Light and Gravity

Various Circumstances Gravitational redshift (dense stars) Gravitational lenses Weak lensing MACHO studies (microlensing)

Gravitational Redshift 1 + z = 1 / sqrt ( 1 – r S / r) where r s is the Schwarzschild radius of the object. r s = 2 G M / c 2 r is the radius from which the light is emitted and z is the redshift seen by a distant observer

Consider the Limits If r = r S, the redshift goes to infinity (the light is effectively climbing away from the event horizon of a black hole) For real stars, r (the emitting surface) is many times r s -- the sun, for example, has r s ~ 3km But its radius is ~ 7 x 10 5 km.

More Relevant A white dwarf has a radius comparable to that of the Earth – say, r = 6000 km. If it is one solar mass, then r s = 3 km. So the gravitaitonal redshift would be 1 + z = 1 / sqrt (1 – 3/6000) 1+ z = so z =

Conclusion Z = would imply a velocity (if interpreted as a Doppler shift) of v = c = x 300,000 km/sec = 75 km/sec which is in the ball park of actual stellar motions in the solar neighbourhood.

An Ideal Place to Test This

Why ideal? The spectrum of the bright normal A0 star Sirius gives us the space velocity of the binary system (suitably averaged over the orbital motions). The spectrum of the white dwarf will register some larger velocity (that is, with a redshift bias) because of the gravitational effects. The difference tells us the story. For a single white dwarf sitting in space, we have no such ability to discriminate.

Gravity as a ‘Lens’ With complete alignment and simple (round) objects, you get a perfect ring. Otherwise, you can get fragmented rings, multiple images, and so on.

Gravitational Lenses Not like conventional lenses! There is no unique focal length, so no focussed image is expected. The behaviour is achromatic (all wavelengths follow the same trajectories)

Achromatic!

If Perfectly Aligned: an ‘Einstein Ring’

As here…

Gravitational Lensing! Gravity focuses and redirects light! (Einstein did not live to see these examples)

The Einstein Cross

Complicating Factors The source and/or the lens does not need to be spherically symmetric or point-like The source and the lens may not be perfectly aligned with the receiver. So the images can be very complex and distorted.

Imperfect Rings and Images

More Generally (L is the lens, S is the source)

In the Solar System

Notice the Sense of the Shift

It Seems to Shift ‘Outward’

A Pattern of Stars

Now Put the Sun in Between The stars should appear to ‘spread out’

Can We Put This to the Test? Two Problems: 1. The shift is absolutely tiny, since even the gravity of the sun is pretty feeble – but Einstein should that it should be (just) measureable. 2. We can’t see the stars when the sun is in the way!

Solution: Wait for an Eclipse! Compare pictures taken (a) with no sun in the way, and later (b) when the sun is there – but eclipsed!

The Planned Expedition (1919)

Photographs Taken

Proven Right!

The Lens Equation

What to Notice There are two roots: in general, we get two images, one one each side of the lens. The function depends on the distances of the lens and the source, plus of course the mass of the lens.

What We Derive If we know where the lensing mass is, then we can derive the location of the source. We can work out the relative distances of the source and the lens if we can measure their redshifts (assumed cosmological). We can then work out the mass of the lens (which may be dark matter)

In the Real World Lenses are not small spheres; nor are sources; and the alignment may be imperfect. So we get complex arcs and distorted images. But we can invert that and map the distribution of dark matter.

Galaxies can’t explain it all! The lensing is caused by dark matter

The Induced Distortions Gravitational lensing can actually be expressed as the sum of two geometrical effects. The lensed image suffers convergence and shear. The convergence term magnifies the background image (increases its size); the shear term distorts it tangentially by stretching it around the lens. These effects are readily seen for strong lensing.

Weak Lensing When the light of a background galaxy passes a foreground lump of modest mass (not a strong lens), its image will be distorted a little. Depending on the number of lensing centers, the impact parameters, the path lengths, and so on, we expect to see some statistical and correlated behaviour in the slightly distorted shapes of background galaxies.

Weak Lensing Exemplified

In the Perfect World… … all the remote galaxies would be E0’s (that is, perfectly round!) and we would notice the slight elongations (and whether these are correlated from galaxy to galaxy, and over what scales) Unfortunately, real galaxies have a variety of ellipticities and intrinsic orientations. So we can only do this statistically,and need to know the intrinsic distribution of shapes (and must assume random orientations in the large)

Operational Problems! There are many, including the fact that: Atmospheric “seeing” and imperfect guiding tend to make small images look rounder Telescope tracking errors and differential refraction can produce elongation in the E-W sense Optical imperfections can lead to axisymmetric structural errors in the images

Still, It Can Be Done -- and out of this we get some idea of the overall distribution of gravitating matter; its ‘filling factor,’ contrast and amplitude; and so on

Microlensing What if the lens is so weak that the bending angle doesn’t lead to the formation of a displaced separate image? And if the lens is itself non- luminous? (a planet, a low-luminosity star [wD or NS], or even a BH?) We see the background source appear to brighten while there is a transit underway (the lens passing over the face of the source). This is microlensing.

Keep the Scales in Mind This doesn’t happen for galaxies, since the necessary relative motion is impossible. But consider stellar-mass remnants in the halo of our galaxy, drifting around at 10s – 100s of km/sec, in front of a background field of stars. This motivated the MACHO project.

We are Looking for MACHOs In astro-speak, MACHOs are MAssive Compact Halo Objects

1. Dim 2. Bright 3. Dim

This is Still Not Simple! 1.Black holes are rare, and only briefly lined up with any particular star. We need to study millions of stars for many years if we hope to catch even a few ‘in the act’. 2.Some stars vary in brightness anyway (eclipsing binaries, pulsating stars). How do we discriminate? 3.Any single event will never be repeated, so we can only work out statistical estimates of black hole masses and numbers.

For Efficiency Find a collection of many stars at some moderately large distance – like a nearby galaxy, say – so they can all be captured in a single big image. Then take picture after picture, year after year, and look for short-lived changes in brightness. Finally, automate the whole process!

One Very Helpful Thing The colour of a star does not change when it is seen through a gravitational lens. [This is because all light behaves the same way under gravity.] By contrast, pulsating stars undergo temperature (and thus colour) changes.

The MACHO Project [monitor the stars in the Large Magellanic Cloud]

A Bonus Companion planets can also microlens…

What Do We Learn from Rapid Variability? First, consider city lights! The alternating electric current comes and goes 60 times a second, too fast for us to notice. The lights are “turning on and off” all the time. (Car lights use direct current from the battery and are steady.) In slow motion: Incandescent lights flickering

Now, Suppose You Could Turn the Sun Completely Off Assume that the whole surface goes completely black, all at the same instant! What would you see? Would it vanish instantly?

No, For Two Reasons First, you would not learn about this event for eight minutes. So there’s an overall delay. Second, the fadeout would take a bit of time because the sun is big. What you would see is shown in the next sequence of panels:

So The Sun’s Light Dies Away… …but it does so gradually, taking about 2- 1/2 seconds to do so, with a growing central ‘blot.’ That’s because the sun is about 2.5 “light- seconds” in size.

Light from the Edge ‘Lags’

Now, What If the Sun Comes Back On? Suppose the sun ‘turns back on (everywhere at once) after about a second, and this on-off cycle repeats? What would we see?

And So On: Over and Over Result: unsteady brightness ….but it never goes out completely!

Summary: If the sun were able to switch on and off very rapidly, over and over, we would see: 1) Concentric thin rings of light and dark, moving outward 2) A fairly steady overall brightness, with only moderate variability!

Imagine “Turning Off” a Galaxy!

If Every Star in M31 Died at Once … the galaxy as a whole would take 100,000 years to vanish gradually! The stars nearest us would disappear first; the ones farther away would be seen to vanish much later. [M31 is about 100,000 light years across]

The Lesson If the light from an astronomical object varies dramatically on some timescale, the “emitting region” can be no bigger than that (expressed in light-seconds or light- years, say).

Relevance to Quasars The first quasars were at quite large redshifts (many even greater ones have been found since). But the quasars themselves were quite bright. If the redshifts are ‘cosmological,’ then the quasars are far away and must be as luminous as entire galaxies.

The Value of Archives Searches back through old plate archives (e.g. in Harvard) revealed that some of the quasars had varied significantly in brightness on ~1 year timescales. This means that the luminous parts of quasars must be emitting the luminous flux of an entire galaxy from a region only ~ 1 light-year in size! [This constraint is somewhat relaxed if you allow for ‘beaming’ of the radiation, so it’s not into 4π steradians.]

For Example

Other Contexts We look at the ‘flickering’ of X-ray sources to set limits on the compact sizes of the regions from which the radiation comes (e.g. a disk of hot gas around an accreting object, like a black hole)