Abell 2218 - A much studied example of weak and strong lensing by a cluster of galaxies
(Shown here: an HST image of the core of the cluster)
Abell 2218 - A much studied example of weak and strong lensing by a cluster of galaxies
(Shown here: an HST image of the core of the cluster)
Table of contents:
Appendix A. Other methods for determining dark matter mass
We estimate the amount of dark matter in Abell 2218 from literature values of the lensing mass of the cluster, its luminosity and intracluster gas mass, and an estimate for the M/L of its member galaxies. The estimate shows that 94%±6% of the total mass consists of dark matter. This amount cannot be brought down to lower than 60%. A rough estimate of the intergallactic dark matter shows that 10%-90% of matter in the member galaxies consists of dark matter. No wrong assumptions or systmatic errors are found in the lensing analysis that can explain the huge cluster mass discrepancy.
Only a theory like MOND would seem to be a viable alternative to dark matter.
Gravitational lensing provides some of the strongest evidence in support of dark matter. Since our aim is to examine the strength of all such evidence, studying the strength of the evidence from lensing is an integral part of our investigation. One of the greatest advantages of gravitational lensing is that, unlike the virial, and X-raymethods, the mass can be determined independent of the dynamical state of the cluster, and in a more direct way.
If this method is proven to be sufficiently reliable in its application to clusters of galaxies, the evidence for dark matter will be greatly strengthened.
The questions I have set myself to answer are: how strong is the evidence from lensing for the existence of dark matter? And can wrong assumptions explain the mass discrepancy?
I will focus on cluster lensing only, since it provides the most reliable results. And given the time constraints on this project I will focus my attention on Abell 2218. I have selected this particular cluster because it is the best studied lensing cluster, and the available literature should therefore provide sufficient data for my analysis.
Here is how I will go about answering these questions: I will attempt to calculate a conservative dark matter estimate from the data which is available in literature.
Finally, I will verify whether or not the amount of dark matter falls within the error bars. If so, the evidence from lensing will prove to be inconclusive, if not, I will look for wrong assumptions or systematic errors that might explain the mass discrepancy. If these attempts should fail I will conclude that the body of evidence in support of dark matter must be quite strong.
1.1 What is gravitational lensing?
The idea of gravitational lensing is actually quite old. It was first predicted correctly by Einstein's General Relativity Theory. The idea is that light rays passing close to a massive object are bent by the object's gravitational field:
Fig. 1-1 Light from a background source passes a massive object and is bent by its gravitational field. The result is that the background source is magnified and is deflected by the angle 
from its true position. If the lens is massive enough, instead of only one image, two or more images appear, depending on the shape of the lensing object.
This effect could be verified soon after the theory became known, but it wasn't until the early eighties that the available instrumentation became powerful enough to detect the first instances of actual lensing.
Gravitational lensing often creates multiple images, and because clusters are rather extended objects, the images are usually deformed to make little arclets or even giant arcs (see Fig. 0).
One of the great benefits of gravitational lensing is that it allows us to estimate the mass of the lensing object in quite a direct way.
For further details the reader may want to consult a recent astronomy textbook, or Schneider '92, which treats the subject in great detail.
If we allow for the lens to be an extended object, and if we assume that the cluster is axially symmetric then the deflection angle is given by:
| (1-1) |  |
| (1-2) |  |
and: |  |
| (1-3) |  |
| (1-4) |  |
we find: |
| (1-5) |  |
| (1-6) |  |
reduces the equation to: |
| (1-7) |  |
>
This means that 
-
, implying that at least two images are created.<, so that and have the same sign, implying that only one image is created. The physical interpretation of this is that the lensing effect is now only strong enough to deflect the image by the angle .
So what are the characteristics of weak and strong lensing?
Strong lensing:
With strong lensing light is bent at a very sharp angle (see fig. 1-2), as a result of a short distance to the lens, and/or a small (see eq. 1-2). Therefore the light only feels the effect of the central regions of the lens. The disadvantage of this method is that, although it is quite reliable, it can only determine the mass of the inner regions of the cluster.
Weak lensing:
Weak lensing can provide a picture of the mass of the whole lens, since light isn't bent so strongly and is able to pass through the outer parts of the object as well. For constructing a mass versus radius profile, this method works well in the outer regions up to about the radius (for Abell 2218 this is 345 kpc) where the strong lensing analysis begins to fail. Unfortunately, the method usually cannot provide the total mass, since the it cannot encompass the whole cluster. What's more, the mass that is calculated within the given radius will then be an underestimate, since the gravitational effect of the mass outside that radius is not taken into account in the analysis (see Squires '96 et al.).
Figure 1-2: Drawn here is a cluster which is acting as a lens to two individual galaxies. It is clear that in the case of strong lensing (red lines), the light rays pass through the cluster, and are only influenced by the mass M(< ) interior to the light rays. As we have seen before, the background object produces only one image in the case of weak lensing (blue lines), so that in reality more than one background object is required. Ideally, then, all the mass is interior to the light rays, and therefore we can find the total mass of the cluster. |
Lensing on other scales
Lensing can occur on many different scales, from planets and stars to superclusters. In the case of planets and stars, the lensing effect is called microlensing, since the image separations are in the micro-arcsecond domain (within our own galaxy it is in the milli-arcsecond domain, though).
Much work is also done with galaxy lenses. The trouble here is that it's very hard to find lenses that we can use for mass determinations. Often, given the lack of a sufficient amount of weakly lensed arclets per individual galaxy, a statistical method is applied over a good number of individual lensing galaxies. Although this does give relatively good results, it's obviously not the most direct and reliable way of detecting dark matter.
2.1 Estimating the amount of dark matter
There are two methods with which to find some evidence for the presence of dark matter in clusters. The first is to look at the overall mass of the cluster at a certain radius (the second method will be discussed in section 2.3). We use the luminosity of the cluster found at that radius, the mass of the X-ray emitting gas in the cluster, and the mass of the cluster found using weak lensing, all at that same radius. We also use the M/L that we would expect for galaxies that consist of stars only. Estimating its value is a little tricky, though. Its value could change dramatically depending on our assumptions of the stellar content of the galaxies. To illustrate this, take as the lower bound the case where the galaxies consist only of O5 stars of class V; then M/L=4.10-5; and take as the upper bound the case where the galaxies consist only of M5 stars of class V; we then have M/L=34. Fortunately, many theoretical and observational studies have been done in this field. We find that the averages for different models lie around 6 to 8, with a tendency to 8 (see Peletier '89, De Jong '95, Squires et al. '96).
We will calculate the expected mass of Abell 2218 at the radius R by:
MX-ray gas + Mstellar = MX-ray gas + (M/L)stellar.Lclus.
Our dark matter estimate then follows from subtracting this expected mass from the total mass that was found using lensing. The error in Mdark is taken at the 100% confidence level.
Note that we will use for the Hubble constant H0=100 h km s-1 Mpc-1 throughout our discussions. This value is merely used as a standard for comparison. We shall see later how our choice of the Hubble constant will affect our results.
The results of our calculations are shown in Tables 1 and 2:
| Lclus | (M/L)stellar | Mstars | MX-ray gas | Mclus | Mdark | |
| R=200 kpc | (6.5±1.5).1011 Lsolar (1) | 8±3 (M/L)solar (2) | (5.2±3.2).1012 Msolar | (6.4±0.5).1012 Msolar (3) | (3.3±0.3).1014 Msolar (4) | (3.1±0.3).1014 Msolar (96%±9%) |
| R=400 kpc | (9±2).1011 Lsolar (3) | 8±3 (M/L)solar (2) | (7.2±4.3).1012 Msolar | (1.7±0.5).1013 Msolar (3) | (3.7±0.1).1014 Msolar (3) | (3.5±0.2).1014 Msolar (93% ±5%) |
Table 1: Listed here are: the range of values found for the luminosity of A2218 at a distance R, the range of mass to light ratios of stars, the expected contribution of stars to the mass of A2218 at the radius R, the contribution of x-ray emitting gas at the same radius, the total mass of A2218 again at the radius R, and the estimated total amount of dark matter with this radius R (in parantheses: the dark matter percentage with uncertainty).
In the first row we take a look at the strong lensing estimate at a sufficiently small radius of R=200 kpc; in the second row we have the weak lensing estimate at a sufficiently large radius of R=400 kpc
| (M/L)galaxy | Mdark, intergallactic | Mdark, total | |
| R=200 kpc | 41±8 (M/L)solar (1) | (2.9±0.4).1014 Msolar (90%±11%) | (3.1±0.3).1014 Msolar (96%±9%) |
| R=400 kpc | 41±8 (M/L)solar (1) | (3.2±0.3).1014 Msolar (85% ±8%) | (3.5±0.2).1014 Msolar (93% ±5%) |
Table 2: This table uses the same data as those presented in Table 1, except that this time we use estimates of the total (observed) M/L of the member galaxies. The resulting dark matter estimate, then, is the amount of dark matter in the intergallactic medium only. The intergallactic and total dark matter estimates are listed in the last two columns.
The result is very interesting, and perhaps a little surprising. We find that the amount of dark matter in the member galaxies is just a fraction of the amount of dark matter in the interstellar medium.
2.2 Can wrong assumptions explain the mass discrepancy?
From Table 2 it is clear that the contribution of the stellar M/L to the value of our dark matter estimate isn't very significant. The uncertainty in the clusters luminosity isn't very significant either. Errors such as including or excluding galaxies and/or foreground stars yield errors of only about 11%.
Unfortunately, we only have one study that estimated the amount of X-ray gas, which is a very important part of the equation. But how great is it's influence? To get a feeling for that we consider two extremes: first we take the case where there is actually less X-ray gas than stellar matter, and we assume a value of 1.1012 solar masses. Second, we take the case where we assume a much higher value, say 1.1014 solar masses. The results are shown in Table 3:
| MX-ray gas | Mdark | |
| R=200 kpc | 1012 Msolar | 3.2.1014 Msolar (98%±9%) |
| 1014 Msolar | 2.2.1014 Msolar (68%±10%) | |
| R=400 kpc | 1012 Msolar | 3.6.1014 Msolar (98%±4%) |
| 1014 Msolar | 2.6.1014 Msolar (71%±5%) |
The lensing estimate of the total mass of the cluster seems quite accurate, especially if we realize that the weak lensing analysis underestimates the mass. The weak lensing analysis assumes that no matter exists outside the radius at which the mass is determined. If, as is the case here (see Squires et al. '96), there is still some mass outside the 400 kpc radius, the gravitational effect of this mass causes a slight underestimation of the cluster mass, which only makes our dark matter estimate larger .
So far we've discussed all the sources of uncertainty seperately, but how much dark matter is there if we assume that all of the extremes in our previous considerations are true: H0=100, L=1.11.Lclus, MX-ray gas=1014, (M/L)stellar=34? If we make a rough estimate of the amount of dark matter, using these parameters, we find:
| Mdark | |
| R=200 kpc | 2.0.1014 Msolar (62%) |
| R=400 kpc | 2.4.1014 Msolar (64%) |
Clearly, this still leaves more than 60% of the total amount of matter unaccounted for.
We will assume that the people we've quoted didn't make any serious mistakes in their calculations, which is very reasonable, also because in different studies the values found are of the same order. And given the fact that the discrepancy is so large that it cannot be taken to fall within the error bars, we have to concentrate on finding systematic errors, i.e. we need to look for assumptions that these studies depend upon, which are either not justified, or could by the same token be reformulated to yield entirely different results.
Since the luminosity can be measured directly, and the possible error due to including or excluding galaxies or forground stars is relatively small, we shall assume that no systematic errors occur here. What is left then is the determination of the mass of the cluster gas, and of the cluster itself. These quantities are determined indirectly, and so obviously more assumptions will go into the calculation.
To narrow down our search for wrong assumptions and/or systematic errors in the determination of the cluster total mass, it is very useful to note that there are two methods for determining the total mass in the cluster: the virial and X-ray methods. They are briefly described in Appendix A. If the results of these two methods are consistent with those derived from the lensing method, we will only need to examine the assumptions that go into all three methods, as only such assumptions would be able to explain the mass discrepancy. Fortunately, as you can see below, the lensing estimate does agree, to within the error bars, with the virial and X-ray estimates:
| Mtot (Virial) | Mtot (X-ray) | Mtot (Lensing) | |
| R=200 kpc | - | (2±1) .1014 Msolar (1) | (3.3±0.3) .1014 Msolar (2) |
| R=400 kpc | 4.40 (3.04-4.44) .1014 Msolar (3) | (2.6±1.6) .1014 Msolar (4) | (3.7±0.1).1014 Msolar (4) |
One of the assumptions that invariably goes into any mass determination is that for the Hubble constant (H0). Since its value has as yet not been fixed, an assumption has to be made for its value, which, everyone agrees, should lie somewhere between 50 and 100. Since the mass turns out to be inversely proportional to the Hubble constant, this means that the dark matter mass estimates that are made using a value of 100 (and therefore the masses listed in this document), are a lower estimate of the actual dark matter mass. Clearly, then, the uncertainty in the Hubble constant cannot account for the mass discrepancy.
Another assumption common to all three methods is related to the fact that we're looking at a projection of the cluster. According to Renyue '97, the X-ray analysis underestimates the mass by about 20%, whereas the lensing analysis overestimates the mass by some 5% to 10%. The intracluster gas estimate is underestimated by about 30%-40%, which falls well within the bounds we set in our discussion above. The projection effect for the virial method isn't given by Renyue '97 or any other studies, but it should be fairly large - but comparable to the X-ray projection effect - given the methods strong dependency on morphology (see Appendix A). But again, looking at the projection effects seems to confirm, rather than challenge, this huge mass discrepancy.
Something that should also affect all three methods is the possibility that Abell 2218 may be in the middle of a merging process (see Girardi et al. '97, Kneib et al. '95, Markevitch '97), which might explain the two clumps at the centre of the cluster. For the virial and X-ray analyses, this means that the cluster isn't exactly in hydrostatic equilibrium.
For the lensing analysis what is important is just the fact that there are two clumps. A model is usually taken where the secundary, much less massive clump is almost disregarded. If this assumption is wrong, then the secundary clump might act as an extra lens, although this isn't likely given their alignment. If this would be so, the masses found using the single lens model would be an overestimate of the real value. Unfortunately, this scenario isn't very likely and it certainly couldn't explain the mass discrepancy. To test this, some non-merging clusters would need to studied.
2.3 Locating dark matter concentrations and estimating mass
Up to now we've only looked at the overal mass of the cluster. Another way of looking at the extent of the dark matter problem is to compare the distribution of mass in the cluster to that of light (which implies stars) and gas. Since the dark matter distribution may not necessarily match the distribution of the visible matter, this may provide another approach to our questions.
Unfortunately, such an analysis can only be done qualitatively, since there are no distribution maps yet for the stellar mass in stars and particularly the x-ray emitting intracluster gas.
Two studies have done some work on this so far. We won't regard this as a definite proof of the presence of dark matter, but rather as a very interesting and significant indication that there might be dark matter there. As such analyses will be done more and more, it will certainly provide some interesting new insights.
Here are the results of these studies: in 1997 Giraldi et al. found the following distribution map:
 |
Fig. 2-1 Shown here is a density map of the core of the cluster. Each of the two points mark the galaxy that lies at the centre of one of the two clumps. The points appear to be significantly offset from the density peaks. (Image taken from Giraldi's paper) |
 Fig. 2-2 Shown here is a map of the mass distribution around the central CD galaxy. The cross represents the optical position of the CD galaxy. The offset from the expected position is very clear. (Map taken from AbdelSalam's paper) |
 Fig. 2-3 The core of Abell 2218 consists of two major clumps (see Fig. 0). The first clump is centred on the CD galaxy (see Fig. 2-2), the second is centred on the bright galaxy that is represented by the central cross on the map. This detailed distribution map again shows a significant offset. There is another interesting thing, though. The peak in the mass distribution that is associated with the central galaxy - which is the brightest and most probably the most massive one - is actually a saddlepoint between the two peaks associated with the crosses on either side of the central galaxy. (Map taken from AbdelSalam's paper) |
Despite the fact that Abell 2218 is probably the best studied cluster there is, I wasn't able to find many studies that presented mass estimates for the cluster's dark matter and/or intracluster gas. For the sake of comparing the results of different studies, some data could not be used, and other data was not reliable. The latter is due to the type of lensing analysis. Some data was obtained using weak lensing where only the strong lensing estimate can be reliable (i.e. for small radii). To illustrate this, I've listed in Table 4 the unreliable weak lensing estimate at a radius of 200 kpc (which is within the strong lensing regime), and the strong lensing estimate which was listed in Table 1:
| Mclus | Mdark | |
| Weak lensing | (1.2±0.1).1014 Msolar (1) | (1.1±0.1).1014 Msolar (90%±11%) |
| Strong lensing | (3.3±0.3).1014 Msolar (2) | (3.1±0.3).1014 Msolar (96%±9%) |
Table 6: This table clearly illustrates how the weak lensing analysis significantly underestimates the total mass enclosed by a small radius of up to about 345 kpc
The dark matter estimates in Abell 2218 average 94%±6%, and as we have seen, the uncertainties and our assumptions cannot account for the mass discrepancy, bringing it down to a still massive 60%. The fact that the main galaxies don't seem to follow the total mass distribution is also an interesting point that cannot be disregarded simply as an error. As such it makes it even harder to find an alternative to dark matter. The evidence from lensing in support of the existence of significant amounts of dark matter in Abell 2218 is therefore quite convincing.
But despite the overwhelming numbers, we cannot make any final conclusions about the existence of dark matter in other clusters, especially since this particular cluster might be more of an exception than a typical cluster. As we noted before, this cluster may be in the middle of a merging process. For greater confidence in our results, more (non-merging) clusters would need to be studied.
Still, certainly for this cluster, only a fundamental error in the assumptions that are made would seem to be able to provide an alternative to dark matter. One such error could be that the laws of physics as we understand them, might not be entirely accurate. This is an idea that has developped into several theories of which
MOND is the most prominent. However, whether this theory could be made to apply to gravitational lensing is not clear.
I for my part have not been able to locate any wrong assumptions or systematic errors that could explain the mass discrepancy.
Many thanks go to our instructor Penny D. Sackett for her constant guidance and support, and for the lessons she taught me about the process of research.
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Markevitch, M., 1 July '97, ApJ 483:L17-L20, Abell 2218: x-ray lensing, merger or both?
Natarajan, P. & Kneib, P., 26 September '98, MNRAS, Probing the dynamics of cluster-lenses
Peletier, R., 1989, Elliptical galaxies: structures and stellar content, thesis
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There are two other methods for estimating the total mass in a cluster:
1. The virial method
The virial method is the oldest method for determining the mass of a cluster. It was first used by Zwicky in 1933. It is based on the virial theorem which is used in many applications of (astro)physics.
The method is based on the virial theorem which in this case states that the kinetic energy in the motions of the member galaxies must be just the amount needed to check the gravitational potential energy of the system:
(A-1) 2<K.E.>=<P.E.>
For the case of a spherical cluster in equilibrium, the equation reads:
| (A-2) |  |
gal(r) is the number density of the galaxies, and 
r and t are the radial and tangential velocity dispersions at each radius r.t cannot be measured, the assumption is usually made that since the cluster is assumed to be spherical, the r= t at all radii. Also, based on observations, an assumption needs to be made about the galaxy density profile (gal(r)). The total mass then follows directly from the equation.
2. The X-ray Method
The X-ray method was developed as an alternative way of determining the mass of clusters, and was mainly intended to test the existing virial method. This method is still being used a lot, and as X-ray telescopes improve, this method will be able to tell us a lot about the structure of a cluster.
The method looks a bit like the virial method in that both are based on forces that maintain some form of equilibrium. But instead of using the motions of the member galaxies, this method looks at the pressure of the intracluster gas. Here's how it works: if we assume that the gas is in hydrostatic equilibrium, then that means that the force due to the gas pressure is equal to the gravitational force on the gas; and this allows us to estimate the total mass of the cluster within a certain radius:
| (A-3) |  |
| (A-4) |  |
| (A-5) |  |
| (A-6) |  |
| (A-7) |  |
, and is the velocity dispersion (for Abell 2218, =1370).| (A-8) |  |
| (A-9) |  |
where |  |
For a look at some methods that are applied for galaxies, see Olof's, Douwe's, and Stefan's pages. And for a look at an alternative explanation for the mass discrepancy in galaxies turn to Peter's page.