4.2.2. The Solar Neighborhood
The structure of the Galaxy can be divided into three distinct kinematical
regions that are defined by particular ratios of rotational velocity
(Vc) to velocity dispersion
(v). These regions
are known as the
thin disk, the thick disk and the halo. Their vertical density
distribution,
(z) assumes the form
![]() | (7) |
where zh is the vertical disk scale height and
Rh is the radial disk scale length.
This exponential form can be derived by
assuming an infinitely thin disk (which is justified by the observations)
together with an isothermal velocity distribution. In the case of
a self-gravitating disk (z) goes
as sech2. The self gravity in this case is provided by the sum
of the stellar distribution and the dark matter distribution.
The Solar Neighborhood is a region of
of radius roughly 300 light years that contains a few thousand stars.
This region contains thin disk, thick disk and halo stars and their
normalization is important to the determination of the mass density
within this region.
The Thin Disk: This is a
coherent and highly rotating
component. The average value of
v / Vc is
0.1 and
the scale height is
100 pc. Since the total disk
diameter is
30 kpc the thin disk has a
thinness ratio
of
.003. Most of the
molecular gas, and hence massive star
formation in the Milky Way, is concentrated in the thin disk.
The Thick Disk: The
existence of this component is still
somewhat uncertain, although it seems to be required to
account for the observed star counts in the solar neighborhood as
well as the observed stellar metallicity as a function of height
above the galactic plane. Zinn and West (1984) also identified
a population of Globular clusters that has disk like kinematics.
Estimates
of zh range from .7 -1.1 kpc with some as high as 1.5 kpc.
These estimates vary according to the particular sample of stars which
is used. A fair tracer of the thick disk population has not been
unambiguously identified (see von Hippel and Bothun 1993).
The value of
v,
however, is well
determined at 45 ± 5 km s-1, making
v / Vc
0.25.
Unfortunately, the thick disk is sometimes confused with the old thin disk,
which has zh
300 pc. The higher value of
zh for the old thin disk
stars is a consequence of their many orbits about the center of the galaxy
and the cumulative effects of small gravitational scattering off of
other stars (see Chapter 3).
This acts like a diffusion process which causes the stars
originally born in the very thin disk to diffuse vertically.
The Bulge/Halo: These are
the spheroidal components of galaxies
that are supported by internal velocity dispersion instead of large
scale rotation. The orbits of the stars can be anisotropic and so
the spheroidal component is not necessarily always round. The bulge
portion of the spheroid is a regime of high stellar density over
a fairly small scale. The halo is best defined by the GCS of
the galaxy. It is a large, very low density collection of old stars.
These stars were the first to form in the galaxy as it was collapsing
(Chapter 5). There is very little rotation in the
galactic halo so
v /
Vc substantially exceeds 1. The overall
scale size of the halo component is difficult to ascertain as there
are few stars which trace it. For spheroidal systems, the fall off
in surface density is
![]() | (8) |
where re is the effective radius defined to enclosed 50% of the light. For the high density bulge, re is likely to be less than 2 kpc (Gilmore et al. 1989; Sackett 1997). For the halo, re lies in the range of 3-7 kpc (Bothun et al. 1991).
The solar neighborhood lies about 8.5 kpc from the center of the Galaxy (see Sackett 1997). At that distance, there are very few, if any, true bulge stars. Hence, the solar neighborhood stars are a mixture of thin disk, thick disk and halo stars. Each of these three components have different kinematical and metallicity distributions and it is possible to assign nearby stars to these three components on the basis of observations. Halo stars in the solar neighborhood are extremely rare (hence the difficulty in finding the population II main sequence from a trigonometric parallax sample) with roughly 1 out of every 500-800 stars belonging to this population. The normalization of the thick disk, however, is quite uncertain with values of 2-15% consistent with various samples. There is covariance between the determination of the normalization and zh with larger values of zh resulting in lower normalizations. This covariance is a consequence of the fitting procedure keeping the total numbers of stars in the thick disk to be relatively constant.
In a highly flattened rotating stellar system, the density distribution in the vertical (z) direction, is a measure of the surface mass density. This situation arises as Poisson's equation for a flattened system assumes the form
![]() | (9) |
As the density increases, then the z-coordinate sees a larger derivative
in the potential which means it experiences a larger gravitational
restoring force in that direction. In practice, this gravitational
restoring force can be estimated by measuring zh and
the vertical
velocity dispersion v
for some well defined sample of stars.
This transformation from equation 9 to observables makes use of a
variant of the collisionless Boltzmann equations.
Since stars are not escaping from this system, the collisionless
Boltzmann equation can be combined with the equation for continuity
of mass to yield (see Binney and Tremaine 1987 for details):
![]() | (10) |
Measurements of the density distribution of stars in the z
direction, (z), combined with
the vertical velocity dispersion then constrains
.
The first
attempt to utilize equation 10 to constrain the solar neighborhood
mass density was by Jan Oort in 1932 who derived a mass density of
0.15 M
pc-3. Of course, Oort had no knowledge of the
presence of another kinematic system, the thick disk, and so he explicitly
assumed that his sample of F stars and K giants was tracing out a
few thin disk scale heights. Under that assumption, Oort found that
the equivalent surface density of the thin disk in an imaginary cylinder
of height 700 pc was 90
M
pc-2.
A more recent determination of these parameters has been done in
a series of papers in the early 80's by John Bahcall and collaborators.
Again using samples of A and F stars, they derive estimates for
0 of 0.18 - 0.21
M
pc-3 and
an equivalent surface mass density out to 700 pc of 75
M
pc-2. The
measured luminosity function of stars in this imaginary cylinder gives an
equivalent surface luminosity density of 15 ± 2
L
pc-2.
This is equivalent to a mean blue surface brightness of
23.0
mag arcsec-2.
Bahcall's sample gives a mean M / L in the solar neighborhood of 5.
Testing for the presence of dark matter in the solar neighborhood now becomes an accounting problem The possible sources of this mass in the solar neighborhood are 1) luminous stars, 2) interstellar gas, 3) stellar remnants (mostly white dwarfs) and 4) dark matter:
Luminous stars: Determinations
of the local luminosity function of stars can yield a mass function
if the bulk of the stars are main-sequence stars where the relation between
luminosity and mass is well-determined. This is the case
in the solar neighborhood. The observed
value of
= 0.044
M
pc-3 is
well short of the required value (e.g., Gilmore et al. 1989).
Gas: Since stars form from
gas with an overall efficiency of
1-20% (depending upon many factors), then every star formation event
should leave behind plenty of gas. The
local volume mass density of gas has been measured at
= 0.042
M
pc-3; not surprisingly, almost identical to the stellar value. The
consideration of stars + gas accounts for one-half the inferred volume
mass density.
Stellar Remnants: Since
stellar remnants are no longer producing
energy they necessarily have high values of M / L and are hard
to detect. This makes an accurate
determination of their local space density quite difficult. Models of
star formation in the solar neighborhood, combined with estimates of
the initial mass function and mass loss rates in post main sequence
evolution can be used to predict the remnant density.
The detectability of white dwarfs is also
dependent upon their cooling rate and hence the inferred remnant density
is strongly dependent upon the assumed age of the galactic disk in the
solar neighborhood. None of these parameters is very well determined.
Currently values of
= 0.01 - 0.03 M
pc-3 are
consistent with the observations, though most determinations cluster around
the upper end. A reasonable upper limit to the density of stellar
remnants is
= 0.044
M
pc-3,
since the galaxy is not
yet old enough to have a mean remnant mass density higher than the observed
mass density of low mass stars.
Adding the observed densities of stars,gas and stellar remnants gives
0 in the
range 0.096 - 0.13
M
pc-3. This implies at least 0.05
M
pc-3 of
dark matter, or roughly 1/3 of the
total dynamically inferred volume mass density. This is taken as
evidence for significant amounts of dark matter in the solar neighborhood.
However, we have ignored the thick disk
contribution to the observed column of 75
M
pc-2.
In a long series of papers, Gerry Gilmore and
collaborators have taken this assumption to task and have persuasively
argued that omission of the thick disk has serious consequences on
the question of dark matter in the solar neighborhood. In their
most recent analysis, Kuijken and Gilmore (1991) derive a surface
mass density of 71 ± 6
M pc-2
(i.e., the same
as the Bahcall value) but conclude that only 48 ± 9
M
pc-2
is associated with thin disk material near the sun. The remaining
1/3 is associated with the thick disk and halo which has not been
accounted for in our previous census. It is difficult to find fault with this
analysis and hence the establishment of the thick disk has essentially
resolved the Oort limit problem thus removing any credible evidence
for the presence of dark matter in the solar neighborhood.