The Age and Size of the Universe: Blog 27: Worksheet 8.1, Problem 3

It is not strictly correct to associate this ubiquitous distance-dependent redshift we observe with the velocity of the galaxies (at very large separations, Hubble’s Law gives ‘velocities’ that exceeds the speed of light and becomes poorly defined). What we have measured is the cosmological redshift, which is actually due to the overall expansion of the universe itself. This phenomenon is dubbed the Hubble Flow, and it is due to space itself being stretched in an expanding universe. Since everything seems to be getting away from us, you might be tempted to imagine we are located at the centre of this expansion. But, as you explored in the opening thought experiment, in actuality, everything is rushing away from everything else, everywhere in the universe, in the same way. So, an alien astronomer observing the motion of galaxies in its locality would arrive at the same conclusions we do. In cosmology, the scale factor, a(t), is a dimensionless parameter that characterizes the size of the universe and the amount of space in between grid points in the universe at time t. In the current epoch, t = \(t_0\) and \( a(t)_0 = 1\) a(t) is a function of time. It changes over time, and it was smaller in the past (since the universe is expanding). This means that two galaxies in the Hubble Flow separated by distance \(d_0 = d(t_0) \) in the present were \(d(t) = a(t)d_0 \)apart at time t. The Hubble Constant is also a function of time, and is defined so as to characterize the fractional rate of change of the scale factor: \[H(t) = \frac{1}{a(t)} \frac{ da}{ dt}|_t \] and the Hubble Law is locally valid for any t: \[v = H(t)d\] where v is the relative recessional velocity between two points and d the distance that separates them.

Part A: Assume the rate of expansion, a = da/dt, has been constant for all time. How long ago was the Big Bang (i.e. when a(t=0) = 0)? How does this compare with the age of the oldest globular clusters (= 12 Gyr)? What you will calculate is known as the Hubble Time.

Since we know \[v = H(t)d\] \[H(t) = \frac{v}{d} = \frac{1}{t} \] So \[ t_0 = \frac{1}{H(t)} = \frac{1}{68.816 \frac{km/s}{Mpc}} = 0.01453 \frac{Mpc \cdot s}{km} \] This is not a very useful number for us, so we can convert it to years: \[ 0.01453 \frac{Mpc \cdot s}{km} \cdot \frac{ 3.086 \times 10^{19} \: km}{Mpc} \cdot \frac{ 1 year }{3.15 \times 10^7}  = 1.385 \times 10^{10} \: years \] This is pretty close! The actual age of the universe is 13.82 billion years. This means the oldest globular clusters formed less than 2 billion years after the big bang.

Part B:  What is the size of the observable universe? What you will calculate is known as the Hubble Length.

Once again, we know \[ v = H(t)d\] So, if the universe is expanding at the speed of light and the speed of light in km/s is \( 3 \times 10^{5} \). \[ d = \frac{c}{H(t)} \] \[ d = \frac{3 \times 10^{5}}{68.816} = 4,356 Mpc \approx 4 \times 10^{3} Mpc \] The actual size of the universe is about \( 3 \times 10^{3} \) Mpc, so we aren't too far off.

Our old friends, Lyman-alphas: Blog 24 & 25, Worksheet 7.2, Problem 4 & 5

This week, we learned about active galactic nuclei and all of their fun applications. One of those applications in particular is especially dear to me, because the summer before tenth grade, I did a research project (and wrote a blog post about it) in a lab where I calculated the distance to ancient galaxies using redshift from Lyman-alpha emitting active galactic nuclei. And wouldn't you know, that's what we're doing in class this week:

Problem 4

One feature you surely noticed in a spectrum was the strong, broad emission lines. Here is a closer look at the strongest emission line in the spectrum:

This feature arises from hydrogen gas in the accretion disk. The photons radiated during the accretion process are constantly ionizing nearby hydrogen atoms. So there are many free protons and electrons in the disk. When one of these protons comes close enough to an electron, they recombine into a new hydrogen atom, and the electron will lose energy until it reaches the lowest allowed energy state, labeled n = 1 in the model of the hydrogen atom shown below (and called the ground state):

On its way to the ground state, the electron passes through other allowed energy states (called excited states). Technically speaking, atoms have an infinite number of allowed energy states, but electrons spend most of their time occupying those of lowest energies, and so only the n = 2 and n = 3 excited states are shown above for simplicity. Because the difference in energy between, e.g., the n = 2 and n = 1 states are always the same, the electron always loses the same amount of energy when it passes between them. Thus, the photon it emits during this process will always have the same wavelength. For the hydrogen atom, the energy difference between the n = 2 and n = 1 energy levels is 10.19 eV, corresponding to a photon wavelength of λ = 1215.67 Angstroms. This is the most commonly-observed atomic transition in all of astronomy, as hydrogen is by far the most abundant element in the Universe. It is referred to as the Lyman α transition (or Lyα for short). It turns out that that strongest emission feature you observed in the quasar spectrum above arises from Lyα emission from material orbiting around the central black hole.

Part A: Recall the Doppler equation: \[ \frac{ \lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}} = z \approx \frac{v}{c} \]Using the data provided, calculate the redshift of this quasar. \[ z =  \frac{ 1410 - 1215.67}{1215.67} \approx \frac{1}{6}  \] So that \[ v \approx \frac{c}{6}\]

Part B: Again using the data provided, along with the Virial Theorem, estimate the mass of the black hole in this quasar. It will help to know that the typical accretion disk around a \(10^8 M_{\odot} \) black hole extends to a radius of r = \(10^{15} \) m. 

Ideally, our emission lines would be infinitely thin and tall, but they have width. This happens because there is some dispersion due to the rotation of the galaxy. This means some of the galaxy will be redshifted and some will be blueshifted. 

So we can use this to find rotational velocity.

\[ z = \frac{ \lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}} = \frac{1415 - 1406}{1406} \approx 0.00640 \]

\[ z = \frac{1415 - 1406}{1406} = 0.0064 \] \[ v \approx 0.0064 c \]

Now we can use the virial theorem: \[ K = - \frac{1}{2} U \] \[ Mv^2 = \frac{G M^2}{R}\] \[M_{BH} = \frac{ v^2 R}{G} = \frac{( 0.0064 \cdot 3 \times 10^{10})^2 (10^{17})}{6.67 \times 10^{-8}} = 5.5 \times 10^{40} \: g \approx 3 \times 10^7 M_{\odot} \] 

Problem 5

You may also have noticed some weak “dips” (or absorption features) in the spectrum:

Part A: Suggest some plausible origins for these features. By way of inspiration, you may want to consider what might occur if the bright light from this quasar’s accretion disk encounters some gaseous material on its way to Earth. That gaseous material will definitely contain hydrogen, and those hydrogen atoms will probably have electrons occupying the lowest allowed energy state.

As the question implies, we are probably looking at instances where light is being absorbed or emitted by hydrogen gas on its way to us. These dips themselves, occurring before the peak, imply that this gas is between the galaxy and us, therefore they are not as redshifted as the distant galaxy. 

Part B: A spectrum of a different quasar is shown below. Assuming the strongest emission line you see here is due to Lyα, what is the approximate redshift of this object?

\[ \frac{ \lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}} = z \approx \frac{v}{c} \]

\[ z = \frac{5650 - 1215.67}{1215.67} = 3.347 \approx 3\]

Part C: What is the most noticeable difference between this spectrum and the spectrum of 3C 273? What conclusion might we draw regarding the incidence of gas in the early Universe as compared to the nearby Universe?

In this galaxy, there are many more variation, especially a larger number of deeper "dips." Based on the large redshift of the galaxy, we know it is quite old and far away. This means the light had to have traveled through a lot more gas to get here, and some of that gas had hydrogen atoms occupying the lowest energy state. Additionally, this could tell us that there was a lot more gas in the early universe. This fits in well with our current ideas of the early universe. 

Deriving Hubble's Law: Blog 26: Worksheet 8.1: Problem 2

In 1929, astronomer Edwin Hubble discovered that almost all distant galaxies exhibit a positive redshift. Furthermore, it appeared that the farther the galaxy, the larger its redshift. Here we will rediscover Hubble’s Law using modern spectroscopic data and supernovae Ia as our distance indicator to these galaxies. The data we will use come from the Sloan Digital Sky Survey (SDSS), a project that aims to comprehensively map the universe. You can access the relevant data products for this exercise at http://goo.gl/fmIvqc

Part A: Below is a list of supernovae observed between 2004 and 2007 and their positions in RA and Dec. You can find the images and spectra of their host galaxies by entering their coordinates in the respective fields. Explore the functions available, including magnifying the image, reading off the photometric measurements (magnitudes in wavebands u, g, r, i, z) of your selected object, and using the ‘Explore’ button to access more quantitative measurements for these objects. In particular, familiarize yourself with the ‘interactive spectrum’ feature.

For each galaxy, we have a pretty extensive spectrum.

Part B: One of the features for determining distances to Type Ia supernovae is its peak absolute magnitude. You explored the peak bolometric luminosities of SN Ia’s in Worksheet 7.1. The peak V-band magnitude for SN Ia’s is about -19.3. Use the apparent peak magnitudes given in the table above to calculate the distance of these supernovae in unit of Mpc.

To find distance, we can use our trusty distance formula :\[d = 10^{ \frac{m - M + 5}{5}} \] for each magnitude. The fourth galaxy did not have a spectrum associated with it, so we did not analyze it.

Part C: We can use the absorption or emission lines of the host galaxy to find their redshifts which, as you found in Question 1), roughly equals the recessional velocity as a fraction of the speed of light. To measure the redshift to each host galaxies, click on ‘Explore’ and then on the link ‘Interactive Spectrum’. Uncheck the boxes Best Fit and Mark Emission Lines. Zoom in on the absorption line labeled Hα, and move your mouse over to the center of the line to read the observed wavelength in Angstroms. The Hα has a rest (i.e. emitted) wavelength of 6563.0 Angstroms. Calculate the redshift, and then derive the radial velocity in kilometers per second, using the relation z = v/c. How close does your redshift measurements compare to the one SDSS reports in the table under the Interactive Spectrum link? Repeat for all the galaxies.

We found redshift and velocity using \[ \frac{ \lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}} = z \approx \frac{v}{c} \] to find: 

Part D: Make a plot of your findings, with distance on the x-axis and velocity on the y-axis. Report the slope of your line in appropriate units. This is the Hubble Constant, H0.

Result: \(H_0 = 68.816 \frac{km/s}{Mpc} \)

Part E: Write an equation for this line in the form of v = ___ D, where v is an object’s recessional velocity and D is the distance to that object. Express your Hubble Constant in terms of units km/s/Mpc. Congratulations, you have arrived at Hubble’s Law!
\[ v = 68.816 \frac{km/s}{Mpc} D \] Now we have a basic law of the universe (kinda) down!

Blog 23: Astronomy and the Harvard Art Museums

This week, we are going to explore the intersection of Astronomy and another field because while Astronomy is the study of the universe, it is not the only subject to study in the universe. Last semester, I talked about Astronomy and Mystery-Solving. This semester I started a job as a tour guide at the Harvard Art Museums (HAM). At HAM, tour guides have to design their own tours based on a central theme connecting 3-5 objects on display in the museums. Themes vary from political to religious to technical. For this blog post I am going to try to propose an astronomy-themed tour based on our collection. All images and information about the objects can be found at the museum's website.

Admittedly, astronomy is not one of the most centralizing themes in art history. However, it has been a constant part of our lives before, throughout, and long after history. The night sky has always been round us, both inciting our imaginations and giving us the tools to understand the world we live in. The oldest astronomy-themed object on this tour is the Long-Case Musical Clock, created by the Dutch craftsman Otto Van Meurs in 1750. The grand, imposing clock sits in a seconds floor gallery of the museum, still keeping accurate time - with a little adjusting from conservationists every set number of years. That is not all the clock does; it also tracks tides, the current position of the sun and moon, gives the date, and plays one of dozens of charming tunes every half hour. Mechanical clocks like these are one of the first iterations of modern computers. This "computer" in particular, heavily showcases society's uses for the skies above. Our entire conception of time, of calendars and schedules, of past, present, and future is based on the motion of stars and planets. We can see that in this piece, a representation of years of evolution of the clock and calendar.

Yet this piece isn't purely functional. It is highly symbolic, covered in classical allusions such as Atlas carrying Earth at the very top of the clock. This piece, used to show the high societal status of its owner, is inherently intertwined with modern cultural norms. This culminates in a symbolic, functional, and aesthetic piece that dominates any room it is in.

More than a century later, Paul Manship (considered the most prominent American Art Deco artist) created about a dozen celestial spheres as a preparation for a grand memorial to Woodrow Wilson. This historically influential tool was also highly functional, allowing for navigation, charting, and seasonal predictions. Even today, it is our foremost model of the night sky.

Celestial Sphere - Paul Manship

However, this piece itself is also not purely functional, it is not even very useful. Its chief function is to memorialize a political figure and does so with many cultural symbols. The constellations and base figures are a mix of Chinese, Babylonian, Assyrian, Latin, and Greek zodiacs to emphasize the international political harmony the artist wanted a viewer to associate with Woodrow Wilson. Once again, Astronomy is used here as a seamless part of history and modern society, and not just for one culture, but for many. 

There are definitely cultural levels of astronomy that move beyond the scientific into the even more cultural and aesthetic. For example, the second floor has a beautiful painting with a puzzling composition that signifies the coming of the dawn, the daily transition from dark to light and a pagan goddess. The astronomical phenomena has become a personified deity, with her own gossip, lovers, and enemies.

The Dawn - John LaFarge

Even Christian religions have used Astronomy as cultural markers. In this stained glass window, angels are represented by the morning stars in a reference to a biblical passage. The stars symbolized heavenly light, purity, and truth.

When the Morning Stars Sang Together and All the Sons of God Shouted for Joy

Even in contemporary art today, these themes persist. De Kooning's piece, Untitled (The Cow Jumps Over the Moon), plays upon a common cultural anchor, a childhood nursery rhyme. From there, De Kooning's questions a viewer's perspective of traditional art and accepted ways of portraying cows and emotions alike. 

Willem de Kooning - Untitled (The Cow Jumps Over the Moon)

Even in art's evolution, Astronomy is used as a grounding point. The night sky has become irrevocably intertwined with our perception of the world, and what better way is there to portray our world than through art?

How far away are those supernovae? Blog 22, Worksheet 7.1, Problem 6

If your telescope can detect optical magnitudes \(m_V < 21\), how far away, in parsecs, can you detect a Type Ia supernova with your telescope? (HINT: The Sun’s absolute magnitude is \(M_V = 4.83 \) .)

What we know is \(M_{ V \odot}  = 4.83 \) and \(L_{WD} = 10^{10} L_{\odot} \).
To solve, we can revisit week 5 to solve this problem. We know that absolute magnitude is related to luminosity: \[ M_{WD} - M_{\odot} = -2.5log \left( \frac{L_{WD}}{L_{\odot}} \right) \] So \[ M_{WD} =  -2.5log \left( \frac{10^{10} L_{\odot}}{L_{\odot}} \right) + M_{\odot}  \]  \[ M_{WD} =  -2.5log \left( 10^{10} \right) + 8.83 = -25 + 4.83 = -20.17  \] Now that we have a (rather large) absolute magnitude and a limit on our ability to detect apparent magnitudes, we can find distance. \[ m - M = 5log(d) - 5 \] \[ d = 10^{\frac{m-M +5}{5}} \] \[ d = 10^{\frac{21 + 20.17+5}{5}} = 1.7 \times 10^9 \: pc \] By comparison, the center of the galaxy is about 8 kilo-parsecs away so we would get a decent range within our galaxy.

Source

Supernovae can be amazingly bright.

Boom: Blog 21, Worksheet 7.1, Problem 4

This week we are going to talk about one of the brightest, yet most transient, objects in our sky: supernovae.

Source

Astronomers believe that most supernovae are the result of 1) the gravitational collapse of a massive star at the end of its main-sequence life or 2) the explosion of one or more white dwarfs, likely caused by the collision between two white dwarfs. In this problem, we’ll focus on the second type of supernovae, or Type Ia supernovae which we are familiar with.

Calculate the total energy output, in ergs, of the explosion, assuming that the white dwarf’s mass is converted to output energy via fusion of carbon into nickel. Note that the process of carbon fusion is not entirely efficient, and only about 0.1% of this mass will be radiated away as electromagnetic radiation (light!). How does this compare to the total binding energy, in ergs, of the original white dwarf? Does the white dwarf completely explode, or is some mass left over in the form of a highly concentrated remnant?

To calculate energy output we can call upon possibly the most famous equation in science: \[ E=mc^2 \] Well, 0.1% of that energy anyway. Because we know the speed of light \( c = 3 \times 10^8)^2 \) and the mass of a white dwarf is \( M_{WD} = 1.4 M_{\odot}\). Plugging everything in gives us: \[E = \frac{1}{1000} (1.4 \times 2 \times 10^{32} ) ( 3 \times 10^{10})^2  \frac{1}{1000} (1.4 M_{\odot} ) ( 3 \times 10^8)^2 = 2.7 \times 10^{51} \: ergs \]

We can find binding energy from the potential energy of the star: \[U = E = \frac{GM^2}{R} \] We also know the radius of the White Dwarf is about twice the radius of the Earth. \[ E_{binding} = \frac{ c_s  M_{WD}^2 }{R_{WD}} = \frac{ (6.67 \times 10^{-8}) (1.4 \times 2 \times 10^{33})^2 }{2 \times 6.4 \times 10^8} \approx 4.1 \times 10^{50} \: ergs \]

There is something off here, the energy output is not equal to the energy input. So how can that be? Thanks to Einstein (e.g. \(E = mc^2 \) ) we know that mass can be converted to energy. In the explosion, some of the mass of the white dwarf gets turned into energy, and the rest is flung out throughout the galaxy. Explosions like these can even help explain why we have heavier elements dispersed through the Milky Way.

Blog 20: The Hubble Tuning Fork

The Hubble Tuning Fork

The Hubble Tuning Fork is a classification system developed by Edwin Hubble in 1926 as a theory of galaxy evolution. Today we view it as outdated, as there is evidence that spiral galaxies do not evolve from ellipticals, but it is a useful visual for galaxy types.

Ellipticals (E0-E7): 

Elliptical galaxies are spheroid or elongated spheres where stars have no uniform rotation around the center of the galaxy. Elliptical galaxies tend to have older, and therefore redder, stars. Elliptical galaxies are classified due to their shape. E0 galaxies appear almost perfectly spherical, at least from our perspective on Earth. E3 galaxies appear as slightly elongated ellipsoids, E5 slightly more, and E7 galaxies are extremely elongated.

Lenticular galaxies (S0):

Lenticular galaxies (charmingly named after the lentil bean) fall between elliptical and spiral galaxies. They have no spiral shape but do contain a bulge and thin disk, similar to a spiral galaxy.

Spiral Galaxies (Sa-Sc) and Barred Spiral Galaxies (SBa-SBc):

Spiral galaxies are galaxies like the Milky Way or Andromeda. They typically contain a disk around which stars revolve, a bulge, and spiral arms. Spiral galaxies contain younger, bluer, stars than ellipticals. Spirals are classified by how tightly their arms are wound with Sa spirals being the tightest wound and Sc being the loosest. Barred spirals have a bar of stars running across the nucleus out of which the arms extrude. Barred spirals are also classified by the distribution of their arms.

Irregular:

Irregular galaxies do not fit into any of these categories and are often the result of two galaxies colliding.

• E0: Image: M87
• E3: Image M86
• E5: Image: M59
• E7: Image
• S0: Image: Messier 102
• SA: Image: NGC 1302
• Sb: Image: M77 
• Sc: Image: NGC 628
• Sba: Image: M83
• Sbb: Image: Ngc 1300
• Sbc: Image: NGC 1365