Astrophysics and Cosmology. Practical Work Autumn

Transcription

1 Astrophysics and Cosmology Practical Work Autumn Laboratoire d astrophysique Ecole Polytechnique Fédérale de Lausanne (EPFL) Goal of this work: the goal of this pratical work is, through the analysis of real data, to get a taste of how modern astrophysical observations are obtained and analysed. These include digital images obtained with the four 8.2m VLTs (Very Large Telescope) of the European Southern Observatory at Cerro Paranal (Chile) and with the NASA/ESA 2.4m Hubble Space Telescope. A tour is proposed of the techniques in use by the astrophysical community, and illustrated through the study of a wide range of science areas, from the photometry of nearby stellar clusters, to the search for faint distant galaxies and clusters of galaxies. The goal is to give the student a quantitative feeling of what is possible and what is not with present-day instrumentation. It is expected at the end of this 14-week introduction and training that the student is sufficiently well prepared to attack efficiently the scientific work to be done during the spring semester and during the master. The data needed for the tasks described in this booklet can be downloaded as a single archive from Save this file in your home directory, unzip it using the command gunzip TP4 autumn.tgz, unpack it using tar xvf TP4 autumn.tar, and finally rm TP4 autumn.tar. You can then directly work in the resulting file structure. This booklet goes along with information on the computing facilities at LASTRO and languages (python) posted at 1

2 Contents 1 Introduction to Data Handling: Basic Concepts and Tools (up to 4 weeks) NGC 613: a nearby spiral galaxy Displaying images and basic measurements Characterizing the VLT images Reducing the VLT images Comparing with (already reduced) HST images Weighting Galaxies and Clusters of Galaxies (up to 3 weeks) Spiral galaxies and rotating disks Elliptical galaxies as isothermal spheres Galaxy clusters A Deep Observation of a Globular Cluster (up to 3 weeks) Aperture photometry Calibration or transforming to the standard system Fiducial sequence and isochrones Galaxy Clusters as Gravitational Lenses (3 weeks) The Cluster as a Gravitational Lens Detecting the weak lensing effect

3 1 Introduction to Data Handling: Basic Concepts and Tools (up to 4 weeks) The goal of this first chapter is to understand the basics of image reduction and to get used to the tools used to process astronomical data and to display them. Reducing data is the first mandatory step when dealing with an observational astrophysical project. Depending on the complexity of the instrument, it can be very simple (e.g., for simple imaging) or it can involve the development of new specific softwares that are as important as the development of the instrument that took the data itself. In the present work we keep things very simple and propose to process nice-looking images of the nearby spiral galaxy NGC 613, and to characterize them. Two sets of data are available, taken with the ESO VLT ( and with the Hubble Space Telescope ( Although all the data are already on your disk, it is proposed to follow the lines of work without skipping steps! This will allow to discover NGC 613 with instruments of increasing quality in terms of spatial resolution. 1.1 NGC 613: a nearby spiral galaxy You can have a quick look at the galaxy we will study, through the ESO Digital-Sky-Survey (DSS) web page ( where a digital map of the whole sky is available. This archive has been constructed by scanning old (but very wide field) photographic plates obtained at Palomar observatory and in Australia. Enter the name of the galaxy in the appropriate field and display it as a gif image. Download it as well on your disk as a FITS image. FITS is the standard format to store data in astrophysics. All data will be processed in this format. 1.2 Displaying images and basic measurements Open IRAF and its graphical display ds9 in order to see the FITS image. Use the command display to change the cut levels and try to see the extended very faint parts of the galaxy. You note that the signal between two objects, also called empty sky regions is not zero. This is due to the fact that the night sky is not so dark as one may think. It is emitting a small but measurable amount of light. This amount of light is sufficient to affect the observations. The sky level is often higher than the object brightness itself! This is why observatories must be located far away from any source of light pollution. Lets carry out basic measurments on these data: 1. Assuming that the Hubble constant is H 0 =65 km s 1 Mpc 1 and using the measured radial velocity V r =1479 km s 1 of NGC 613, compute the distance to the galaxy. The Hubble law for the expansion of the Uniserve is applicable to NGC 613. Why can t you use it for large distances, like several thousands of Mpc (Mega parsecs; 1 pc = 3.26 light years)? 2. What is the pixel size (in arcseconds) of this image? Using straightforward trigonometry, compute the diameter of the galaxy, in light-years and in parsecs. How does this compare with our own Milky Way? 3. Using the iraf command imexa, measure the resolution of the image in arcseconds, i.e., the Full-Width-Half-Maximum of the (unsaturated) stars in the image. How does this translates in parsecs or light years? What is the largest globular cluster in the Milky Way? What is its size (have a look on the web!)? Would you be able to resolve it if it was in the image you have of NGC 613? Same question for the planetary nebula M 57. 3

4 1.3 Characterizing the VLT images 1. The list of the observations is on your disk as an ASCII file. Have a look at it. Display the raw images on NGC The pixel size of the FORS1 instrument (see the ESO web page) is 0.2 arcsecond. What is the size of the galaxy using these new and much deeper observations. Deeper means that much fainter light levels are reached. 3. Measure the resolution (called seeing ) in arcseconds in the three images taken in the blue (B), green (V) and red (R) filters. Do this using several isolated stars in order to get a precise mean value and an estimate of the error bar on your measurement. Is there a difference in resolution between the three filters? The three images are taken one after the other, with little changes of the atmospheric conditions between each exposure. If you measured a different seeing in the images, give a possible explanation for it. Note that the atmosphere acts as a prism on light rays. 4. Compute the theoretical diffration limit for the VLT which has a diameter of 8.2m. What are the values in arcsecond for the three filters B (4300Å), V(5500Å), R(6500Å). 5. Compare the behaviours of the measured and theoretical resolutions as function of wavelength. Is the measured trend compatible with the theoretical expectations? You guess correctly that the answer is no. Try to imagine why. 1.4 Reducing the VLT images A raw astronomical image, right after it is acquired at the telescope can be interpreted as a signal D(x,y), which gives the intensity or the flux level recorded, as a function of the position (x,y) on the CCD array. It can be written as: D(x, y) = [I(x, y) + Sky(x, y)] F (x, y) + B(x, y), (1) where I(x, y) is the signal of sicentific interest, Sky(x, y) is the sky level, F (x, y) is the pixel-to-pixel response of the camera, called the flat-field. B(x, y) is called the bias level. It is an arbitrary positive constant set in the hardware of the camera, aimed at avoiding any negative value in the image, due to the readout noise. The goal of data reduction is to extract I from the data D, given a number of calibration frames that we will use in the following: 1. Display the bias images and the flat fields. What is the mean flux level in the bias images? In the flat fields? Take the mean of the 2 biases, 3 biases, 4 biases. Compare the noise with one single bias. By how much is it improved and how does this factor compare with the number of biases in the mean bias image? 2. Using python, plot the standard deviation in the 4 biases you just created (i.e., single or combined), as a function of the number of images in the combined bias. 3. Subtract the best mean bias from each flat field. 4. Normalise the flat fields so that their mean level is Flat field the images following the equation above. 6. Subtract the sky from all the images. This can be approximated by a constant value measure in the areas free of any object. Measure it in many places in the image. Compare the mean value of the sky level with the sky noise (standard deviation of the pixels in a part of the image with sky only). Is the sky level really constant accross the image? Compare the sky level at different location in the frames and compare its variation with the sky noise. 4

5 7. Bring all the images to a common resolution using the convolution task gauss in IRAF and assuming the stars are perfect gaussians. To do that, convolve each image with the appropriate Gaussian, so that all images have the same resolution. Of course the common resolution is the worse one of the three. 8. Use the task rgbsun in the color context of IRAF to construct a true-color image using the B, V and R frames. To do this, first type color into IRAF (you will see the prompt change from cl> to co>), then the command epar rgbsun. Change the option swap to yes, and procede as usual for the rest. Don t give any extension for the output filename. Once the task has run, you can open and convert the colour image using The Gimp. An alternative to IRAF s rgbsun is to use the stiff software (unrelated to IRAF), that directly outputs a TIFF file. Where do you think the star forming regions are? Where do you think dust is present? Why? Remember that dust absorbs blue light more than red light and that hot young stars have a higher temperature than the others (this of the black body radiation). 9. The color image can be complemented by a color map of the galaxy. Compute the ratio between the R and B image and take the logarithm. In that way, you obtain an image of the color of the galaxy. High R/B traces red regions while a lower ratio traces blue regions. 1.5 Comparing with (already reduced) HST images 1. Reducing images obtained with the Hubble Space Telescope is very similar to the reduction of VLT images. Because space data are more stable in time than ground based data (e.g., constant weather condition), the reduction process can be automatized easily. The Space Telescope Science Institute has developped a pipeline that does this work for you and reduced images can be retreived directly on the web. We have downloaded these data for you. Have a look at them. 2. Measure the resolution in each image. The pixel size is arcsec. Does the resolution match the theoretical one this time? Why? What is the smallest detail you can resolve in NGC 613, in parsecs? 3. Convolve all the images to the same resolution and compute the true-color image and the color map of the galaxy. Compare with the VLT image. 4. A supermassive black hole with a mass between 10 9 M and M resides in the center of this galaxy. Calculate the radius of the black hole, assuming that it is defined as the distance at which a photon can not escape the black hole anymore. This radius is called the Schwarzschild radius r s. What is the Schwarschild radius of the black hole in parcsecs? Would you be able to see it with the HST (assuming it can still emit light!)? What would be the diameter of the telescope needed to see it? Can you think of any (realistic) technical solution to reach such high angular resolutions? 5

6 2 Weighting Galaxies and Clusters of Galaxies (up to 3 weeks) There exist a large number of ways of estimating masses of objects at cosmological distances, such as galaxies and clusters of galaxies. All but one are indirect and model dependent. They rely on the measurement of the radial velocity of different parts of galaxies. We will review these methods, in a much simplified way, and see how another more recent method called gravitational lensing (the deviation of light by massive bodies) can be used as a model-independent way to carry out mass measurements. We will see that both methods are sensitive to dark matter, but gravitational lensing does not need to assume any prior knowledge on how this dark matter is distributed in galaxies to give precise mass measurements. 2.1 Spiral galaxies and rotating disks There are essentially two types of galaxies. Elliptical galaxies, that can be seen (at least to the zero order) as a cloud of stars, and spiral galaxies that include a rotating disk made of stars, dust and gas, and of a bulge that is comparable to a small elliptical galaxy. Both types of galaxies often harbour a central black hole that can be modeled as a point mass given its small size compared with the whole galaxy, and are thought to reside in a large halo of dark matter which precise shape has not been unveiled yet and which is often taken as spherical in most theoretical models of galaxies. Spectrographs are used to measure the Doppler velocity shift, either along the long or small axis of the galaxy. This radial velocity measured at a distance r away from the center of the galaxy is called the rotation curve. It is related to the mass contained within a circle of radius r. Since the system is in equilibrium, the centrifugal force seen by a test particle of mass m at radius r is compensated by the centripetal force (i.e. the gravitational force): m v2 circ r = F disk + F halo + F CM (2) where the three forces involved are produced respectively by the disk, dark matter halo and central mass components of the galaxy. This circular velocity v circ at distance r must be multiplied by the sinus of the inclination i of the spiral along the line of sight before any comparisons can be made with the observations (Doppler shifts measure only radial velocities). No measurements can be made of face-on galaxies since sin i = 0 in that case. ( π ) v r = v circ cos 2 i = v circ sin i (3) The gravitational potential generated by the disk at distance r from the galaxy s center can be expressed as: G M disk Φ disk = r2 + (a + z ) 2 + cte G M disk + cte (4) r2 + a2 where M disk is the total mass of the disk, a is its scale length, and z is its thickness. We can take z = 0 to make things easier and since disks in spiral galaxies are very thin. G is the constant of gravitation. The potential generated by the dark matter halo is expressed as a function of its mass density ρ 0 and its scale length r 0 : 6

7 Φ halo = 4πG ρ 0 r 2 0 { [ ( ) ] 2 1 r 2 ln r ( ) } 0 r r 0 r arctan + cte (5) r 0 Finally, the potential due to the central mass of the galaxy is given by: Φ CM = G M CM r + cte (6) The gravitational force can then be computed from F = m Φ. We will attempt to use this simple three-components model of the velocity curve of a spiral galaxy to weight it. 1. First of all, orders of magnitude are necessary and we need to work out the correct units. Based on your knowledge of spiral galaxies, or through the web, try to estimate all the necessary parameters in the model. What are typical sizes for disks, and halos of galaxies? What are their typical masses? Estimate, even roughly, the density ρ 0 of a dark matter halo. Express r in kpc, the velocity in km s 1, masses in M and mass densities in M pc 3. The mass of the sun is M = kg, 1 pc = m, and G = N m 2 kg We will now plot the velocity curve v circ as a function of the distance r from the center. You can do all this directly in python, using matplotlib. Plot each of the three components separately for a set of parameters that spread a plausible range for real galaxies. Plot also the total velocity v circ. This should help you define which component (disk, halo, or central mass) is responsible for the flatness of the rotation curve at large r. 3. Looking at the image of NGC 3198 estimate roughly the inclination of the disk. Radial velocities have been measured in function of the distance r from the center. Plot the observed rotation curve of NGC 3198, and estimate all the model parameters, in order to match the data (knowing that NGC 3198 lies 9.2 Mpc away from us). Do not forget the inclination! What are the masses involved? You can compute them through numerical integration (scipy.integrate) of the mass densities, which are given by the Poisson s law 2 φ = 4πG ρ. Remember the coordinates are not cartesian. What is the total mass of the galaxy in solar masses? Approximate its error bars, by estimating the range of models with different parameters that match the data equally well. 2.2 Elliptical galaxies as isothermal spheres Elliptical galaxies are essentially composed of stars and contain very little quantities of gas and dust. Therefore they can be approximated as large clouds of stars. One can see such a system as a gas of particles, where each particle is a star, and where the interaction between the particle is described by the gravitational force. If this system is in a steady state, one gets, by applying statistical mechanics, the following expression (known as a statement of the virial theorem) : 2T + U = 0 where T = 1 2 M v2 and U = 1 G M 2 (7) 2 r g where T is the kinetic energy, U is the potential energy, v 2 is the mean-square speed of the system s stars, M is the mass of the galaxy, and r g is the gravitational radius. In many stellar systems, one can approximate r g r h /0.4, where r h is the median radius, which is defined to be the radius within which lies half the system s mass. We shall suppose that the considered galaxy has a nearly spherical symmetry and that the velocity distribution of the stars is gaussian, then we can express the mean-square speed in function of the radial velocity dispersion σ r : 7

8 v 2 = σ 2 = σ 2 x + σ 2 y + σ 2 z = 3 σ 2 r (8) Observations show that spectra of elliptical galaxies are very similar to those of red giant stars. And indeed, red giants are known to be the main contributors to the visible light in elliptical galaxies. An interesting point is that the observed absorption lines in the galaxy spectra appear broader than in the stellar spectra. This is simply explained by the fact that the stars are moving around within the galaxy, with velocities that follow a given velocity distribution. Because of this velocities and of the Doppler effect, one observes broadened absorption lines in the galaxy spectra. By examining to which extend these lines are broadened, one can estimate the velocity dispersion in the galaxy, and eventually the mass of the galaxy. Have a look at the spectrum S of the K0 III star and at the spectrum G of an elliptical galaxy. You could do this by using the splot command in IRAF (part of the onedspec package, so type onedspec first). Or you can do this in python, using the module pyfits to read the FITS file as well as it s header, and matplotlib to draw interactive plots. From the equation of the Doppler shift (λ λ 0 )/λ 0 = v r /c, one has: v r c = dλ λ = d ln(λ) = ln(10) d log(λ) (9) This means that if you want to sample a spectrum linearly in velocity, you have to sample it in function of the logarithm of the wavelength. This is why the S and G spectra are sampled in log λ. Identify the following important lines in the spectra: λ [Å] element Ca, K band Ca, H band CH molecules, G band H β Mg band Na doublet H α Calculate the sampling of the spectra (i.e. 1 pixel =? km/s). If you are not using python to read the FITS file, you can convert the contents of the FITS file into columns of a plain text file using the wspectext command of IRAF. If F (v r ) is the radial velocity distribution in the galaxy, one can express the observed galaxy spectrum as the convolution of the star spectrum with this velocity distribution: G(v r ) = F (v r ) S(v r ) (10) Determine the radial velocity dispersion σ r in the galaxy by supposing that the velocity distribution is gaussian. If working with IRAF, you can use the gauss command to convolve the stellar spectrum with a gaussian function, which σ is directly linked to the radial velocity dispersion σ r (be careful with the units, gauss requires σ to be expressed in pixels). If working 8

9 in python, scipy.ndimage.filters has the easy-to-use gaussian convolution functions you need. Try several values of σ to estimate which one gives the best match between G and F S. Using IRAF, this could simply be done by computing G (F S) with the imarith command, and then examining the results with imstat. Using python, you can think of even more advanced possibilities. From the best-fit radial velocity dispersion estimate the mass of the galaxy (expressed in solar masses) if its gravitational radius is r g = 0.8 kpc. 2.3 Galaxy clusters One can weight a whole cluster of galaxies in a similar way than we did for the elliptical galaxy. Whereas, this time, we consider whole galaxies as our theoretical gas particles, in order to apply the virial theorem. Spectroscopy of the galaxies enables us to determine their mean radial velocities. For a cluster containing N galaxies, the radial velocity dispersion is then given by: σ 2 r = 1 N 1 N (v r,i v r ) 2 (11) i=1 where v r is the mean radial velocity of the galaxies. From the radial velocity dispersion you can infer the mass of the cluster. The redshift of an object is defined as z = (λ λ 0 )/λ 0. From the spectra of 20 galaxies of the galaxy cluster Abell 370, estimate the redshift of each galaxy, and then the mean redshift of the cluster. Again, you could use python or the IRAF command splot to plot a spectrum. With splot you can zoom on the spectrum by hitting w, and then twice e. A gaussian fit of a given absorption line is obtained by pressing k on each edge of the line. Use several absorption lines in each spectrum to get a better estimate of its redshift. Convert the redshifts into radial velocities. Determine the radial velocity dispersion and the mass of the cluster (r g = 0.62 Mpc). Given the fact that the cluster contains approximately one hundred galaxies, what can you conclude about its mass? The measured radial velocities are rather high. Is there any correction we should have thought about, when converting the redshifts into velocities? Or, equivalently, how should radial velocities be computed in a (large enough) comoving volume? 9

10 3 A Deep Observation of a Globular Cluster (up to 3 weeks) Globular star clusters (GC s) usually contain about 10 5 stars. The stars are spherically distributed, and the central densities are about ten times larger than in open clusters. The globular clusters are among the oldest stars in the Milky Way, and are therefore of great importance for studies of stellar evolution. Very small heavy element abundances, down to about 10 3 times the solar value, have been detected in some halo globular clusters. They therefore give important information about the production of elements in the early Universe and during the formation of the Milky Way. There are about globular clusters in the Milky Way. The H-R diagrams of stellar clusters, can be constructed in a self-consistent way without knowledge of the exact distances to them. Since the dimensions of a typical cluster are small relative to its distance from Earth, little error is introduced by assuming that each member of the cluster has the same distance modulus. As a result, plotting the apparent magnitude, rather than the absolute magnitude only amounts to shifting the position of each star in the diagram vertically by the same amount. By matching the observational main sequence of the cluster to a main sequence calibrated in absolute magnitude, the distance modulus of the cluster can be determined, giving the cluster s distance from the observer. This method of distance determination is known as main-sequence fitting. Rather than attempting to determine the effective temperatures of every member of a cluster by undertaking a detailed spectral line analysis of each star, it is much faster to determine their color indices (B-V). With knowledge of the apparent magnitude and the color index of each star, a color-magnitude diagram (CMD) can be constructed. B and V CCD images of the metal-poor Galactic GC M15 (= NGC 7078) have been obtained from the ESO FORS2 imager attached to one of the four VLT units. The images have already been reduced (i.e. bias substracted and flat-fielded): construct a calibrated CMD for M15; describe its morphology, i.e. link the different regions of the diagram with the phases of stellar evolution; remove the interstellar reddening, assuming E B V = 0.10; fit the main-sequence with a calibrated zero-age main-sequence (ZAMS) in order to find the distance of the cluster; estimate its age by fitting isochrones to the CMD fiducial sequence (isochrone = curve connecting the positions of stars of various masses and same composition and age); locate the main-sequence turn-off point (TO), i.e. the point where the stars are currently leaving the main-sequence; estimate its radius (hint: see the very beginning of this practical work); estimate its luminosity (hint: determine M15 apparent magnitude first); estimate its mass assuming the cluster consists in solar-type stars only (rough approximation!). 3.1 Aperture photometry Aperture photometry is the most straightforward way to compute magnitudes in uncrowded or moderately crowded stellar fields. The instrumental magnitude m of a star is computed as follows: 10

11 ( ) Ctot ac sky m = 2.5 log + m 0 (12) t exp where C tot is the total count (star + sky) in the photometry aperture; a is area of the aperture in pixels squared and is roughly equal to πr 2 where r is the radius of the photometry aperture in pixels; c sky is the estimated sky value in counts per pixel squared computed in a sky annulus centered on the star; t exp is exposure time and m 0 is zero point offset for the magnitude scale. In practice the aperture photometry is carried out using the sextractor software. An introduction to this software will be given, but some information as well as the required configuration files can be found on our online documentation on computing at LASTRO. 3.2 Calibration or transforming to the standard system Conversion from instrumental (b, v) magnitudes to the calibrated (B, V ) magnitudes follows a simple linear model: b = B + b 0 + b 1 X + b 2 (B V ) (13) v = V + v 0 + v 1 X + v 2 (B V ) (14) where X is the airmass ( sec z where z is the altitude of the star measured from zenith). For the extinction coefficients b 1 and v 1, one can use the average atmospheric absorption coefficients measured at Paranal : b 1 = 0.26 [mag / airmass], and v 1 = 0.17 [mag / airmass]. The constants b 0 and v 0 and the color coefficients b 2 and v 2 can be determined using standard stars, that is stars with determined magnitudes and colours, observed with the same instrument and filters and under identical conditions as the M15 cluster. You will find this material in the directory 3 M15/STD. 3.3 Fiducial sequence and isochrones The so-called fiducial sequence is a smoothed sequence of M15 obtained by e.g. P.R. Durrell and W.E. Harris (1993, AJ 105, 1420), which just represents some kind of best fit to the observed sequence. You can use it to check the quality of the fit of your magnitudes to the standard system. To determine the ages, your need isochrones corresponding to the metallicity of M15, namely [F e/h] = 2.15, which corresponds to a fraction of heavy elements Z = Such isochrones were computed by Vandenberg (1985, ApJS 58, 561) for similar metallicities and are stored in the file vb85iso.txt. See the file ReadMe.txt for a detailed description of this file. 11

12 4 Galaxy Clusters as Gravitational Lenses (3 weeks) In this practical work, we will make use of deep images of the massive galaxy cluster Abell 370 (A370 in the following). You have been given three frames obtained with the ESO VLT through three filters. They consist of numerous short exposures that have already been reduced as explained in Chapter 1 of this booklet. The reduced exposures have also been combined into one single frame that allow to see very faint objects. Among the numerous objects in the images are distant galaxies, nearby galactic stars and a massive galaxy cluster. We will use the images in order to estimate the mass of the galaxy cluster, to study the color distribution of galaxies in the cluster and to detect the effect of weak gravitational lensing on galaxies in the background of A The Cluster as a Gravitational Lens Following Einstein s equivalence principle between gravity and inertial forces, all bodies even with no mass, are under the influence of gravitation. As a direct consequence, a photon can be attracted by a massive body such as a galaxy or a galaxy cluster. As an optical lens does, a mass modifies the path followed by photons and therefore acts as a gravitational lens. The angle by which the light rays are bended by an object with a mass M can be calculated in the context of Einstein s theory of general relativity, as: ˆα = 4GM c 2 ξ, (15) where ξ is the impact parameter of the light ray (see Fig. 1). Compute the value of ˆα for the mass of the sun. The impact parameter can be approximated by the radius of the sun. Fig. 1 gives a schematic description of the phenomenon of gravitational lensing, where a source S is lensed by a massive object in the lens plane (labeled L). The source is not seen at its real angular position β by the observer O, but is displaced by an angle α that depends on the true deflection angle ˆα. Figure 1: Schematic view of the gravitational lensing effect. The observer O sees the image L of the source S at a position θ on the plane of the sky. The source S is never observable. The lensing galaxy is also called the deflector, hence the indices used to label the different distances. If we assume that the distances between the different objects are known show that, when the source, the lens and the observer are almost perfectly aligned (i.e., β 0): 12

13 α = ˆα Dds D s (16) When the source is perfectly aligned with the observer and then lens (i.e., β = 0), it is imaged into a ring called an Einstein ring. Using Fig. 1 and simple geometrical considerations, show that the Einstein radius is: θ E = 4GM c 2 D ds D d D s (17) All the above is true only when the distances involved in the calculations are angular diameter distances. As the reference used to measure distances changes with redshift (i.e., the ruler you are using to measure distances is changing with the redshift), angular diameter distances are given under the form of an integral. The distance between two objects at redshifts z 1 and z 2 (z 1 < z 2 ) is given by: D 12 = z 2 f K (x 12 ) (18) with 1/ ( K ) K sin x K > 0 f K (x) = x K = 0, 1/ K sinh ( K x ) K < 0 (19) and x 12 = c H 0 z2 z 1 K ( H0 c ) 2 (Ω M + Ω Λ 1) (20) dz (1 ΩM Ω Λ )(1 + z) 2 + Ω M (1 + z) 3 + Ω Λ, (21) where H 0 is the Hubble parameter, Ω M is the density of matter of the universe, normalised to its total density and Ω Λ is the density associated to the vacuum. From the most recent studies using distant supernovae, the current values of these parameters are Ω M = 0.3 and Ω Λ = 0.7. The value of H 0 is still poorly known. The currently more accepted (but probably biased) value is H 0 = 72 km s 1 Mpc 1. As the integral above can not be calculated explicitly, integrate it numerically to compute the distance between us (z 1 = 0) and an object at redshift z 2. You could code your own elementary integration, or use one of scipy.integrate s numerous possibilities. Make a graph showing the distance as a function of redshift. Also plot the Hubble relation for comparison. Of course this relation is valid only at very low redshifts. At z > 1 it would even imply that galaxies have velocities larger than the speed of light. The Hubble law, in its original form does not take relativistic corrections into account. Using the three VLT images of A370, make a color image, as was done for NGC 613. You can use for instance stiff or the log scale option of the IRAF task rgbsun. Later you can fine-tune the color scale using the color editor in xv or gimp in order to emphasize the color contrast between the different objects in the field of view. You notice immediately an overdensity of galaxies that seem to have the same color. These galaxies belong to the A370. Give a plausible reason for them being of the same color. 13

14 An arc-like structure is seen close to the center of the cluster. This arc is not physically in the cluster but corresponds to a gravitationally lensed galaxy in the background of the cluster. The redshift of the arc is z = 0.72 while the redshift of the cluster is z = Compute the angular diameter distances to the cluster and to the arc as well as the distance between the cluster and the arc. The pixel scale in the image is 0.2 arcseconds. Considering the arc as a part of an Einstein ring, measure the angular Einstein radius of the cluster. Give an estimate of the mass of the cluster. If the lensing mass is considered as an isothermal sphere, its Einstein radius can be expressed as a function of the velocity dispersion following: θ E = 4πσ2 v c 2 D ds D s (22) Give an estimate of the velocity dispersion of the lens. How does this compare with the velocity dispersion measured in the second chapter for the same object? If you find a difference, try to explain it. 4.2 Detecting the weak lensing effect The large arc is a heavily distorted galaxy in the background of A370. The distorsion is so large that one speaks of strong lensing. When the alignment between the lens, the observer and the background source is not perfect, the distorsion is not so obvious any more and one speaks of weak lensing. Try to find other objects in the field of view that seem to be affected by weak gravitational lensing. Give their (x, y) coordinates. We will now carry out a photometric study of the cluster using the V and R images. A color-magnitude diagram will be constructed in a way similar to the case of the globular cluster M15. However, since galaxies are not point sources, their flux can not be integrated anymore using circular apertures. Sextractor is a program that carries out photometry through elliptical apertures. Use it to obtain photometric catalogues of the objects in the cluster in V and R. Write a python script that reads the two photometric catalogues and matchs them in order to infer a V R color for each object. Plot the color magnitude diagram V R vs. R. Note that we do not need to perform a photometric calibration in the present case as we will work with color differences between objects. Identify the cluster member galaxies on the color magnitude diagram. Remember that they are among the brightest objects in the field of view and that they are all at the same redshift. Create a catalogue containing the position of each object, its V and R magnitude, its ellipticity and FWHM. Plot a diagram with the FWHM of each object versus its ellipticity. Identify the stars in this diagram. Based on this, create a new photometric catalogue where the stars have been removed. The galaxies in the background of A370 are gravitationally lensed by the cluster. Create a catalogue with only the galaxies in the background of A370, and a catalogue with only the galaxies of the cluster. For the two catalogues plot the mean ellipticity of galaxies as a function of angular distance to the center of the cluster. Do you detect the effect of gravitational lensing? 14