The first step in the ladder is the Earth-Sun distance. We take this number (the Astronomical Unit, AU) as already known.
The most reliable method of distance determination is of course the geometric one, based on triangulation. In this case we do not need to know anything about the light sources we measure (well, in fact we need to know a little bit, but just in order to make some minor corrections). In a first approximation we simply take advantage of the motion of the Earth around the Sun and measure the apparent displacement of a nearby star projected against the more remote ones. To make these measurements we need at least one year. Please refer to Appendix 1 and the figure that goes with it. In this way we obtain the trigonometric parallaxes.
Armed with trigonometric parallaxes we can start learning about the stars. This is a process that started in the 19th century and has not finished yet. One important thing that was learned in the early 20th century is that if you plot the intrinsic brightness of stars (which can be derived only after we measure their distances) as a function of their surface temperatures, you find that most of the stars fall along a band in the luminosity-temperature diagram, which is called the ``main sequence". Stars spend most of their lives as Hydrogen burners at some position along the main sequence. The exact positions of stars in the luminosity-temperature diagram depend on the stellar ``ages", masses and chemical compositions. See the luminosity-temperature diagram in Appendix 2. The main sequence is indicated by the points labeled ``1'' for different stellar masses.
Concerning the luminosity-temperature diagram, note that in practice we do not use temperatures, but colors that are a suitable function of the temperature. For example (B-V), where B and V are magnitudes measured through Blue and Visual filters. In this work we will use two other magnitudes: U for Ultraviolet and I for Infrared. The transmission curves of these 4 filters (U,B,V,I) as a function of wavelength have peaks of maximum transmission at (roughly) 3600, 4200, 5500 and 9000 Angstroms, respectively. See Appendix 3. The hotter the star, the smaller becomes the color (B-V). It is zero for a star of spectral type A0 V (for example Vega, which is the brightest star of the constellation Lyra). The color (B-V) is negative for even hotter stars. It is customary to refer to hot stars as ``blue" and to cooler stars as ``yellow" or ``red". Note that a blue star has a negative (B-V) color because astronomers have defined magnitudes in such a way that they become smaller for brighter objects. Yes it is a nuisance, but get used to it, because this definition will not be replaced by a more rational one before you finish your studies.
It is fortunate that we can measure trigonometric parallaxes of the individual stars in groups called ``clusters". You probably have seen or at least heard about the Pleiades; that is a nice example of a nearby cluster. Its distance is slightly more than 100 parsecs, or in other words the parallaxes of individual stars in the cluster are slightly smaller than 0.01 arc second.
Since the stars in a cluster are all at essentially the same distance, we find a main sequence even if we plot just the apparent magnitudes as a function of the (B-V) color. This is very useful for our ``ladder project". Unless the chemical compositions are very different, we expect the main sequences of different clusters to fall at the same absolute brightness for each intrinsic color (B-V). Then any difference in the apparent magnitudes can be immediately translated into a difference in the distances. To know the trigonometric parallax of one cluster is then enough to get the distances to all clusters with well-defined main sequences in the (apparent magnitude) vs. (B-V) diagram. The use of the main sequence as a ``standard candle" is a typical example of an astrophysical method of distance determination.
When we look in the direction of a star cluster, we may also see foreground and background stars projected upon the cluster. How can we decide if a given star is or not a physical member of the cluster? In the case of nearby clusters it is most efficient to consider the proper motions and radial velocities of the stars in the cluster. For a definition of these concepts see Appendix 4. The members of the cluster move together in space (otherwise they would disperse very soon), and therefore all of them must have similar proper motions and radial velocities. Another way to find non-member stars is to plot apparent visual magnitudes V as a function of colors (B-V): a non-member star will be very unlikely to fall within the cluster main sequence. It will be too bright if it is in the foreground, or too faint if it is in the background. However this method is not so reliable as that based on the space motions of stars, because in some clusters there are a few stars located above the main sequence: the giants and supergiants. But do not worry, there are no supergiants in the Pleiades. Although in fact Astrophysics would be a bit easier if there were.
In our application of the ladder we consider clusters located at much larger distances, and soon we find it necessary to take into account the existence of matter between the stars: especially dust, which produces a certain degree of obscuration or extinction. See Appendix 5. We need to correct the observed magnitudes, like V, to the value they would have in the absence of interstellar extinction. This can be done because of the properties of interstellar extinction: when expressed in magnitudes, it is roughly inversely proportional to wavelength. The extinction is therefore stronger at shorter wavelengths, and this produces a ``reddening" of the stellar light. We measure the reddening and then calculate the corresponding amount of extinction. For example, if we call E(B-V) the reddening or ``color excess" suffered by the color (B-V), empirical studies have shown that the extinction AV in magnitudes is given in almost all cases by
|
which allows us to calculate V corrected by extinction: V0 = V - AV. We can estimate the E(B-V) using the two-color diagram, (U-B) plotted as a function of (B-V). Using colors, which are in fact magnitude differences, distances do not play a role. In the case of a cluster, the observed position of the upper branch of the main sequence lies to the right and below the position of an unreddened main sequence. The length of the displacement along the reddening line that is needed to put the observed upper branch on top of the unreddened one allows us to get the value of E(B-V).
After correcting for the effects of interstellar extinction, we can go back to the V vs. (B-V) diagram and obtain the distance modulus difference between two clusters directly from how many magnitudes we need to displace one main sequence until it coincides with the other. See Appendix 6.
After we have collected the distances to many clusters, we can obtain the intrinsic brightnesses of many more stars (all those that belong to the clusters). Having collected so many stars we have a good chance to get a few of those stars that belong to less populated regions of the luminosity-temperature (or luminosity-color) diagram. These stars have evolved away from the main sequence, becoming more luminous. They are less frequent than main sequence stars because they do not spend much time in their new evolutionary stage. We call these bright evolved stars ``giants" or ``supergiants". See Appendix 7. The fact that they are very bright means that we can see them even if they are very distant. Unfortunately they do not have all the same intrinsic brightness. However, if we can find a way to know how their intrinsic brightness depends on other, easily observable properties, then we can use them as another step in our distance ladder.
Nature has provided us with a very useful ``instability strip" across the luminosity-color diagram. In fact there are several such strips, but let us keep things as simple as possible. Whenever a star enters this instability strip, in its evolution away from the main sequence, it begins to pulsate, in a similar way as a pendulum keeps oscillating as it swings past the position of minimum potential energy in alternative directions.
Probably the star would prefer to stay quiet, but once it enters the instability strip it is forced to pulsate. And as soon as the star leaves the instability strip, the pulsations are damped and stop. The pulsation produces periodic changes in the size and the surface temperature of the star, and therefore also the stellar brightness varies, with a period of several days or weeks. A few centuries ago, astronomers noticed a few bright stars pulsating in this way, and after some time decided to call them ``cepheids" in honor of one of the first known cases, Delta Cephei. During the 20th century a theory of stellar pulsation has been under development to explain this behavior, but we do not need to discuss it right now. What we need to know is that the period of pulsation depends on the inverse of the square root of the density in the star: the denser the star, the smaller is the pulsation period. It can be shown that this relation implies the existence of a period-luminosity relation: the longer the period of pulsation, the more luminous is the star. There are many kinds of pulsating stars, but the cepheids are a very homogeneous group, with extremely stable periods, very similar light curves, and following a very well-defined period-luminosity relation. See Appendix 8. It is comparatively easy to discover cepheids, and once you know the period of the light variation you have information about the luminosity of the cepheid.
We need to explain a little about period determinations. We start from a list of observations of the variable stellar brightness at different times. The simplest method of period determination is practical because we have computers: we simply test all possible periods until we find the right one. How do we notice that we have got the right one? Because after the observations have been ordered according to phase, the light curve is not a chaotic zig-zag but actually looks like a smooth curve. For the computer to perform its duty, we need a quantitative estimator of the smoothness of a light curve, so that the computer can show us which period produces the optimum value of the estimator. We will use an estimator built by summing the squares of the magnitude differences between observations made at adjacent phases. The best period (which is presumably the real one) causes this estimator to drop to a minimum. Having clarified this point, we go back to the cepheids.
You may wonder how could astronomers decide that cepheids obey a period-luminosity relation before it was possible to get the distances to many of them. The reason is that many cepheids were discovered in a satellite galaxy of our Milky Way: the Large Magellanic Cloud (LMC). Although the discoverers did not know the distance to the LMC, they knew that all the cepheids in this satellite galaxy are at the same distance from us. So in fact they discovered a relation between the average apparent magnitudes and the periods of the LMC cepheids. Only much later it was possible to get the intrinsic brightnesses of just a few cepheids in Milky Way clusters, and thus an accurate distance to the LMC could be finally calculated, some 50 years after the LMC cepheids were discovered.
With current technological advances (the Hubble Space Telescope and the new 8-m and 10-m telescopes) we can discover cepheids at very large distances, up to 25 or may be 30 Mpc, and in the next few years there will be a lot of activity in the determination of distances to many galaxies where cepheids can be found. Using the distances to all these galaxies we will be able to understand the properties of other brighter astrophysical objects (for example supernovae), and a new step in the distance ladder will become firmly established. You may have a chance to participate in this new development. But first you must finish this Praktikum correctly.
Some people may find it difficult to imagine how essential good distances are for the development of our scientific ideas about the Universe. Before 1924 (when cepheids were discovered in the Andromeda spiral galaxy) astronomers did not know if galaxies are small satellites of our Milky Way, our Galaxy being the dominant constituent of the Universe, or if other galaxies are (as we know now) independent accumulations of luminous and dark matter, of comparable importance to our own galaxy, and at much larger distances from us than previously thought. The realization that we live in a Universe of galaxies triggered explosive advances in Astrophysics and led to the discovery of the isotropic expansion of the Universe and the modern development of Cosmology. We will make a very short excursion into Cosmology. We will estimate the Hubble parameter H0, in km/s per Mpc, which gives the rate of expansion of the Universe. The inverse of H0 can be taken as a very rough estimate of the time elapsed since the Universe was all together in the ``big bang". In order to estimate H0 we need to know (a) the radial velocity of a galaxy and (b) its distance. As the final challenge of the Praktikum you will be asked to estimate the age of the Universe.
(A) The HIPPARCOS parallax of the Pleiades open cluster permits to obtain its distance.
(B) Having found cepheids in other, more distant clusters, we measure the distance moduli of these clusters relative to the Pleiades by superposing the unevolved sections of the cluster main sequences. Here we must make corrections for the interstellar extinction and reddening. We will apply this procedure to the particular case of the open cluster NGC 6087, to which the cepheid S Normae belongs.
(C) Knowing the average apparent visual magnitudes of other cluster cepheids, and the distance moduli, we obtain the absolute visual magnitudes of the cepheids. Plotting them as a function of the pulsation periods, we find the period-luminosity (or period - absolute magnitude) relation. But we see that most of these local cepheids with reliable distances have rather short periods, and we would prefer to have our period-luminosity relation defined by many long-period cepheids, which are brighter and therefore can be detected at larger distances from us.
To solve this, we compare the local cepheids with those in the Large Magellanic Cloud (LMC), which have a more convenient frequency of long periods. We obtain the distance to the LMC and use it as our standard candle: we measure distances to more distant galaxies relative to the distance of the LMC.
(D) We apply our method to a galaxy in the Fornax cluster: NGC 1365. The full work would involve the following steps:
(a) acquisition of CCD images of the galaxy at several epochs.
(b) image processing, discovery of variable stars.
(c) photometric reductions, magnitude estimates.
(d) period determination and confirmation as cepheids.
(e) estimation of average magnitudes and colors on a standard system.
(f) correction for extinction and distance estimate, using the LMC cepheids as standard candle.
Since we do not have enough time for the full procedure, this last phase of the Praktikum will be restricted to steps (d) to (f). In the final step we estimate the Hubble parameter H0 and the age of the Universe.
3.1. Take the relevant volume of the HIPPARCOS catalogue (old style, not recommended) or go to
http://cdsweb.u-strasbg.fr/cats/cats.html (modern style)
and search for as many of the Pleiades as you can find, knowing the coordinates of the cluster, RA2000=3h 47m, Dec2000=+24° 07'. Bear in mind that the proper motions should be: ma around 20 mas/yr, and md around -45 mas/yr. The angular radius of the cluster is about 2 degrees.
You should find some 40 members of the Pleiades. Make a file ``pleiades.dat'' with the information extracted from HIPPARCOS. Run the program ``hipar'', which plots a histogram of the parallaxes. Discuss if some of the individual parallaxes should be rejected. Can we test if some of these objects do not belong to the cluster? Make a preliminary estimate of the cluster distance, derive from its angular size its physical size in pc, and consider if depth effects within the cluster can explain the different parallaxes determined by HIPPARCOS. After deciding which stars, if any, have to be rejected, calculate the average parallax and express the distance in parsecs and in magnitudes (distance modulus: the difference between the apparent and the absolute magnitude). Give an estimate of the uncertainty in the distance to the Pleiades.
IDL program: hipar.pro . Data: pleiades.dat
Summary of tasks: at the end of 3.1 you must have a list of Pleiades members, with parallaxes and proper motions, and notes justifying eventual rejections of certain stars as members. You must have calculated the average parallax of the Pleiades, and expressed the resulting distance in pc and as a distance modulus. You must have also the size of the cluster in pc.
3.2. Consider now the cepheid S Normae, located
in the cluster NGC 6087.
Period = 9.75 days
Average Visual magnitude = 6.42 (apparent magnitude)
Color excess E(B-V) = 0.18
The interstellar extinction correction is AV = 3.2 E(B-V). Calculate the average visual magnitude V0 that would be observed in the absence of interstellar extinction.
In order to obtain the absolute visual magnitude we need the distance to the cluster. We will estimate the difference between the distance moduli of NGC 6087 and the Pleiades by comparing their main sequences. Run the IDL program ``clusters''. We have files with UBV photometric data for the Pleiades (50 stars) and NGC 6087 (29 stars). For this comparison we must use corrected stellar magnitudes, because each cluster suffers a different amount of reddening and therefore of extinction. The colors U-B and B-V can be used to estimate the amount of reddening correction to be applied. We do not need the individual reddenings for each cluster; it is enough to know the difference in E(B-V). We determine this difference by comparing the positions of the selected stars in the (U-B), (B-V) plane. There will be a certain amount (DE(B-V)) that will force the stars from both clusters to occupy the same place in the two-color diagram. We will estimate this DE(B-V) by trial and error.
Having obtained the DE(B-V) we can calculate the DAV, which must be 3.2 times the DE(B-V). Can you explain why? After applying this differential extinction correction, we assume that the only difference that remains between the main sequences of the 2 clusters derives from their different distances to us. We will estimate the difference in distance moduli by trial and error until we get a good superimposition of both main sequences. Please estimate the uncertainty in the distance modulus difference, which has two main sources: the error in the DAV, and the error in the main sequence comparison, due to the natural dispersion and the errors in the photometry.
At the end of this step we calculate the distance modulus of NGC 6087 and express the distance in parsecs. We also calculate the average absolute visual magnitude of S Normae. How many times brighter than our Sun is this star? Remember that the absolute visual magnitude of the Sun is +4.8. What would be the average apparent visual magnitude of S Nor at a distance of 25 Mpc, assuming no interstellar extinction? At what distance would a star like our Sun have that same apparent magnitude?
IDL program: clusters.pro . Data: cluple.dat, clu6087.dat
Summary of tasks: at the end of 3.2 you must have an estimate of the following quantities: the average visual magnitude of S Nor that would be observed in the absence of interstellar extinction, the DE(B-V) between the Pleiades and NGC 6087, the difference between the distance moduli of the Pleiades and NGC 6087, the resulting distance modulus of NGC 6087, its distance in pc, the average absolute visual magnitude of the cepheid S Nor, its luminosity in solar luminosities, its average apparent visual magnitude if it were at a distance of 25 Mpc, and the distance at which the Sun would have that same apparent visual magnitude.
3.3. In principle we should repeat the
previous work for each of several
cepheids that belong to open clusters or associations, but we will
simply take the results from the literature, e.g. Feast and Walker 1987.
We have a file cepgal.dat with the periods and absolute visual mags of
19 cepheids. Using the program ``zeropo" we plot these numbers (they
appear as plus signs) and add our result for S Normae, which should
appear as a diamond and fall of course on the period-MV
relation if all went well. Now we introduce the LMC cepheids: we will
determine interactively the distance modulus to the LMC by adjusting
(trial and error) the LMC period-mV relation to our local Galactic
period-MV relation. The LMC cepheids will appear as squares in
our plot. We use files plr.dat and spc.dat with data for LMC cepheids
taken from Madore (1985). We must adopt a value for the E(B-V)
of the LMC cepheids: take 0.12 (the program corrects for extinction),
and then estimate the LMC distance modulus and its uncertainty.
IDL program: zeropo.pro . Data: cepgal.dat, plr.dat, spc.dat
Summary of tasks: at the end of 3.3 you must have an estimate of the distance modulus of the LMC. Express the corresponding distance in pc.
3.4. Now we are ready to jump to the Fornax cluster
of galaxies. We will use recent data on 43 cepheids discovered with
the HST by Silbermann et al. (1998). For your convenience and comfort
we have painstakingly prepared 43 files, one for each variable star.
Their names are cepv#.dat, with # from 3 to 52 (not all the variables
in Silbermann et al. (1998) were used). Each file gives the visual
magnitudes at certain times of observation. This information is used
by the program ``lafkin" to test all possible periods, by the
method of Lafler and Kinman, and find the period that produces
the smoothest light curve, causing the function ``theta" to
become minimum. You have to verify if the selected period produces
a typical cepheid light curve; the program is not powerful enough to
make this evaluation without your help, and it might happen that the
discovered variable is not a cepheid but, for example, an eclipsing
binary. Or it might happen that the program finds more than one
suitable period, in which case you have to select the best one.
Sometimes the best period does not give the most convincing light
curve, and a slightly worse period can be selected if its light curve
looks better.
The time interval between the first and last observation in Silbermann et al. is 49 days. This immediately implies that the method of Lafler and Kinman cannot discriminate between any periods longer than 49 days. Can you explain why? Even so, Silbermann et al. have listed 5 cepheids with periods longer than 50 days. How did they manage? Would you reject those data as unreliable?
After you are convinced that each variable is a cepheid, please estimate on the light curve its average visual magnitude. Also give an estimate of the average infrared (I) magnitude, using the light curves provided in the work of Silbermann et al. Edit the file n1365.dat and add your results into it: period in days, average V, average I, for each cepheid.
Now we use the last program, ``plr", to estimate the distance modulus of NGC 1365 relative to the LMC. Give the name of the file with your data (n1365.dat) and the program will ask for two relative distance moduli: one for the V and another for the I magnitudes. Select them by trial and error until you are satisfied with the fits of both period-apparent mag relations. Note that in this case we have not applied any reddening correction; we are comparing the observed apparent mags. If the cepheids in NGC 1365 suffer more reddening than those in the LMC, we will notice because there will be a difference in the relative distance moduli for V and I; the reason being that the interstellar extinction is weaker in the infrared. If there were a difference between the V and I relative distance moduli, we could estimate what happens as we let the wavelength increase to infinity, because we know AI = 0.48 AV, given the shape of the interstellar extinction curve as a function of wavelength (remember, we said before that the extinction is inversely proportional to wavelength, in a first approximation). Bearing this in mind, estimate the delta mag that corresponds to infinite wavelength, and using the LMC distance modulus obtained in step 3 give the distance modulus of NGC 1365 and express the distance in Mpc. Estimate the accumulated uncertainty in the final result.
Knowing the redshift of the Fornax cluster, we could use the distance we have determined to derive the Hubble parameter H0. However this is not a good idea because there are uncertainties in the local velocity field (where ``local" means now a few tens of Mpc). Therefore we will use the following information: the Coma cluster of galaxies is 5.7 ± 0.5 times as distant as the Fornax cluster, and Coma has a redshift of 7185 ± 60 km/s. With this information plus your distance to Fornax, estimate H0 and its uncertainty. Finally estimate the age of the Universe (1/H0) and its uncertainty.
IDL programs: lafkin.pro, plr.pro . Data: cepv#.dat, plr.dat
Summary of tasks: at the end of 3.4 you must have a file with the periods and average V and I magnitudes for cepheids in NGC 1365. You must have obtained the distance modulus relative to the LMC, and the final result: the cepheid distance to NGC 1365, with its uncertainty. You must have got also your estimate of H0 in km/s per Mpc, and its inverse, expressed in years.
Freedman et al. 1994, ApJ 427, 628: The HST Key Project on the extragalactic distance scale I. The discovery of cepheids and a new distance to M 81
Jacoby et al. 1992, PASP 104, 599: A critical review of selected techniques for measuring extragalactic distances
Madore and Freedman 1991, PASP 103, 933: The cepheid distance scale
Freedman and Madore 1990, ApJ 365, 186: An empirical test for the metallicity sensitivity of the cepheid period-luminosity relation
Feast and Walker 1987, ARAA 25, 345: Cepheids as distance indicators
Turner 1986, AJ 92, 111: Galactic clusters with associated cepheid variables. I. NGC 6087 and S Normae
Madore 1985, Procs. of IAU Coll 82 (Je 109 in the USM library) p. 166: Cepheid variables as extragalactic distance indicators
Lafler and Kinman 1965, ApJS 11, 216: method of period determination
Johnson and Mitchell 1958, ApJ 128, 31: the color-magnitude diagram of the Pleiades cluster