by Jaan Einasto
The basic problem in the modelling of the Galaxy is the choice of galactic parameters. The most important parameters are the distance of the Sun from the centre R0, the Oort constants A and B, the Kuzmin constant C, the circular velocity near the Sun V0, and the matter density in the solar vicinity 0. Additional parameters are the ratios of velocity dispersions in the radial, tangential and vertical directions, and some other observed quantities. The circular velocity as a function of the distance from the galactic centre, V(R), can be found from radio observations of the neutral hydrogen through the function of differential rotation U(x) as follows:
where x = R/R0, and R is the distance from the centre. Radio observations allow us to derive directly the gradient of the differential rotation near the Sun, W = –1/2(dU/dx) at x = 1. This gradient can also be considered as a galactic parameter.
In earlier models of the Galaxy (Kuzmin, 1952a; Schmidt, 1956) some fixed values for principal galactic parameters were used, such as R0, V0, A, B, and 0. As noted above, there are actually more observable quantities available. All these quantities are connected by certain formulae which follow either from the definition of parameters or from the theory. To make use of these additional observed quantities, the system of galactic parameters can be found using all quantities and connection formula, and a balanced system of parameters can be found by the method of least squares. This method was developed in detail by Kutuzov (1965). The theory of the determination of the balanced system of galactic parameters was the subject of his PhD thesis. I calculated a preliminary system of parameters in the model of the Galaxy, using data published until the end of 1961 (Einasto, 1965).
Observational data accumulated rapidly, and I prepared together with Sergei Kutuzov a new system of galactic parameters for the Commission 33 Meeting of the XII General Assembly of the International Astronomical Union (IAU) in Hamburg 1964 (Einasto & Kutuzov, 1964). In the new system we used 10 independent parameters and 6 relations between them, which were used to find by the method of least squares the optimal set of parameters. I applied to attend the Assembly, but my application was not approved by Soviet authorities. Our text was printed, and I made a trip to Moscow to hand over our preprint to Professor Ogorodnikov, who was allowed to participate in the General Assembly. He presented my results on the new system of galactic parameters.
A few weeks later one of the leading experts on the structure of our Galaxy, Bart Bok, made a visit to the Sternberg Astronomical Institute of Moscow University. I regularly visited Sternberg Institute and was informed on his planned seminar talk. So I made a second trip to Moscow to listen the talk by Bart Bok. He reported scientific news from the IAU General Assembly, and to my surprise gave a rather detailed overview also of my new system of galactic parameters. After the talk I had a long discussion with Bart — this was our first personal meeting. He was interested in the work of Tartu astronomers, in particular that by Kuzmin.
In the late 1960’s I prepared a second model of the Galaxy using recent observational data and an improved method to calculate the system of galactic parameters. This new model was presented at the Commission 33 Meeting of IAU General Assembly in Brighton (Einasto, 1970a). The model consists of three populations, flat disk, disk, and halo, using the modified exponential profile. Tables for the gravitation potential, circular velocity and other main parameters as functions of z and R (in cylindrical galactocentric coordinates) were published separately by Einasto & Einasto (1972a).
In this model I used a novel method to find the main galactic parameters, Ro and V0, based on ideas applied first by Eggen et al. (1962) in their study of the formation of the Galaxy, as well as ideas used in the calculation of the Kuzmin (1956b) model. As written above, radio observations of the 21-cm line allow us to find the differential rotation function U(x). The circular velocity is then calculated from equation 3.2. But radio observations allow us to find the rotation and the mass distribution function only for inner galactic regions, R ≤ R0. For larger distances, R > R0, the mass distribution can be found by smooth extrapolation. Figure 3.3 shows two variants of extrapolation for different values of V0. In both cases the function U(x) is identical. If we use smooth extrapolation with different V0 we get also two different values of the limiting radius of the model, Rlim; this radius is defined as the distance where the spatial density is a hundred times lower than near the Sun.
It is well-known that there are no stars with velocities in the direction of galactic rotation which exceed the solar velocity by about 65 km/s or more (Oort, 1928). This velocity, ∆υ, is often called the Oort’s limiting velocity. Oort (1928) assumed that this velocity corresponds to the escape velocity near the Sun. However, as shown by Kuzmin (1956a,b), this assumption ignores the finite dimensions of the Galaxy, and should be interpreted as the velocity needed to reach the boundary of the Galaxy, Rlim. Using the mass distribution model we can calculate the gravitational potential and find apogalactic distances, Rapogal, of stars moving with the velocity Va = V0 + ∆υ in the direction of the rotation of the Galaxy.
Fig. 3.3 Left panel: the extrapolation of the mass distribution function beyond the Sun’s distance R > R0 (dashed lines) with different values of V0. The limiting radii of models, Rlim, are indicated. Right panel: the dependence of the limiting radii, Rlim, and the apogalactic distances, Rapogai, on the circular velocity near the Sun, V0. Two cases of smooth extrapolation of the mass function with different Rlim are shown. Apogalactic distances are given for two values of the Oort’s limiting velocity, ∆υ (Einasto, 1970a).
The left panel of Fig. 3.3 shows the mass distribution function (the mass of an equidensity ellipsoidal layer of unit thickness), μ(a) = 4πεa2(a), where a is the semi-major axis of the equidensity ellipsoid with the axial ratio ε = b/a,and (a) is the spatial density. The function is calculated for two values of V0. The right panel of Fig. 3.3 shows the dependence of the limiting radii, Rlim, and the apogalactic distances, Rapogai, on the circular velocity near the Sun, V0. The circular velocity V0 is calculated for two values of the limiting radii, Rlim, and of the Oort’s limiting velocity, ∆υ.
If the adopted circular velocity is too small, then the Galaxy has a rather sharp edge, and Rlim is small, but the gravitational potential is also small and apogalactic distances Rapogal of stars moving with the velocity Va are large, much larger than Rlim. If the adopted circular velocity is too high, then we have the opposite situation, as seen in Fig.3.3. Both distance coincide when we use the true value of the circular velocity for this mass distribution. This method I applied in the calculation for the Kuzmin (1956a,b) models and my own models (Einasto, 1965, 1970a). In these models no dark halo was included, which led to a rather high value of the circular velocity near the Sun, V0 = 250 km/s.
About ten years later I applied the same method using a much more accurate mass distribution model including a dark halo (Einasto et al., 1979a). The result was V0 = 220 km/s, in very good agreement with values found by other independent methods. This value is presently generally accepted.
A new system of galactic parameters was discussed by Commission 33 of IAU in the late 1980’s using the same approach. The accepted system (Kerr & Lynden-Bell, 1986) was rather close to the system we presented in 1964. It was very difficult to get permission for visits outside the USSR, so I was not able to take part in the preparation of the new system in 1986.
The local structure of a galaxy is best known for our own Galaxy. But our position inside the Galaxy makes it difficult to see its overall global structure. Global information on stellar populations is better known for external galaxies. Thus it was natural to continue the study of physical and dynamical properties of galaxies using as test object our neighbour, the Andromeda galaxy M31. This task was realised in a series of papers (Einasto, 1969b, 1970c; Einasto & Rümmel, 1970b). In the construction of models of external galaxies a number of other problems need to be solved.
3.2.4 Mass-to-luminosity ratios of stellar populations
In the late 1950’s and 1960
’s astronomers having access to large optical telescopes continued to collect dynamical data on galaxies. The most extensive series of optical rotation curves of galaxies was made by Margaret and Geoffrey Burbidge, starting from Burbidge & Burbidge (1959); Burbidge et al. (1959), and including normal and barred spirals as well as some ellipticals. Rubin & Ford (1970) derived the rotation curve of M31. For all galaxies the authors calculated mass distribution models. The problem with these models was the difficulty of extrapolating mass distribution to large distances from the galactic centre.
Also radio observations of the neutral hydrogen 21-cm line accumulated. van de Hulst et al. (1957) found that the neutral hydrogen emitting the 21-cm line extends much farther than the optical image. They were able to measure the rotation curve of M31 up to about 30kpc from the centre. Roberts (1966) made a new 21-cmhydrogen line survey of M31 using the National Radio Astronomy Observatory’s 300-foot telescope. He calculated a mass distribution model for M31, using analytical expressions for the rotation curve, similar to the Bottlinger model and suggested by Brandt (1960), Brandt & Scheer (1965). As in the case of the Parenago model of the Galaxy, results for the mass distribution at large distances from the centre depend critically on the choice of parameters of the velocity law.
In these models data on the distribution of light in stellar populations was not used. My goal in modelling M31 was to make use of these data. In my preliminary model of M31 (Einasto, 1969b) I used the following populations: nucleus, core, disk, and flat disk. Photometric data were available for all of them. The problem was how to get mass-to-luminosity (M/L) ratios to find the mass distribution of populations. In principle, the rotation data can be used for this purpose, since in various regions of the galaxy different populations dominate. But there exists also a second possibility to use independent data: velocity dispersions of the nucleus and the core are measured, and detailed spectroscopic observations of the stellar content are available to find the stellar luminosity function, which allows one to estimate the M/L. For the nucleus of M31 such observations were made by Spinrad (1966), who found M/L = 16.7.
Einasto (1969b) calculated mass distribution models for both M/L variants. If rotation data by Roberts (1966) are identified as circular velocities, then for the nucleus and core very small values for the mass-to-luminosity ratio were obtained, M/L ≪ 1. For the disk rotation the data suggest M/L ≪ 9, and for the flat population M/L ≃ 80. These values are in conflict with other data, found on the basis of the luminosity function of respective stellar populations. For the nucleus we have every reason to accept Spinrad data, and for the disk and flat disk we can trust the estimate by Öpik (1922a), who found for the solar vicinity M/L ≃ 3. On the other hand, if we apply the Spinrad value M/L = 16.7 also for the core and the disk, then this leads to too high a value for the circular velocity at small distance from the centre, V ≃ 380 km/s at R ≃ 1 kpc.
To understand the very low value of M/L found from radio data at small distances from the centre, Einasto & Rümmel (1970a) investigated the radial velocity field of M31. This analysis showed that small rotational velocities, obtained from radio data, are due to the low angular resolution of radio data, which smear also rotation velocity profiles at small distances from the centre.
To avoid these difficulties, I calculated a new model of M31 (Einasto, 1970b), which included the nucleus, bulge, disk, flat disk, and halo. I used new spectral determination (M/L ≃ 42) of the nucleus by Spinrad (personal communication), and new rotation data by Rubin & Ford (1970). To find the spatial distribution of the halo I used data on the distribution of globular clusters in M31. The spatial structure of the flat disk was found from the distribution of neutral hydrogen. Mass- to-luminosity ratios for populations were found from the rotation curve in regions where the particular population dominates. The M/L values, obtained for the bulge, the disk, the flat disk and the halo, were still very uncertain.
Thus my experience with these preliminary models of M31 emphasised that the problem of finding proper values of M/L for stellar populations is not so easy. Additional information is needed to find better values.
One possibility is to use additional spectrophotometric data for regions outside the nucleus and in other compact systems — star clusters and nuclei of galaxies. Spinrad et al. (1970) and Spinrad & Taylor (1971) found that star clusters M67, NGC 188 and NGC 6791 are super-metal-rich, which suggests a high M/L value. Spinrad et al. (1969) studied M7 giants in the nuclear bulge of the Galaxy and found that the metal abundance is higher than in stars near the Sun. Spinrad et al. (1971) investigated colour changes and absorption-line variations over the inner disk of M31, and in central regions of M32 and NGC 4472. They made a preliminary synthesis of the disk population of M31 and found that the mass-to-luminosity ratio of the disk is equal or ever slightly greater than the nuclear value, M/L ≃ 45. On the other hand, the metal abundance of the disk of M31 and NGC 4472 is close to the solar level, in contrast to a high level at their nuclei. Thus it is difficult to accept so high values of M/L for the disk.
The second possibility is to use dynamical data on compact stellar systems like star clusters. It is well-known that stars form in various star-formation clouds, which later evolve to stellar associations and clusters. Later associations and loose clusters are dissolved by various perturbations. Probably all field stars of galaxies formed in just such way. So I started to collect dynamical data on star clusters of various type and age. I expected that for a certain set of colour, metallicity and age, star clusters have mass-to-luminosity ratios which are close to galactic populations with similar physical properties.
Data on velocity dispersions in star clusters were just starting to be available. For metal-poor systems like globular clusters very low mass-to-luminosity ratios were obtained, for instance, M/Lv = 1.7 ± 0.4 in solar units for NGC 6388 by Illingworth & Freeman (1974). Freeman & Munsuk (1972) found for relatively young LMC clusters NGC 1835 and NGC 2210 even lower values, M/Lv = 0.2. For Omega Centauri Poveda & Allen (1975) found a significantly larger value, M/LB = 3.2, than for other globular clusters.
Using a compilation of velocity dispersion measurements in nuclei of galaxies of various luminosity Einasto & Kaasik (1973) and Einasto (1973) found a clear dependence between the total luminosity and the velocity dispersion: more luminous galaxies have higher values of central mass-to-luminosity ratios.
3.2.5 Evolution of galaxies
To summarise, my experience suggested that different data and methods lead to rather different values of M/L, and it is not clear which data can be trusted. Thus there exists a need to bring all estimated M/L values to a coherent system. It is clear that the luminosity of a population depends on its chemical composition and the age. A similar problem exists in compact stellar systems, such as star clusters. To understand how M/L depends on the composition and on the age of the population, evolution models are needed. For this reason I started in the late 1960’s to develop my own evolution model. The description of the evolution model is given in my Doctor of Sciences thesis (Einasto, 1972b). The model is similar to the model developed by Tinsley (1968), but my model was developed completely independently from the Tinsley model and some details were different. I started to develop my model before the model of Tinsley was published, thus I had to invent all details of the model calculation myself.
Models of the evolution of stellar populations and galaxies are based on stellar evolution tracks, star formation rates (as a function of time), and the initial mass function (IMF). From literature I found stellar evolution tracks for 16 star mass values between 0.05 and 60 Solar masses. To model the evolution of star systems, tracks were tabulated for 19 evolution stages, either directly from published tracks, or by interpolation. Each stage in all tracks corresponds to a certain phase of the evolution. For each stage the age, the bolometric luminosity and the effective temperature were given. Tracks were found for three values of metal content: direct data were available for stars of normal metal content, Z = 0.02. By ex
trapolation tracks were found for metal-rich stars, Z = 0.10, and for extremely metal-poor stars, Z = 10−5. This dataset allows through interpolation to find isochrones — the luminosity-temperature diagrams for a stellar population of an arbitrary age and composition. Luminosity functions, colours and mass-to-luminosity ratios were calculated in the UBVRIJKL colour system. Model populations were found for 14 epochs: 0.01, 0.03, 0.1, 0.3, 1,…, 15, 20 billion years.
For IMF I used the Salpeter (1955) law F(m) = a m−n, where m is the mass of the forming star, and a and n are parameters. This law cannot be used for stars of arbitrary mass, because in this case the total mass of forming stars may be infinite. Thus I assumed that this law is valid only in the mass interval from M0 to mu, the lower and upper limits of the forming stars, respectively. My calculations showed that the mass-to-luminosity ratio Mi/Li of the population i depends critically on the lower mass limit of the IMF, M0.
An independent check of the correctness of the lower limit is provided by homogeneous stellar populations, such as star clusters. Here we can assume that all stars were formed simultaneously, the age of the cluster can be estimated from the HR diagram, and the mass derived from the kinematics (velocity dispersion) of stars in the cluster. Such data were already available for some old metal-poor globular clusters, for some relatively young medium-metal-rich open clusters, as well as for metal-rich cores of galaxies. I compared the results of stellar population modelling with direct dynamical data for central regions of galaxies (velocity dispersions) (Einasto & Kaasik, 1973). As a further test I studied the rate of star formation as function of the density of stellar populations in M31 (Einasto, 1972c). The results confirmed the Schmidt (1959) law: the star formation rate is proportional to the density squared.
