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Milky Way Galaxy

The Mass Density of the Local Galactic Disk

One of the great problems in astronomy is measuring the distribution of mass within the Milky Way Galaxy, or within any other galaxy.  Astronomers can only infer the mass density of the more distant regions of our Galaxy from the velocities of the stars.  For the local Galactic disk, however, astronomers can derive the mass density by counting the nearby stars.  Until recently, these two methods gave somewhat different values, with the mass density derived from star counts less than that derived from stellar motions.  This was a local version of the missing mass problem, where the dynamics of stellar motion within a galaxy imply a larger mass than does the light emitted by the galaxy.  The fundamental problem is that starlight does not trace the mass of a galaxy.

Estimating a mass density by counting stars is a difficult enterprise, even locally, where state-of-the-art instruments permit the detection of all stars within about 5 parsecs of the Sun.  The problem is not as simple as counting all the starlight, and then making a simple conversion to mass.  The power generated by a star is not proportional to the mass; rather, it increases dramatically faster than the mass.  The most massive main-sequence stars have of order 1,000 times the mass of the least massive main-sequence stars, but the most massive stars have luminosities that are of order 10 billion times those of the least massive stars.  On top of this, the least-massive stars dramatically outnumber the most-massive stars.  For these reasons, the starlight we see from the local stars is predominately from the most luminous, most massive stars, but the mass density of the local Galactic disk is predominately placed in the stars that are less luminous and less massive than the Sun.  To get a local mass density from star counts therefore requires detecting all of the dim local stars.

Even with the detection of all stars within about 5 parsecs, deriving a density is difficult, because assigning a mass to a star is difficult.  What can be observed for most stars are their distances, their proper motions, their spectra, their Doppler shifts, and their brightnesses, from which astronomers can derive the number densities of stars of different luminosities (called the luminosity function).  The only stars for which a mass can be directly measured are stars in spatially-resolved binary systems.  To go from number density to mass density requires a link between the luminosity and the mass of a star.  Astronomers have created models that provide this link by combining the insights from numerical simulations of stellar evolution with the relationship between mass and luminosity displayed by stars in binary systems.  With these models, astronomers can convert the luminosity function into a mass function, which gives the number density of stars of a given mass.

The mass function is generally parameterized as

dN/dmm−α,

where N is the number density of stars, m is the stellar mass, and α is an index that is a function of the stellar mass.  For stars greater than 1 solar mass, α is constant, with a value of approximately 2.35, but for stars less than 1 solar mass, α is approximately 1.3, and it may vary with m.  Because α > 1, most stars in the Galaxy have a mass that is at the minimum for the thermonuclear fusion of hydrogen (about 0.075 solar masses), but most of the mass density is in stars with masses for which α ≈ 2, which means stars of about one-quarter of a solar mass.  Recent work on the accurate modeling of the mass function suggests that the mass density of stars and brown dwarfs at the Galactic plane is about 0.05 solar masses per cubic parsec.[1,2]

The more direct method of deriving the mass distribution is from the dynamics of the stars in the Galactic disk.  This is the method used to derive the masses of galaxies, including the mass of our own Milky Way.  It can be applied to the Galactic disk because most of the stars in the Galactic disk are bound to the disk.

The stars surrounding the Sun fall into two camps: the low-velocity stars confined to the Galactic disk, and the high-velocity stars moving out of the disk.  The stars confined to the Galactic disk, which comprise most of the disk stars, are bound by the gravitational field generated by the stars and gas within the disk.  The high-velocity stars, on the other hand, are members of the Galactic halo.  Most are old, having been formed when the Galaxy was young, but some are young runaways, having escaped from disrupted binary systems and from open star clusters, and others are neutron stars, kicked out of the disk by the supernovae that created them.

The bound, low-velocity stars oscillate up and down in the Galactic disk; the velocity with which they move perpendicular to the Galactic plane is directly related to the depth of the Galactic disk's gravitational potential; more to the point, the square of the average stellar velocity perpendicular to the Galactic plane at the Galactic plane is directly proportional to the disk's gravitational potential.  Because the first derivative of the gravitational potential with distance away from the Galactic plane is proportional to the mass density within the disk, one can derive the mass density of the disk from the drop-off of stellar velocity with altitude above the Galactic plane.

Roughly, the mass density is proportional to V2/D2, where V is the velocity perpendicular to the galactic plane at the galactic plane and D is the thickness of the disk.  This relationship, which is related to the Jean's length for the Galactic disk, shows that the density one derives depends strongly on the distance one measures for each star, because D increases proportionally with the stellar distance scale R, and V is inversely proportional to the distance scale, making the mass density proportional to R−4.  A careful analysis using data from the Hipparcos satellite gives a mass density of 0.076±0.015 solar masses per cubic parsec.[3]  Taking into account the density of interstellar gas, which is poorly estimated at about 0.03 solar masses per cubic parsec, and the mass that appears to reside in white dwarfs, this value is consistent with the mass density within stars found from star counts.

This current agreement between two different methods of deriving mass density is a recent result; around 1990, dynamical estimates gave mass densities that are larger than the recent value by a factor of 2.  Under those results, the Galactic disk has a dark-matter problem, with 50% of the mass in the disk unseen.[4]  The sources of this conflict were the inaccuracies in distance measurements of the stars, the difficulty of measuring the luminosity function of the low-luminosity stars, and the difficulty of assigning the correct mass to a main-sequence star of a given luminosity.  The accurate distance and proper motion measurements of stars by the Hipparcos satellite, improvements in the modeling of low-mass stars, the resolution of many binary star systems through interferometry, and the new ability to see all of the nearby stars have brought the results from the two methods of measuring density into harmony.

[1]Chabrier, Gilles.  “The Galactic Disk Mass Budget.  I. Stellar Mass Function and Density.”  The Astrophysical Journal 554 (20 June 2001): 1272–1281.

[2]Chabrier, Gilles.  “The Galactic Disk Mass Budget.  II. Brown Dwarf Mass Function and Density.”  The Astrophysical Journal 567 (1 March 2002): 304–313.

[3]Crézé, M., Chereul, E., Bienaymé, O., and Pichon, C.  “The Distribution of Nearby Stars in Phase Space Mapped by Hipparcos: I. The Potential Well and Local Dynamical Mass.”  Astronomy and Astrophysics 329 (1998): 920–936.

[4]Bachall, John N.  “Self-Consistent Determinations of the Total Amount of Matter Near the Sun.”  The Astrophysical Journal 276 (1 January 1984): 169–181.

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