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# Relativity and Cosmology

This is a Blog on relativity and cosmology for engineers and the like. You are welcome to comment upon or question anything said on my website (http://www.relativity-4-engineers.com), in the eBook or in the snippets I post here.

Comments/questions of a general nature should preferably be posted to the FAQ section of this Blog (http://cr4.globalspec.com/blogentry/316/Relativity-Cosmology-FAQ).

A complete index to the Relativity and Cosmology Blog can be viewed here: http://cr4.globalspec.com/blog/browse/22/Relativity-and-Cosmology"

Regards, Jorrie

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### Gravitational Lensing & Einstein Rings

Posted May 02, 2007 1:00 AM by Jorrie

Although Einstein predicted in 1915 that gravity will bend light and Eddington confirmed it in 1919, it took until the 1930s before the phenomenon of gravitational lensing was properly understood and calculated. In this mini-series gravitational lensing will be shown in a few easy to follow equations.

Einstein did (in 1915) calculate the bending of light around a star like our Sun, using the relativistic null geodesic equation for light,(1) as approximated for a low gravitational field. Figure 1 shows graphically how the Sun bends light when observed during a total solar eclipse (highly exaggerated for clarity).

Fig. 1:

The magnitude of the angle of deflection is approximated by:

where M is the deflector mass in kg and R is the distance from the center of the mass in meter, assuming that the mass is spherically symmetric. If we plug in the mass of the Sun (M ~ 2x1030 kg) and the distance of closest approach equal to the radius of the Sun (R ~ 7x108 m), we get Einstein's original result for "Sun grazing" rays being deflected by 1.75 arc seconds.

This is exactly twice the value that Newton's dynamics predicts, which can be obtained by calculating the hyperbolic (Kepler) orbit of a particle grazing the Sun at speed c. The reason for this difference is that Newton's theory is compatible with the equivalence principle (half the effect), but not with curved space, the other half.

Figure 2 below shows the geometry for using this deflection in gravitational lensing. The astronomer observes the light coming from the distant source as ring of light (the Einstein ring) around the deflector. This happens because the gravitating mass bends light less and less the farther from it, which is opposite to a convex lens, which bends light more farther from its center.

Fig. 2

The magnitudes of the angles are approximated by:

where φ is as defined before, α0 is the angle between the deflector and the image(s), Ds is the distance to the source and Dd the distance to the deflector. Eq. 2 is a purely geometrical approximation where the angles are all very small (arc-seconds). The figure is again exaggerated for clarity.

The ring of light appears many times brighter than the source, because the intensity per area of the observed ring is the same as that of the source without the lens, but the ring's surface area is much larger. The dotted circle in the center of the image represents the un-lensed surface of the source, as observed.

We cannot observe such Einstein rings around the Sun during a total solar eclipse, because the deflection is too small for our distance from it. It can however be seen around galaxies and clusters of galaxies at large distances. This effect makes it possible to observe distant galaxies having a chance alignment with a foreground one, with magnification, allowing useful measurements at distances where it would not have been possible.

One of the most useful applications of the Einstein ring is that it gives astronomers a tool for "weighing" a distant galaxy. They can measure angle α0 and distances Ds and Dd, the latter pair by means of their respective redshifts. Using equations (1) and (2), the mass M of the deflector lens follows easily:

Let's put in some values, say: α0 = 1 arcsec, Dd = 1 billion ly and Ds is double that. I find: M = 1.5x1041 kg, or 75 billion solar masses. This may sound outrageously large, but it's actually less than half the estimated mass of our own Galaxy.

In case you want to check my calcs, here's values from the spreadsheet: G=6.67E-11, c=3.00E+08m/s, alpha_0=1 arcsec=4.85E-06rad, Ds=2Gly=1.89E+25m, Dd=1Gly=9.47E+24m, M=1.50E+41kg. One solar mass=2E+30kg.

Many such Einstein rings have been observed, although not as perfect as in this idealization. The approximation used holds only for the case where the source is closely lined up with the lensing deflector. It is much more likely that the source will not be well lined up with the deflector. This situation will be discussed in part II of this mini-series.

Notes:

(1) For information on the orbital equation for light, see Tests of Relativity on the website Relativity 4 Engineers. The eBook Relativity 4 Engineers contains more details on the geodesics followed by light and other particles with mass.

(2) The equations are all converted to "engineer-friendly" form from standard texts on relativity. Special credit to George K. Francis et al. (http://new.math.uiuc.edu/superball/pb.pdf)

-J

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#1

### Re: Gravitational Lensing & Einstein Rings

05/02/2007 11:36 PM

Very cool Jorrie!

I have heard of this before, but didn't know about the weighing. I assume that the weighing would include any dark matter.

Here is a picture I found with Google:

Some of the others were not very convincing.

StandardsGuy

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#2

### Re: Gravitational Lensing & Einstein Rings

05/03/2007 12:12 AM

Hi SG

You wrote: "I assume that the weighing would include any dark matter."

Yep, dark matter would tend to clump together with the bright (baryonic) stuff, actually overwhelming it by at least 3:1. Any lensing observed must be ~75% caused by dark matter! It is sometimes not even possible to observe the lensing galaxy.

-J

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#3

### Re: Gravitational Lensing & Einstein Rings

05/03/2007 12:24 AM

Hi Jorrie, as Stanndardsguy said, cool! Question: you said "This happens because the gravitating mass bends light less and less the farther from it, which is opposite to a convex lens, which bends light more farther from its center."

If this happens in a grav. lens, the deflections must actually shrink the image, not magnify it! Or what do I miss?

SL

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### Re: Gravitational Lensing & Einstein Rings

05/03/2007 1:15 AM

Hi SL, you don't miss much, do you?

You're right - I showed it in my fig. 2 without saying anything about it. If you look carefully, you'll see that the thickness of the ring is less than the radius of the dotted circle in the center of the image.

The "magnification" is due to the total surface are of the ring being larger than the area of the "straight image". In distributed lensing systems, e.g., clusters of galaxies, it is possible to get enlargement of a single source. More about that later.

-J

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#6

### Re: Gravitational Lensing & Einstein Rings

05/03/2007 3:11 AM

Very well Jorrie.

I have read on the gravitational lens, but the form as you explain it, it helps to understand me, I hope to help with my small contribution. AstroSeti, matter and energy dark, watched through a gravitational lens.

El cúmulo bala
Bullet Cluster, 1E0657-56, se encuentra, según su corrimiento al rojo, a unos 4 000 millones de años luz en la constelación de Carina y consiste en dos cúmulos de galaxias en colisión. Crédito: Rayos X: NASA/CXC/CfA/M.Markevitch et al.; Óptico: NASA/STScI; Magellan/U.Arizona/D.Clowe et al.; Mapa de lente gravitatoria: NASA/STScI; ESO WFI; Magellan/U.Arizona/D.Clowe et al.

Dark matter that appears in blue color in the image detects of indirect form thanks in order that gravitational lens that is observed on the bottom galaxies.

Will be able to be seen Galaxies older than the great explosion?

TAVC

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#8

### Re: Gravitational Lensing & Einstein Rings

05/03/2007 4:36 AM

Hi Grage, nice picture you posted!

You asked: "Will be able to be seen Galaxies older than the great explosion?"

No, we cannot observe anything even close to the age of the universe. The oldest light we can see is from ~300 thousand years after the Bang - the CMB radiation.

-J

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#9

### Re: Gravitational Lensing & Einstein Rings

05/04/2007 7:40 AM

Hi Jorrie. The sharp angles that you show for the deflections can obviously not be quite like that. Does it mean that the value for R in your equation 1 is to the sharp point of to the rounded point? and how is the angle measured?

Regards,

SL

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#10

### Re: Gravitational Lensing & Einstein Rings

05/04/2007 9:13 AM

Hi SL, you asked: "Does it mean that the value for R in your equation 1 is to the sharp point of to the rounded point? and how is the angle measured?"

R is literally the point of closest approach, along the curved path. For the extremely small angles over vast distances, the sharp bending and the distance to the bend is a perfectly good approximation for R.

If by "the angle measured" you meant the deflection angle φ, that is not directly measured. It is in fact the angle between the two asymptotically straight paths far from the lens and is calculated from other measurements.

If you meant the angle α0, it is measured by the observer, usually from a photographic plate (or CCD image), knowing the characteristics of the telescope used.

Hope this clears is.

-J

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#11

### Re: Gravitational Lensing & Einstein Rings

05/06/2007 11:21 AM

Hi Jorrie, tx, its cleared for now.

BTW, I've eventually decided to subscribe to this good forum (CR4). I am following yours and other posts with interest. Tx for a great Blog!

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#12

### Re: Gravitational Lensing & Einstein Rings

05/07/2007 1:38 AM

Hi SL, you wrote: "BTW, I've eventually decided to subscribe to this good forum (CR4). I am following yours and other posts with interest. Tx for a great Blog!"

Contrats and welcome as a member of CR4!

Its always a pleasure to discuss things with you.

-J

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#5

### Re: Gravitational Lensing & Einstein Rings

05/03/2007 2:00 AM

Did you mention that astronomers couldn't figure out why quasars were usually double or quadruple before they figured this out?

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### Re: Gravitational Lensing & Einstein Rings

05/03/2007 4:31 AM

Hi vermin, so far I dealt with the "perfect rings" only, but yes, in practical situations what you said is true. Will go over to that in a later part of the mini-series.

-J

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