The kinematical properties of the LMC provide important clues to its structure. Observations have therefore been obtained for many tracers. The kinematics of gas in the LMC have been studied primarily using HI (e.g., Rohlfs et al. 1984; Luks & Rohlfs 1992; Kim et al. 1998). Discrete LMC tracers which have been studied kinematically include star clusters (Freeman, Illingworth & Oemler 1983; Schommer et al. 1992), planetary nebulae (Meatheringham et al. 1988), HII regions and supergiants (Feitzinger, Schmidt-Kaler & Isserstedt 1977), and carbon stars (Kunkel et al. 1997b; Graff et al. 2000; Alves & Nelson 2000; van der Marel et al. 2002). A common result from all these studies is that the line-of-sight velocity dispersion of the tracers is generally at least a factor ~ 2 smaller than their rotation velocity. This implies that the LMC is kinematically cold, and must therefore to lowest approximation be a disk system.
To understand the kinematics of an LMC tracer population it is necessary to have a general model for the line-of-sight velocity field that can be fit to the data. All studies thus far have been based on the assumption that the mean streaming (i.e., the rotation) in the disk plane can be approximated to be circular. However, even with this simplifying assumption it is not straightforward to model the kinematics of the LMC, because it is so near to us. Its main body spans more than 20° on the sky and one therefore cannot make the usual approximation that "the sky is flat" over the area of the galaxy. Spherical trigonometry must be used, which yields the general expression (van der Marel et al. 2002):
![]() |
(5.1) |
with
![]() |
(5.2) |
In this equation, vlos is the observed component of the
velocity along the line of sight. The quantities
(,
)
identify the position on the sky:
is the angular
distance from
the center and
is the
position angle with respect to the center
(measured from North over East). The kinematical center is at the
center of mass (CM) of the galaxy. The quantities
(vsys, vt,
t)
describe the velocity of the CM in an inertial
frame in which the sun is at rest:
vsys is the systemic velocity along
the line of sight, vt is the transverse velocity, and
t is
the position angle of the transverse velocity on the sky. The angles
(i,
)
describe the direction from which the plane of the galaxy
is viewed: i is the inclination angle (i = 0 for a face-on
disk), and
is the
position angle of the line of nodes, as
illustrated in Figure 6. The line-of-nodes is the
intersection of the galaxy plane and the sky plane. The velocity
V(R') is the rotation velocity at cylindrical radius
R' in the disk plane. D0 is the distance to the
CM, and f is a geometrical factor. The quantity s =
± 1 is the `spin sign' that determines in
which of the two possible directions the disk rotates.
![]() |
Figure 6. Schematic illustration of the
observer's view of the LMC disk. The plane of the disk is titled
diagonally out of the paper. The inclination i is the angle
between the (x, y) plane of the sky, and
the (x', y') plane of the galaxy disk. The x'-axis
is the line of nodes, defined as the intersection of the (x,
y) plane of the sky and the (x', y') plane of the
galaxy disk. The angle
|
The first term in equation (5.1) corresponds to the internal
rotation of the LMC. The second term is the part of the line-of-sight
velocity of the CM that is seen along the line of sight, and the third
term is the part of the transverse velocity of the CM that is seen
along the line of sight. For a galaxy that spans a small area on the
sky (very small
), the second
term is simply vsys and
the third term is zero. However, the LMC does not have a small angular
extent and the inclusion of the third term is particularly
important. It corresponds to a solid-body rotation component. Given
the LMC transverse velocity implied by equation (4.1), it
rises to an amplitude of 71 km s-1 at
= 10°, which
significantly exceeds the amplitude of the intrinsic rotation
contribution (first term of eq. [5.1]) at that radius. The
fourth term in equation (5.1) describes the line-of-sight
component due to changes in the inclination of the disk with time, as
are expected due to precession and nutation of the LMC disk plane as
it orbits the Milky Way
(Weinberg 2000).
This term also corresponds to a solid-body rotation component.
The general expression in equation (5.1) appears complicated,
but it is possible to gain some intuitive insight by considering some
special cases. Along the line of nodes one has that
sin( -
) = 0 and
cos(
-
) = ± 1, so that
![]() |
(5.3) |
Here it has been defined that
los
vlos
- vsys
cos
vlos - vsys. The quantity
vtc
vt
cos(
t -
) is the component
of the transverse velocity vector in the plane of the sky that lies
along the line of nodes; similarly,
vts
vt
sin(
t -
)
is the component perpendicular to the line of nodes. Perpendicular to
the line of nodes one has that
cos(
-
) = 0 and
sin(
-
) = ± 1, and
therefore
![]() |
(5.4) |
Here it has been defined that wts =
vts +
D0(di / dt). This
implies that perpendicular to the line of nodes
los
is linearly proportional to
sin
. By
contrast, along the line
of nodes this is true only if V(R') is a linear function of
R'. This is not expected to be the case, because galaxies do not
generally have solid-body rotation curves; disk galaxies tend to have
flat rotation curves, at least outside the very center. This implies
that, at least in principle, both the position angle
of the
line of nodes and the quantity wts are uniquely
determined by the observed velocity field:
is the angle along
which the observed
los are best
fit by a linear proportionality
with sin
, and
wts is the proportionality constant.