
Does SN1994i appear to be a Type Ia or a Type II supernova? _______________________________

Part 8: Absolute Magnitudes of Supernovae This exercise is based on one developed by Lindsay Clark (see http://www.astro.princeton.edu/~clark/SNLab.html), and has been adapted for our use.

On the accompanying pages are the light curves of nine Type Ia supernovae that appear in galaxies during the years 1994-1996 (Type 1a supernovae are a particular type of exploding star). The units on the x-axis are days, and on the y-axis are apparent magnitude. Each supernova was monitored for several weeks so that its rise to maximum light and its subsequent decline in brightness are well-determined. For each, the apparent magnitude at which the supernova was brightest can be determined from the light curve. Our goal is to determine the absolute magnitude of each supernova.

Record the brightest magnitude reached for two of the supernova in the table below.

Supernova	Host Galaxy	z	Recession velocity (km s-1)	Distance in Mpc for Ho=72 km s-1 Mpc-1	App Mag (mmax)	Absolute Magnitude (M)
1994S	NGC 4495	0.01610	4,463		14.8
1995D	NGC 2962	0.00655	1,998
1994ae	NGC 3370	0.06700	1,279
1995al	NGC 3021	0.00514	1,536
1995ac	Anon	0.04900	14,615
1995bd	UGC 3151	0.01520	4,758
1996X	NGC 5061	0.00681	2,034
1996bl	Anon	0.3480	10,598
1996bo	NGC 673	0.01650	5,063
Mean Absolute Magnitude
Median Absolute Magnitude

The recession velocities of the host galaxy of each supernova were measured from the redshift of spectral lines. The recession velocity must be measured from the host galaxy, rather than from the supernova itself, because the gas from which the supernova spectrum arises is expanding explosively from the original supernova progenitor.

Use the value of the Hubble Law to compute the distance of the host galaxy (and the supernova) from the recession velocity. Remember that the distance = velocity / H_o.

Calculate the absolute magnitude from the distance and the apparent magnitude using the inverse square law. Recall that the distance in parsecs is related to the difference between the absolute and apparent magnitude as follows:

Distance in parsecs = 10^(m-M+5)/5

In this case, we know the distance and apparent magnitude and want to determine the absolute magnitude. The expression can be rewritten as

M = m – 5 log(distance in parsecs) + 5

Use a calculator to obtain the logarithm of the distance in parsecs (the distance in parsecs is 10⁶ times larger than the distance in megaparsecs. The absolute magnitudes of the supernovae should be negative, since the supernovae are very bright compared to ordinary stars. Remember that astronomical magnitudes become smaller, and then negative, as objects become brighter. Each 5 magnitudes is a factor of 100 in brightness.

How does the brightness of a Type Ia supernova compare to the brightness of the Sun? The Sun's absolute magnitude is approximately 5. (The Sun's apparent magnitude is -26.)

What can you conclude about the absolute magnitudes of Type Ia supernovae?

What is peculiar about the supernova 1995bd? What explanation might account for its difference from other Type Ia supernovae?

How might your conclusions about Type Ia supernovae be useful for determining the distances to other galaxies?

Part 9: Reflection (5 minutes, max) – Write a short statement (no more than five sentences) summarizing the basic ideas covered in this Exploration about galaxies and the methods astronomers use to determine the distances to galaxies.