Physics 121 November 23, 2009

Today’s class: • Schwarzschild Metric (continued from last class) • Hubble’s Law

Logistics:

• No class Wednesday, November 25 • The next homework assignment will be posted in a

couple days; due on December 4. • There will be cosmology handouts distributed at the next

class meeting (November 30), possibly posted earlier.

ΔS2 =

General Relativity: Schwarzschild Metric

According to general relativity, the spacetime interval at distance r from a star of mass m is given by the Schwarzschild Metric:

r 2m 1

− Δt2

r 2m 1

1

−

Δr2 – r2Δθ2–

Deflection of Light Time Delay of Light The Schwarzschild metric predicts that light passing near mass m will be deflected. (We haven’t proven this. But it’s true.)

Arthur Stanley Eddington, a British astronomer, realized that this prediction could be tested by looking for stars’ positions to “move” slightly as their light went close to the Sun. But there was a problem: you can’t see stars when they are close to the sun. His solution: observe the stars during an eclipse of the Sun.

This is a negative of Eddington’s photograph of the 1919 eclipse, taken on Principe (off the west coast of Africa).

In comparing positions of stars on this photograph with their positions when they are far away from the Sun, he found that their positions had changed as predicted by the Schwarzschild metric.

Another prediction of the Schwarzschild metric is that, according to distant observers, a pulse of light takes longer to travel through space near mass m than it would in flat space.

This has been observed! • Radar signals bounced off of Venus take longer to get there and back when they travel near the sun.

• Radio pulsar signals can be delayed by tens of microseconds when traveling near binary star companions to pulsars.

pulsar

companion star

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Binary Pulsar: Doppler Shifts

PSR J1903+0327: A pulsar (neutron star) in a 95 day binary orbit with another star. We don’t yet know much about the companion star.

(You’ve seen this plot in your homework.)

Binary Pulsar: Orbital Delays

The top plot shows delays in the time pulses are received due to the position of the pulsar in its orbit.

The middle plot shows what is left after subtracting off a model of the orbit and pulse propagation using a metric for flat space. An extra delay of 47 microseconds is seen when the pulsar is furthest behind its companion star.

The bottom plot shows what is left when we use a model based on the Schwarzschild metric. The orbit and pulse propagation are fully explained.

Reference: Champion et al. 2008, Science, 320, 1309.

Black Holes Black Holes

ΔS2 = 1− 2m Δt2 – 1 Δr2 – r2Δθ2

r 1− 2m

r As we discussed last class, on of the implications of the Schwarschild metric is that lines on the “graph paper” of space become closer and closer together at smaller and smaller values of r, meaning that light travels less and less distance in a given amount of time (according to a distant observer) as it gets closer to the central mass. At r=2m, the “graph paper lines” have no separation at all, and light can’t move forward any farther. This is the event horizon of a black hole. From our (outside) perspective, things head toward the event horizon but never quite get there, and the clocks on those things become infinitely slow as they approach the event horizon. Oddly enough, something falling into a black hole, because its clock has slowed down, doesn’t feel things becoming infinitely slow. In fact, this thing can fall through the event horizon according to its own observations. We just can’t see it go there. And, once it is there, it can’t send us any signals of any kind.

This all sounds quite ludicrous! How can we find black holes?

We need to look for places where there is a lot of mass packed into a small volume.

This all sounds quite ludicrous! How can we find black holes?

We need to look for places where there is a lot of mass packed into a small volume.

In essentially all cases of interest, we can measure (or at least estimate) the mass of something by observing orbits of stars or other things surrounding it. We haven’t talked about orbits in this class (and we’ll leave the details for Physics 122 and beyond), but let’s mention one essential feature: the heavier something is, the faster things orbit around it. If we can measure or estimate the speed of something orbiting it, and the size of the orbit, we can estimate its mass. If there is a lot of mass in a small area, it is a good guess that it is a black hole.

Also important, of course, is that black holes themselves don’t emit light. Once we’ve found a lot of mass somewhere, we can usually either attribute it to (i) a whole lot of stars, or (ii) a black hole. If case (i) is correct, there should be a lot of light. If there isn’t a lot of light, then we conclude it is a black hole.

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Black Holes

Three (types of) places we firmly believe that black holes have been observed:

• Active galactic nuclei (AGN) (An example was on Physics 121 exam #4.) Intense emission of light, radio waves, and X-rays shows that there is a lot of energy being produced near the cores of some galaxies. Observations of matter close in to the galaxy cores strongly suggests that these galaxies harbor super massive black holes around one billion times the mass of our Sun. One nice detail in these observations is that the signals are redshifted, as expected for light traveling outward through the Schwarzschild metric.

• The center of our Galaxy (see next slide) Observations of orbits of stars near the center of our Galaxy show them to be rapidly orbiting something we cannot see. Analysis of the star orbits shows that whatever they are orbiting has a mass of around four million times the mass of our Sun. If this were composed of four million stars, it would emit copious amounts of light, infrared radiation, etc. The fact that we don’t see this leads us to conclude it is a black hole.

• Compact binary systems (see later slides)

The Center of our Galaxy

Reference: Prof. Andrea Ghez, UCLA www.astro.ucla.edu/~ghezgroup/gc

This picture shows a false-color image of stars near the center of our Galaxy. The picture was taken using infrared light because visible light can’t get through the dust that permeates the Galaxy between the center and us.

The next two slides show computer animations of observations of the region around SgrA* (Sagittarius A*).

The Center of our Galaxy The Center of our Galaxy

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- i l i i l l

Compact Binaries

In the late stages of stellar evolution, stars pass through a phase in which they grow tens or hundreds of times larger than their normal size.

When a star in a close-in binary system goes through this phase of evolution, some matter on its surface gets pulled toward its companion star.

If the companion star is a compact star (white dwarf, neutron star) then this in-falling matter loses a lot of energy as it hits the star’s surface, putting out a lot of light and/or X-rays.

If the companion star is a black hole, we can see some light/X-rays as the matter spirals toward it, but the intense emission characteristic of matter hitting the surface is missing.

Figure: NASA The current best guess is that stars greater than eight times the mass of the sun eventually evolve into black holes.

Cosmology

The Hubble Ultra Deep Field

High resolution spectrum of the sun McMath P erce Te escope, K tt Peak; Nat ona So ar Observatory

400 nm 500 nm 600 nm 700 nm

infrared ultraviolet

The Sun’s Spectrum

Absorption spectrum of the sun: a cartoon picture

• white light (all colors) is generated slightly below the surface of the sun

• atoms on the thin outer layers selectively absorb some wavelengths, leaving black lines on the spectrum due to the absence of light; the other light passes right through

(Note: in reality, white light is generated throughout the sun, and absorption of light of selected wavelengths also occurs throughout the sun.)

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“ ”

400 nm 500 nm 600 nm 700 nm

infrared ultraviolet

Moving Star: Doppler Shift

400 nm 500 nm 600 nm 700 nm

infrared ultraviolet

Emitted Spectrum

Observed Spectrum

What Data Really Look Like

400 nm 500 nm 600 nm 700 nm

440 nm 450 nm 460 nm

440 nm 450 nm 460 nm

Light Intensity

Intensity more-or-less constant for most wavelengths…

…except that it is much smaller where there are spectral absorption lines

30,000 parsecs 30 kiloparsecs

0.030 megaparsecs

Artist’s impression of the Milky Way galaxy.

1 Megaparsec

The local group of galaxies. Motion of these galaxies is dominated by gravitational interactions with the Milky Way and Andromeda galaxies.

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Galaxies more distant than the local group....patterns emerge....

DISTANCE IN MEGAPARSECS

12 Mpc

160 Mpc

200 Mpc

400 Mpc

600 Mpc

These “reference lines” show where spectral lines would be found in a galaxy at rest (v=0)

This is the spectrum of the galaxy pictured at right. The most prominent lines are the Calcium “H” and “K” lines.

The faint white arrows show how much the H and K lines are shifted for each galaxy.

• Nearly all galaxies have lines shifted in the same direction; they are all moving away from us

• The rate at which they are moving away increases with increasing distance.

Edwin Hubble Measured distances and velocities to many galaxies, and was the first to notice that velocities of galaxies were proportional to their distances from us. The further away a galaxy, the faster it moves. This is now called “Hubble’s law”

How exactly did Hubble measure velocities and distances?

Velocities: Doppler shifts of spectral lines. These are reliable measurements.

Distances: These were harder. Hubble observed a particular kind of star called a “Cepheid variable.” These are very bright stars which have the peculiar property that they regularly get brigher and dimmer with periods of, typically, a few days. The important thing about Cepheids is that all Cepheids with a given variation period have the same intrinsic luminosity—so, if you see two Cepheids with the same period, but one is four times dimmer than the other, you know it must be two times farther away. (Recall that intensity is inversely proportional to distance squared.)

Observations of Cepheids are very good to get relative distances to different galaxies (example: galaxy A is twice as far as galaxy B), but they were not as good (in Hubble’s day) for getting absolute distances (example: galaxy A is 100 Mpc from us). This is because, although it was known that Cepheids of a given period all have the same luminosity, it was not known what the numerical value of that luminosity actually was.

Hubble’s law: v = H0 d

velocity of a distant galaxy receding away from us

distance to the distant galaxy

Hubble constant H0=71 pcMskm

In fact, because of problems with the Cepheid distance scale, Hubble’s distances are substantially incorrect, and his value for H0 is wrong--see homework for details.

All distant galaxies are moving away from us!

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All distant galaxies are moving away from us — and from each other!

How can this be?

The fabulous balloon demonstration

Key idea:

Space itself is expanding!! In the language of general relativity, the “graph paper” we use to measure distances is expanding over time.

The Big Bang

The universe is expanding as we move forward in time.

Imagine going backward in time.

The universe was smaller and smaller, and denser and denser, the further we go back..

We call the starting point of this process the “big bang.”

The Metric for an Expanding Universe

We can describe the expansion of the universe by defining a “scale function”, R(t). By definition, R=1 right now. At earlier times t, when the universe was smaller, R(t) was smaller. As the universe expands, R(t) is getting larger.

We can take the flat-space metric,

Δs2 = Δt2 – (Δx2 + Δy2 + Δz2)

And modify it to allow for an expanding (or contracting) universe:

Δs2 = Δt2 – R2(t)(Δx2 + Δy2 + Δz2)

The Big Bang

We have no idea what happened before the big bang, or how to even ask that question. We have no idea why the big bang happened. However, there is very strong evidence that it did, indeed, take place:

• Hubble’s law tells us that the Universe is rapidly expanding

• The “cosmic microwave background” is a leftover remnant of the light particles that were bouncing around the Universe shortly after the big bang.

• Nuclear reactions in the early stages of the big bang, when the Universe was hotter and denser than the interior of a star, form Helium (and other elements) at more-or-less predictable rates. The amount of Helium now seen in the universe agrees with expectations from the big bang model.

• As we “look back in time” at distant galaxies, we see fewer and fewer atoms of elements with high atomic numbers. These elements are only formed in stars, and their relative absence billions of years ago is evidence that the age of the Universe is billions of years.

• As we “look back in time” at clusters of galaxies, they are less and less clustered together. Again, this is evidence that the Universe has evolved from a relatively uniform “soup” of particles into its present state where matter is concentrated into small, dense galaxies.

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What next?

Hubble’s law is just the beginning for describing the motions of galaxies and the evolution of the Universe.

For example, since galaxies are gravitationally attracted to each other, you might expect this mutual attraction to slow down their velocities over time, possibly even reversing course and bringing everything back together in a “big crunch.”

Through recent observations, we have learned (to everybody’s surprise) that galaxies are, on average, accelerating away from each other: their velocities are actually getting larger, not smaller, over time.

How do we know this? What other tools do we have to probe the Galaxy? And what else can we say about the evolution of the Universe beyond Hubble’s law?

Stay tuned....

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