Dark Matter and Cosmic Web Story

by Jaan Einasto

  Modern studies by Kuijken & Gilmore (1989c,a,b) have confirmed the early results by Oort (1932), and the results by Kuzmin and his collaborators. Gilmore et al. (1989) gave a very detailed analysis of the problem and concluded that the mass density near the Sun is 0.10 Solar masses per cubic parsec. Using Hipparcos satellite data Creze et al. (1998) and Holmberg & Flynn (2000) found the local dynamical mass density 0.102 ± 0.010 M pc3, very close to the estimate of the density of visible disc matter. Thus we can say that there is no evidence for the presence of large amounts of dark matter in the disk of the Galaxy. If there is some invisible matter near the galactic plane, then its amount is small, of the order of 15 percent of the total mass density. The Gilmore et al. (1989) review as well as the papers by Creze et al. (1998) and Holmberg & Flynn (2000) have no references to Kuzmin or other Tartu astronomers papers, thus their results are completely independent.

  3.1.3 Galactic models by Parenago, Kuzmin, and Schmidt

  In the 1950’s several authors elaborated models of the Galaxy using new observational data on the kinematics and structure of stellar populations of various types. For our understanding of the structure of the Galaxy most important were models by Parenago (1948, 1950,1952), Kuzmin (1952a, 1953,1954,1956a,b), and Schmidt (1956).

  Galactic models are given by functions which describe the gravitational potential, the spatial density, the shape of the Galaxy, the circular velocity, and the projected density. In dynamical models some other functions are used, such as the phase density, which describes the velocity distribution of stellar populations. Functions are connected by equations, which follow from their definitions, laws of gravitation and dynamics. The density is connected to the gravitational potential by the Poisson equation, the circular velocity and the projected density are integrals of the spatial density, etc. If one of the principal functions and equations connecting principal functions are given, then all other functions can be calculated. So the modelling can be reduced to the choice of the principal function and the connecting equations.

  Several authors have chosen the spatial density as the principal description function. It is well-known that the density decreases when we move from the centre of a galaxy to its outskirts. Oort (1932) in his model of the Galaxy used a number of ellipsoids of constant density to represent the change of density with changing distance from the Galactic centre. Instead, (Öpik (1915) applied a Gaussian law to express the density distribution in the vertical direction. Thinking about the relation between the circular velocity and the spatial density, Kuzmin (1952a) found a simple solution: the square of the circular velocity can be calculated by integrating an ellipsoidal density distribution of an arbitrary density law. It is also possible to find the inverse solution — to calculate from the circular velocity the density distribution by solving the respective integral equation. Kuzmin used this approach to calculate from Babcock (1939) rotation data the density distribution for the Andromeda galaxy. It was war time and difficult to publish scientific papers, thus Kuzmin published his results only in the Almanac (Calendar) of the Tartu Observatory in Estonian in 1943. For our Galaxy he used the new method almost ten years later (Kuzmin, 1952a). Models with non-homogeneous ellipsoids were used independently by Wyse & Mayall (1942) for the Andromeda galaxy, and Perek (1948) for our Galaxy.

  Parenago (1948, 1950, 1952) in his model of the Galaxy accepted a circular velocity law used by Lindblad (1933). Kuzmin (1954) analysed Parenago’s model and came to the conclusion that this circular velocity law can be applied only in a restricted range of the distance from the Galactic centre. If applied to the whole range of distances then this leads to slightly negative densities at the outskirts, and to a zero total mass of the Galaxy. Negative masses can be avoided if the circular velocity law is used only up to the distance which corresponds to the zero spatial density, and to consider the corresponding distance as the outer radius of the Galaxy, taking the density outside this radius equal to zero. Such a model was suggested by Idlis (1956, 1957).

  In his model of the Galaxy Kuzmin (1952a) used data on the rotation of young populations. These populations have small velocity dispersions and rotate with almost circular velocity. Accepting for the Galaxy rotational symmetry and flat shape Kuzmin solved the integral equation to calculate the spatial density.

  My first experience of galaxy modelling was in 1952, when I made calculations for the Kuzmin model of the Galaxy. I suggested an improvement to the model: to apply the observed rotation velocities only until the galactocentric distance where rotation data were available. From these rotation data the spatial density was calculated. At larger distances the density was smoothly extrapolated to zero value at the probable outer radius of the Galaxy. The circular velocity in this region was calculated from the density. In this way negative spatial densities can be easily avoided.

  Kuzmin was not satisfied with his first model of the Galaxy. He wrote (Kuzmin, 1956b): “It is known that the velocity distribution of stars in all subsystems of the Galaxy is triaxial, i.e. can be represented by triaxial velocity ellipsoid. Theoretical explanation of this phenomenon meets still certain difficulties. The theory of stationary stellar systems, being successful in explaining the basic properties of stellar motions, gave biaxial velocity distribution. According to Jeans it was usually assumed that the phase density of a stationary axisymmetric stellar system is a function of two integrals of motion — the energy integral and the angular momentum integral. In this case the velocity distribution is biaxial. In order to have a triaxial velocity distribution we must suppose that the phase density is a function of three integrals of motion. The existence of such kind of integrals is related with the restrictions on the gravitational potential of the stellar system. For the existence of the energy and the angular motion integrals it is needed the stationarity and the axial symmetry of the potential. For the existence of the third integral, being different from previous integrals, some other additional restrictions are needed.”

  I suggested further improvements when I made calculations for the Kuzmin (1956b) model. Kuzmin describes these improvements as follows: “The extrapolation of the mass function must be done smoothly without jumps. In addition there must be taken into account three conditions. First, the mass function must vanish when the radius a approaches to a certain upper limit Rm which can be considered as the Galactic outer radius. Second, the mean radial gradient of the mass function must be in agreement with the observed radial gradient of stellar density in a wide neighbourhood of the Sun. And third, the potential corresponding to the mass function must be chosen in a way, that the Galactic outer radius Rm will be in agreement with the real upper limit of stellar velocities in the vicinity of the Sun. In other words, stars moving in the vicinity of the Sun with maximum galactocentric velocities (Oort’s limiting velocity) must reach their apogalaxy just at the boundary of the Galaxy. Usually it is wrongly supposed that the upper limit of velocities corresponds to the escape velocity, but this does not take into account the finite dimensions of the Galaxy.” I used this improvement also in my models of the Galaxy, see Fig. 3.3.

  In his new model Kuzmin (1953, 1954, 1956a) added one more restriction, posed by the existence of the third integral of motion. He found differential equations for the gravitational potential. Solving Poisson equations, he derived a formula to find the matter density, following conditions that the model has finite mass and the density is nowhere negative. He found a simple density law where the isodensity surfaces differ somewhat from ellipsoids of revolution. Their meridional sections are more curved near their extremes than the ellipses with the same eccentricity. The mass distribution can be approximately expressed by a sum of two components, one almost spherical, and the other as a flat disk. In this limit the density distribution can be considered as a flat disk with the property that the gravitational potential above the plane is equal to that of a point mass at a distance –z0 on the z-axis below the plane, while the potential below the plane is equal to that of a point mass at a distance z
0 on the z-axis above the plane. This mass distribution is called the Kuzmin disk, and it has found wide use in the dynamics of galaxies.

  Models by Parenago, Kuzmin and Idlis were published in Russian and were known only to a small circle of astronomers outside the Russian speaking community. A more popular model of the Galaxy was elaborated by Schmidt (1956). This was the first model where new rotation data based on observations of neutral hydrogen at 21-cm wavelength were used. The density of the Galaxy was approximated by 4 non-homogeneous and 9 homogeneous spheroids. Homogeneous spheroids represent 4 main populations: population I, F-M stars, high-velocity F-M stars and unknown objects. The main Galactic constants used in the model were as follows: the distance of the Sun from the Galactic centre, R0 = 8.2 kpc, and the circular velocity near the Sun, Vc = 216.5 km/sec. Schmidt calculated the gravitational potential for the whole Galaxy, and the escape velocity near the Sun, Ve = 286.5 km/sec. The difference between these velocities is approximately equal to the limiting velocity 65 km/sec, obtained by Oort in the direction of the rotation of the Galaxy. The matter density near the Sun was found to be 0.093 Solar masses per cubic parsec.

  For many years the Schmidt model was used as a standard in Galactic studies.

  3.2 New Galactic models

  3.2.1 Search for better models

  I wrote my PhD thesis on kinematics of stars and returned to galactic modelling again in the late 1950’s. Soon a good review on models of galaxies by Perek (1962) was available and I read it carefully. What surprised me was the approach of most authors to the modelling — to model the structure of galaxies of different morphological type, completely different methods were used. For spiral galaxies rotation curves were used to find the mass distribution, but photometric data were mostly ignored. In models of elliptical galaxies the projected surface density profile was the basic source of information, and the dynamical calibration was made either from the central velocity dispersion or from relative motions of companion galaxies. In modelling of both external galaxies as well as our Galaxy, data on stellar populations were mostly ignored.

  Actually all galaxies consist of similar basic populations: disk, bulge, halo, in spiral galaxies also a population of gas and young stars forming the spiral structure. The differences between galaxies of different morphological type lie basically in the proportion of various populations. When populations are explicitly used in models, then galaxies of all morphological types can be modelled in a similar way. The comparative study of galaxies of different morphological type can give information of the evolution of galaxies.

  Thus I had the feeling that in order to make optimal use of available data one has to proceed as follows:

  • To use all data of interest to construct models of galaxies of various morphological type;

  • To apply identical methods to model galaxies of all types.

  To model the structure of galaxies in more detail several problems must be solved. First: Which descriptive functions are to be used? In mass distribution models the spatial density, the projected density, and the circular velocity are the basic functions. The gravitational potential (and escape velocity) can be easily calculated on the basis of these functions. If we include data on velocities of individual populations then we can calculate hydrodynamical models.

  The second basic problem is: What connection formulae are to be used which relate different description functions? The forms of these functions depend on the shape of galaxies. When we look for the structure of regular galaxies, the natural assumption is that galaxies are flattened systems having rotational symmetry and symmetry with respect of the plane of the galaxy. Just these assumptions were made by most authors, including Oort, Kuzmin, Schmidt and others.

  Fig. 3.2 Discussing galactic models with Sergei Kutuzov and Grigori Kuzmin, late 1960’s (author’s photo).

  The third problem is connected with the choice of the principal descriptive function. If we consider only regular galaxies of axial and planar symmetry, all basic description functions are connected by simple integrals. If one function is given, all the other functions can be calculated. The question is: Which function is to be used as the principal one? If the choice is not optimal, one can have problems like Parenago had getting a model with negative densities on the outer regions. Our experience told us that the optimal choice is the spatial density, since all other functions are integrals of the spatial density. If some other function is chosen as the basic one, then other functions can be found by solving integral equations, which makes the problem more complicated.

  And the final question: Can we formulate some simple physical criteria that the principal description function must satisfy? In the first calculations for the Kuzmin model in 1952 we used direct observational values of the rotation velocity to calculate the spatial density, and extrapolated it graphically, demanding that the function itself and its radial gradient be smooth functions of the distance. It is clearly better to use some mathematically defined expression to represent the density law. But how to make a choice between various expressions? After some thinking I came to the conclusion that some physically motivated criteria for the density law are needed. Mathematically these criteria can be expressed as criteria for the density and for integrals over the density multiplied by the radial distance at some power (moment of the density). It seems natural that at least the following criteria must be fulfilled:

  • The spatial density is non-negative and finite;

  • Some moments of the spatial density are finite, at least moments which define the central gravitation potential, the mass and the effective radius of the system;

  • At large distances from the centre the density smoothly approaches zero value;

  • The description functions have no breaks.

  Together with my collaborator Sergei Kutuzov we analysed various aspects of galactic modelling (Einasto, 1968a,b,c,d; Kutuzov & Einasto, 1968; Kutuzov, 1968). A summary in English of methods of galactic modelling was published separately (Einasto, 1969a).

  3.2.2 Generalised exponential model

  Our analysis of various expressions for galactic description functions led us to the conclusion that the best approximation of the spatial density of all stellar populations can be obtained by a generalised exponential model (Einasto, 1965):

  where 0 is the central density, a is the semi-major axis of the equidensity ellipsoid, ac is the core radius, and N is the structural parameter, which allows one to vary the shape of the density profile. The cases N = 1 and N = 4 correspond to the conventional exponential and the de Vaucouleurs models, respectively. This model satisfies all conditions mentioned above, allows a natural extrapolation of the density for large distances from the centre of the galaxy, and fits observed density profiles of known galactic populations very well.

  In comparing various density profiles used to express the same observational density distribution I noticed that we get very different values of scaling parameters to measure the mass and the radial extent of the model, if we use parameters of the density law, such as the central density, 0, the core radius, ac, or similar other parameters as the virial radius. Then I started to think whether it would be possible to define such scaling parameters, which characterise the mass and the radius of the model in a way that is less dependent on the particular form of the profile. I came to the conclusion that the most stable results give the parameters defined by moments of the mass density profile — the mass M, and the harmonic mean radius, a0. These parameters are related to parameters shown in 3.1 as follows: , and αc = kα0; here ε = b/a is the axial ratio of the equiden- sity ellipsoid, and h and k are dimensionless normalising constants depending on the shape parameter N. Tamm et al. (2012) calculated the relations between the harmonic mean radius, and various other characteristics (the core radius ac, and the radius a−2, where the logarithmic slope of the profile equals the isothermal value –2). For large N values the ratios of these radii to the harmonic mean radius vary between 0.1 and 10.

  In a
ll our models of galaxies, starting from the models of M31 by Einasto (1969b) and Einasto & Rümmel (1970c), we have always used the harmonic mean radius to characterise the radial extent of the model, and the density profile 3.1.

  Trial calculations with this profile were made already in the late 1950’s, and used in my first model of the Galaxy (Einasto, 1965). A similar expression was used by Sérsic (1963) to represent the projected (surface) densities of galactic populations. In all subsequent models of galaxies we used this profile to represent the spatial density of stellar populations. In the 1960’s and early 1970’s computers were not very powerful, and it was customary to make preliminary estimates of density profile parameters by graphical comparison of observed profiles with a series of standard profiles. For this purpose tables of the generalised exponential profile were published for a wide range of the structural parameter N by Einasto & Einasto (1972b).

  This profile is now called the “Einasto profile”, and the shape parameter N the “Einasto index”. For discussions of properties of this profile see, among others, Dhar & Williams (2010); Chemin et al. (2011), Retana-Montenegro et al. (2012).

  3.2.3 Our Galaxy, system of galactic constants

  Detailed local structure is known only for our own Galaxy, thus I computed a model for the Milky Way first. The Galaxy was represented by a sum of three main populations: the flat disk consisting of young stars and interstellar gas, the disk consisting of stars of medium age, and the halo, consisting of metal poor stars, like stars in globular clusters. The structural parameter N of these populations was estimated on the basis of analogy with similar external galaxies. Also I used radial gradients of the density in the solar vicinity of main populations. These gradients are related to velocity dispersions of populations, and can be taken into account in the calculation of the model, using the modified Strömberg equation (Einasto, 1961). In the calculation of the model I considered the main populations as representatives of a number of similar populations which have close spatial structure, kinematics, and physical properties (mean age, chemical composition and colour).