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A&A manuscript no.
(will be inserted by hand later)
Your thesaurus codes are:
08.05.1; 08.11.1; 10.11.1; 10.15.1; 10.19.1
ASTRONOMY
AND
ASTROPHYSICS
September 15, 1999
Vertical motion and expansion of the Gould Belt
F. Comer'on
European Southern Observatory, Karl­Schwarzschild­Str.2, D­85748 Garching bei M¨unchen, Germany
email: fcomeron@eso.org
Received; accepted
Abstract. The kinematics of the Gould Belt is con­
sidered taking into account its orientation in space and
the motions of its member stars parallel and perpendicu­
lar to the galactic plane. An analysis of Hipparcos data for
these stars, complemented with published radial velocit­
ies, shows that there is a mild gradient along the galactic
plane in the velocity component perpendicular to it. The
maintenance of the arrangement of Gould Belt stars form­
ing a plane, even for times that are at least a considerable
fraction of the vertical oscillation period of stars around
the galactic plane, is a rather strong constraint on any kin­
ematical models of the Gould Belt. It is shown that such
a constraint can be satisfied if the stars had initial velo­
cities linearly dependent on their positions in the plane
of the Belt. Adopting such linear patterns and the epicyc­
lic approximation to galactic orbits, analytical expressions
are derived that allow the calculation, for any age of the
Gould Belt, of the direction of its nodal line, its inclina­
tion, the values of the Oort constants A, B, C, and K,
the gradient of the velocity component perpendicular to
the galactic disk, and the direction of the axis of oscilla­
tion of the stars of the Belt perpendicular to the galactic
plane. The evolution of all these quantities is calculated for
several cases: a purely circular motion of the Gould Belt
stars around the galactic center; the radial expansion from
a small volume or over an extended area; the expansion
along a line; and an initial rotation to the Gould Belt stars
around an axis perpendicular to its plane. Pure expansion
models seem to be ruled out by observations, as none of
them, under any combination of initial parameters, is able
to simultaneously reproduce all the observed values of the
orientation, the Oort constants, and the characteristics
of the vertical motion. Nevertheless, a good agreement is
found between measured values of the quantities defining
the orientation and kinematics of the Gould Belt, and the
predictions of the rotation model. This model is the only
one among those considered here able to account for the
large observed offset between the nodal line of the Gould
Belt with respect to the galactic plane and the axis of ver­
tical oscillation of its stars. The best fit is achieved for an
Send offprint requests to: F. Comer'on
age of the Gould Belt of (3:4 \Sigma 0:3) \Theta 10 7 years, consist­
ent with individual ages determined for Gould Belt stars.
The implications that the rotation model has concerning
the possible origin of the Gould Belt is briefly discussed.
It is found that the disruption of a rotating, star form­
ing giant molecular cloud is unlikely to be at the origin
of the Gould Belt, due to the significant tilt with respect
to the direction perpendicular to the galactic plane that
it should have had.
Key words: Stars: early­type, kinematics; the Galaxy:
kinematics and dynamics, solar neighbourhood; open
clusters and associations: Gould Belt
1. Introduction
The most distinctive and intriguing feature of the Gould
Belt system of young stars and star forming regions is its
tilt of 20 ffi with respect to the galactic equator. Several
models for the origin of the Gould Belt exist in the lit­
erature, such as those based on the fragmentation of a
large molecular complex, on propagating star formation
triggered by supernovae, the compression of gas during
the passage of a spiral density wave, or the collision of
high velocity gas with the galactic disk (see P¨oppel 1997
for a comprehensive review). However, few of these mod­
els have explicitly addressed the question of the origin of
the tilt. Observations of the distribution of the stars in the
Gould Belt show that the tilt is preserved at least over the
distance from ae Ophiuchi to the Orion molecular complex,
the two star forming regions which delimit the extent of
the Belt in the galactocentric and antigalactocentric dir­
ections (roughly coinciding with its apsidal line), implying
a coherence over a lengthscale of about 700 pc. This co­
herence, and the very existence of the Gould Belt as a
single entity, have been eventually called into question by
authors who noted that the distribution of stars in the
Belt is apparently dominated by a few major structures
which might have independent origins, and whose arrange­
ment along a tilted disk may be casual (Franco et al. 1988,
L'epine & Duvert 1994; see also Guillout et al. 1998). In­

2 F. Comer'on: Vertical motion and expansion of the Gould Belt
deed, most of the early­type stars that make the Gould
Belt outstanding belong to the OB associations of Or­
ion, Perseus OB2, and Scorpius­Centaurus­Lupus (Blaauw
1991, de Zeeuw et al. 1999). However, the existence of a
distributed population of young stars not associated with
any of the massive star forming complexes is revealed for
instance by the sky distribution of young, chromospheric­
ally active low mass stars detected in the ROSAT all­sky
survey (Neuh¨auser 1997, Guillout et al. 1998). Moreover,
the precise three­dimensional picture of the stellar distri­
bution in the solar neighbourhood provided by the Hip­
parcos satellite clearly shows the existence of a distributed
population of B stars that depict the Gould Belt as a disk
with its members spread well outside the boundaries of the
known OB associations. As shown by Torra et al. 1999, the
kinematical peculiarities distinctive of the Gould Belt are
preserved even when the stars belonging to the domin­
ant associations are excluded from analyisis. On the other
hand, the overall age of the Gould Belt, although very
uncertain (published estimates range between 20 and 90
million years; see Torra et al. 1999 for a review of determ­
inations found in the literature), is in any case at least a
considerable fraction of the vertical oscillation period of
stars around the galactic plane under the effects of the
galactic gravitational potential, implying that the coher­
ent structure of the Belt applies not only to the position
of its components, but to their motions as well.
In this paper I intend to make a schematic exploration
on the implications that the maintenance of the coherence
with time of the structure of the Gould Belt has when com­
bined to its kinematics, stressing the importance of stellar
motions perpendicular to the galactic plane when inter­
preting measured velocity gradients. I will first present
evidence for a systematic pattern of the vertical compon­
ents of stellar motions, revealed by the Hipparcos data. Us­
ing the epicyclic approximation to the orbits of the stars
in the galactic potential, I will then discuss the conditions
under which an ensemble of stars, having an initial pat­
tern of peculiar motions and being distributed on a tilted
plane, can still define a plane as their trajectories evolve.
The time evolution of the tilted plane with respect to the
galactic equator, in particular its inclination, the position
of its nodal line, and the different velocity gradients ex­
pected from its member stars, will be studied under the
assumption of different initial patterns of motion. Finally,
a global interpretation of the Gould Belt kinematics based
on the actually observed orientation and velocity patterns
will be discussed.
2. Observed vertical motions in the Gould Belt
The kinematic structure of the Gould Belt has been ex­
tensively studied for nearly one century now (see Frogel &
Stothers 1977 for an exhaustive review, and Torra et al.
1999 for references to more recent, mostly pre­Hipparcos
work). Most of the studies have explored the kinematical
peculiarities of this structure on the basis of the velocity
components of the stars in the directions of the galactic
plane, as well as their gradients in those same directions.
These motions reveal a rather complex expanding pattern
associated to the Gould Belt. Torra et al. 1999 have car­
ried out a comprehensive study of the kinematics of the
Gould Belt using Hipparcos data, complemented with ra­
dial velocities and Str¨omgren photometry which enabled
them to obtain space velocities and ages for a large sample
of O and B stars. Similar analyses have been published by
Lindblad et al. 1997 and PalouŸs 1998. These studies are
also restricted to motions in the directions of the galactic
plane only, and I will refer to their results for the forth­
coming discussions on this aspect.
Studies on the velocity component perpendicular to
the galactic plane have been much less abundant, prob­
ably due to the subtler systematic patterns that may be
expected and to the little use of radial velocities in this
case, as nearly all the stars lie at low galactic latitudes.
Fortunately, the proper motions of unprecedented qual­
ity provided by Hipparcos make it possible now to carry
out meaningful studies based on proper motions of stars
with precisely known distances, out to a distance from the
Sun comparable to the whole extent of the Gould Belt. To
complement the results of the works referred to in the pre­
vious paragraph, I have prepared a sample of Hipparcos
stars fulfilling the following requirements:
-- Spectral types not later than B2.5, to ensure that only
very young stars with the age of the Gould Belt or less
are included.
-- Relative error in the trigonometric parallax below 30%.
The second constraint implies in practice that nearly
all the stars are found within a distance of 400 pc from the
Sun, which matches fairly well the size of the Gould Belt.
At that distance, the average standard error in the Hip­
parcos proper motion for the stars in the sample, 0 00 0009
yr \Gamma1 , results in a velocity error of only 1.7 km s \Gamma1 , and is
below 1 km s \Gamma1 for most of the stars in the sample. There­
fore, no constrain has been set on the proper motions at
the time of selecting the stars. To obtain space velocities,
the sample has been complemented with radial velocit­
ies taken from the catalogues of Barbier 1989, Andersen
& Nordstr¨om 1983a, 1983b, 1985, Duflot et al. 1995, and
Fehrenbach et al. 1997. It should be noted however that,
with most of the stars lying at galactic latitudes jbj ! 25 ffi ,
the radial velocity adds only a minor contribution to the
velocity component perpendicular to the galactic plane
discussed here. For this reason, the 20 stars without pub­
lished radial velocities in the sample of 323 stars selected
from the Hipparcos catalog have been retained, arbitrarily
assigning them a zero radial velocity.
As shown by Torra et al. 1999, most of the stars with
ages below 3 \Theta 10 7 years in the solar neighbourhood (which
approximately includes the sample used here) belong to
the Gould Belt on the basis of their spatial distribution.

F. Comer'on: Vertical motion and expansion of the Gould Belt 3
Therefore, kinematical contamination by stars not belong­
ing to that structure is not expected to be significant in
the present sample.
Let us assume that the perpendicular component of the
velocity, W , has systematic variations along the galactic
plane, expressed as @W=@x, @W =@y, where x is directed
towards the galactic center and y towards the direction of
galactic rotation. Such derivatives of W allow the defini­
tion of an axis of vertical oscillation of the stars around
the galactic plane, G, which has the property that W is
proportional to the distance of the star to it:
W = jG \Theta Rj z (1)
where the z subindex indicates the component along the
direction perpendicular to the galactic plane, and R is
the vector perpendicular to the axis of vertical oscillation
joining it with the star. Eq. (1) can be written as
W = G x y \Gamma G y x + jGjR 0 (2)
with x, y now being the components of the heliocentric
position vector of the star in the directions defined above.
R 0 is the distance of the nearest point of the axis of vertical
oscillation to the Sun, which can be positive or negative
depending on whether the x coordinate of that point is
positive or negative. The sense of G is defined so that, if
G y ? 0, then the stars located in the hemisphere, limited
by the plane defined by G and the z direction, that faces
the galactic center are moving towards decreasing z. The
components of G are thus
G x = @W
@y
; G y = \Gamma @W
@x
(3)
An expression like Eq. (2) can be written for each star
in the catalog, setting up a system with the unknowns
G x , G y , and jGjR 0 that can be solved by least squares.
Before doing that, however, it is necessary to consider the
contribution to W due to the peculiar motion of the Sun,
which otherwise would be engulfed in the last term on the
right hand side of Eq. (2). Following Torra et al. 1999, I
use W fi = 8:0 km s \Gamma1 as the value to add to the observed
W before using it as the input for the left hand side of Eq.
(2).
The solution of Eq. (2) for the sample of stars used
here reveals indeed a significant systematic pattern in the
vertical motions of the stars under study, yielding the fol­
lowing values:
jGj = 6:5 \Sigma 1:8 km s \Gamma1 kpc \Gamma1
ff = 337 ffi \Sigma 20 ffi
R 0 = 210 \Sigma 105 pc
where ff is defined by
Fig. 1. The vertical component of the velocity, W (uncorrected
for reflected solar motion), as a function of the distance to the
axis of vertical oscillation. The cosmic dispersion of W consid­
erably exceeds the systematic difference of velocities over the
distance interval considered here. For this reason, the velocit­
ies averaged over 100 pc bins are plotted here, rather than the
individual velocities. The error bars represent the error on this
average.
tan ff = G y
G x
The result obtained for R 0 is very sensitive to the ad­
opted value of W fi , as one obtains R 0 = 45 \Sigma 99 pc when
decreasing W fi by only 1 km s \Gamma1 . The values obtained
for jGj and ff are nevertheless practically independent of
this choice. The gradient in W can be appreciated in Fig­
ure 1, where its average over 100 pc­wide bins is plotted
as a function of the distance to the axis of vertical oscil­
lation. To test the real existence of such a gradient and
its orientation, the system of Eqs. (2) has been solved re­
peatedly by adding each time a random component com­
pressed between ­10 km s \Gamma1 and 10 km s \Gamma1 to the W
component of each stellar velocity, and by removing from
the sample stars with W ? 10 km s \Gamma1 or W ! \Gamma30 km
s \Gamma1 , so as to ensure that the results obtained are not a
consequence of a spurious effect or of stars with a highly
deviant behaviour. Results within or near the above in­
tervals have been consistently obtained every time for jGj
and ff. It should be noted that at b = 0 ffi (which is roughly
the case for most of the stars in our sample) Eq. (2) can
also be written as ¯ b = k(G x sin l \Gamma G y cos l) + jGjR 0 ú 00 ,
where ¯ b is the proper motion in galactic latitude, k is
a conversion factor, and ú 00 is the trigonometric parallax.
Since ¯ b can be determined directly from Hipparcos proper
motions, the determination of G x and G y is practically in­
sensitive to systematic effects introduced by random errors
in the parallaxes of the stars in the sample.
The meaning of the results found here will be better in­
terpreted in the framework of the models for the internal
kinematics of the Belt, and their discussion is therefore

4 F. Comer'on: Vertical motion and expansion of the Gould Belt
deferred to Section 4. For the time being, it should be
noted that the orientation of the axis of vertical oscilla­
tion is markedly different from the direction of the nodal
line where the Gould Belt intersects the galactic plane,
which runs approximately along the 105 ffi \Gamma 285 ffi direction
(Comer'on et al. 1994, Torra et al. 1999). As will be seen,
the offset between both directions has important implica­
tions on the internal dynamics of the Gould Belt. On the
other hand, it will be shown as well that the rather low
value of jGj implies that the Gould Belt plane is near its
maximum tilt at present. This feature was also noted by
Frogel & Stothers 1977, on the basis of the absence of any
significant gradient of the W component in the direction
perpendicular to the galactic plane.
3. Kinematics
In the epicyclic approximation used to describe the orbits
of stars moving in a separable potential, the equations of
motion are usually written in a rotating cartesian reference
frame whose axes are respectively directed towards the
galactic center, the direction of galactic rotation, and the
North galactic pole. The equations of motion can then be
written as
x(t) = X 1 + C cos(Ÿt + OE) (4a)
y(t) = Y 1 + 2A c X 1 t + 2!C
Ÿ sin(Ÿt + OE) (4b)
z(t) = D cos(št + /) (4c)
where I use a notation similar to that of Comer'on et al.
1997 1 , adding equation (4c) to describe the motion per­
pendicular to the galactic plane. The angular velocity of
an object in a circular orbit around the galactic center at
the position momentarily occupied by the Sun is ! which,
by definition, is also the angular velocity of the chosen
reference frame with respect to an inertial one. A c is the
usual Oort constant for the case of pure galactic differen­
tial rotation, Ÿ is the epicyclic frequency in the galactic
plane, and š is the oscillation frequency perpendicular to
the galactic plane for orbits whose amplitude is well below
the scale height of gravitating matter in the galactic disk.
X 1 and Y 1 + 2A c X 1 t describe the position of the guiding
center of the epicyclic orbit, C is the amplitude of the epi­
cycle in the galactocentric direction, D is the amplitude
of the vertical oscillation, and OE and / define the position
of a star in its orbit at an instant t.
1 I use here ! to denote the angular velocity of circular
galactic rotation, motivated by the later use
of\Omega to refer to
the longitude of the ascending node in the present paper. The
subindex ''c'' has been added to A to indicate the value of
the latter in the case of pure galactic differential rotation. The
amplitudes C, D are written in that way to distinguish C from
one of the Oort constants appearing later.
Developing the arguments of the trigonometric func­
tions in Eqs. (4), and using the values of the coordinates
and velocities at the initial instant t = 0, it is easy to
show that Eqs. (4) can be written in the following com­
pact form:
x = S x 0 + T —
x 0 (5)
where x = (x(t); y(t); z(t)), x 0 = (x(0); y(0); z(0)), —
x 0 =
( —
x(0); —
y(0); —
z(0)). The elements of the matrices S, T are:
S =
0
B
B
B
@
! \Gamma A c cos Ÿt
! \Gamma A c
0 0
2A c !
! \Gamma A c
(t \Gamma sin Ÿt
Ÿ
) 1 0
0 0 cos št
1
C
C
C
A
T =
0
B
B
B
B
@
sin Ÿt
Ÿ
cos Ÿt \Gamma 1
2(! \Gamma A c ) 0
2!
Ÿ 2
(1 \Gamma cos Ÿt) 1
! \Gamma A c
( ! sin Ÿt
Ÿ \Gamma A c t) 0
0 0 sin št
š
1
C
C
C
C
A
If the stars had a common origin so that the present
age of the system is t, then the initial positions are related
to the present ones and to the initial velocity pattern by
x 0 = S \Gamma1 (x \Gamma T —
x 0 ) (6)
On the other hand, if the initial positions of stars were
distributed on a plane tilted with respect to the galactic
plane, then their initial positions fulfilled the relationship
P 0 (x 0 \Gamma x r0 ) = 0 (7)
where P 0 is the vector perpendicular to the plane and x r0
defines the location of the plane with respect to the origin
of coordinates. Replacing Eq. (6) in Eq. (7),
P 0
\Theta
S \Gamma1 (x \Gamma T —
x 0 ) \Gamma x r0
\Lambda
= 0 (8)
It is then possible to show that, if the pattern of initial
velocities can be expressed as a linear function of the ini­
tial coordinates of the stars, then the tilted plane remains
as a tilted plane as the positions of its stars evolve with
time under the influence of the galactic potential. Let the
linear combination be expressed in a general form as
—
x 0 = U(x 0 \Gamma x c ) + —
x c (9)
Using Eq. (6), one obtains after some algebra:
—
x 0 = (US \Gamma1 T + I) \Gamma1 \Theta
U(S \Gamma1 x \Gamma x c ) + —
x c
\Lambda
(10)
where I is the identity matrix. Replacing this in Eq. (8),
P 0
\Theta
S \Gamma1 x\GammaS \Gamma1 T(US \Gamma1 T+I) \Gamma1 [U(S \Gamma1 x\Gammax c )+ —
x c ]\Gammax r0
\Lambda
= 0
(11)

F. Comer'on: Vertical motion and expansion of the Gould Belt 5
This can be expressed simply as
P(x \Gamma x r ) = 0 (12)
which is again the equation of a plane, whose perpendic­
ular vector is now
P = P 0
\Theta
S \Gamma1 \Gamma S \Gamma1 T(US \Gamma1 T + I) \Gamma1 US \Gamma1 \Lambda
(13)
and the present location of the plane is defined by
Px r = P 0
\Theta
S \Gamma1 T(US \Gamma1 T + I) \Gamma1 (Ux c \Gamma —
x c ) \Gamma x r0
\Lambda
(14)
Given the geometry of stellar initial positions and ve­
locities, it may be more convenient to write them in a
reference frame whose axes are aligned parallel or perpen­
dicular to the plane, and whose origin is chosen so as to
simplify the expression of the initial law of motion. In such
a reference frame, in which the position vector is denoted
by ¸, the initial law of motion can be written as
—
¸ 0 = M¸ 0 (15)
where the relation between x 0 and ¸ 0 is
x 0
=\Omega 0 i 0 ¸ 0 + x c (16)
The rotation of axes is explicitly decomposed as the
product of two rotations whose matrices are
\Omega 0
=
0
@
cos\Omega 0 \Gamma
sin\Omega 0
0
sin\Omega 0
cos\Omega 0
0
0 0 1
1
A
i 0 =
0
@
1 0 0
0 cos i 0 \Gamma sin i 0
0 sin i 0 cos i 0
1
A
The initial inclination of the plane with respect to the
galactic plane is i 0 , and the longitude of the nodal line
defined by the intersection of both planes
is\Omega 0 . An ex­
pression analogous to Eq. (16) can be written for the ve­
locity:
—
x 0
=\Omega 0 i 0
—
¸ 0 + —
x c (17)
The initial law of motion expressed in the usual epi­
cyclic motion base is thus
—
x 0
=\Omega 0 i 0 Mi \Gamma1
0\Omega
\Gamma1
0
(x 0 \Gamma x c ) + —
x c (18)
which gives the expression of U
U
=\Omega 0 i 0 Mi \Gamma1
0\Omega
\Gamma1
0
(19)
The time evolution of the nodal line and the inclination
are thus given by that of the vector P, whose components
are
P /
(sin\Omega sin i; \Gamma
cos\Omega sin i; cos i) (20)
This derivation allows a rather straightforward connec­
tion between the matrix U and the local values of the Oort
constants, which can be measured from the observations.
Developing Eq. (5) with the use of Eq. (9), one obtains:
x = (S + TU)x 0 \Gamma T(Ux c \Gamma —
x c ) (21)
Taking the time derivative of Eq. (21),
—
x = ( —
S + —
TU)x 0 \Gamma —
T(Ux c \Gamma —
x c ) (22)
Using Eq. (21) to isolate x 0 , and replacing it in Eq.
(22), one obtains
—
x = ( —
S+ —
TU)(S+TU) \Gamma1 \Theta
x+T(Ux c \Gamma —
x c )
\Lambda
\Gamma —
T(Ux c \Gamma —
x c )
(23)
Taking spatial derivatives now, one obtains
Q = ( —
S + —
TU)(S + TU) \Gamma1 (24)
where the elements of Q, Q ij , are
Q ij = @ —
x i
@x j
(25)
It should be noted that Eq. (24) is valid for any system
of stars for which the epicyclic approximation applies, and
whose initial positions and velocities are related by Eq.
(9). The additional condition (7), implying that the stars
are distributed in a plane, has not been used in deriving
Eq. (24). This condition must be implicitly included in the
expression of the spatial derivatives that will be actually
used here in the calculation of the Oort constants, namely:
\Gamma @ —
x i
@x j
\Delta
p = @ —
x i
@x j
\Gamma @ —
x i
@z
P j
P 3
(26)
where the p subindex denotes the derivatives measured on
the plane, and x j is either x or y. Using these definitions,
the Oort constants become
A = 1
2 (Q 12 + Q 21 \Gamma Q 13
P 2
P 3
\Gamma Q 23
P 1
P 3
) (27a)
B = 1
2 (Q 21 \Gamma Q 12 \Gamma Q 23
P 1
P 3
+ Q 13
P 2
P 3
) \Gamma ! (27b)
C = 1
2 (Q 11 \Gamma Q 22 \Gamma Q 13
P 1
P 3
+Q 23
P 2
P 3
) (27c)
K = 1
2 (Q 11 +Q 22 \Gamma Q 13
P 1
P 3
\Gamma Q 23
P 2
P 3
) (27d)
to which one can add the derivatives involving the velocity
component perpendicular to the galactic plane, giving the
components of the axis of vertical oscillation (see Eq. (3)):
G x = Q 32 \Gamma Q 33
P 2
P 3
(28a)
G y = \GammaQ 31 + Q 33
P 1
P 3
(28b)

6 F. Comer'on: Vertical motion and expansion of the Gould Belt
4. Initial patterns of motion
Undoubtedly, an important reason why no clear interpret­
ation on the kinematical structure of the Gould Belt has
emerged yet is that, whatever the initial pattern of velocit­
ies may have been at the time of its formation, it must have
been severely distorted under the effects of galactic differ­
ential rotation during its lifetime. In principle, it should
be possible to use the presently observed space velocities
of the stars and their ages, together with a model of the
galactic potential, to trace their orbits back in time and
find both their initial positions and velocities. In practice,
this would require a knowledge of velocities, distances, and
ages much more accurate than is available at present in
order to reach any reliable conclusions.
A different way to approach this problem consists of
proposing different initial kinematical patterns and space
distributions for the stars of the Gould Belt, and following
their evolution with time until the best agreement with the
present observations is reached. In case that a satisfactory
solution cannot be found for any possible age of the Belt,
the model is then discarded. This is the approach followed
by Lindblad 1980, and extended by Westin 1985, to eval­
uate the suitability of models based on a radial expansion
from a small volume or on the gravitational perturbation
produced by a spiral arm to reproduce the observed val­
ues of the Oort constants. They concluded that none of
those models provided an acceptable fit to the available
observational material. Comer'on et al. 1994 suggested an
expansion from a line, rather than a point, as a possible
way to achieve a better fit to the Oort constants derived
from their data. Recently, PalouŸs 1998 has presented the
result of N­body numerical simulations suggesting that the
dissolution of an unbound rotating system of stars with an
age of 3 \Theta 10 7 years is the most favoured model in explain­
ing the observed values of the Oort constants. Such an age
is consistent with that derived from the individual ages of
its component stars.
The development presented in Section 3 is well suited
to the exploration of different kinematical models for sev­
eral reasons. First, its predictions include the orientation
of the Belt and the characteristics of the vertical motion,
in addition to the Oort constants, as a function of time.
Secondly, the development being fully analytical, it is easy
to explore a wide range of parameters when trying to ob­
tain a best fit, and to ensure that some models can be
really discarded for any set of initial parameters. A pos­
sible drawback is the basic assumption that the initial
patterns of motion have the generic form given by Eq.
(9), what in principle is a very restrictive condition. How­
ever, the fact that such patterns are able to maintain the
arrangement of stars on a tilted plane with time, and that
this is indeed an observed feature of the Gould Belt, sug­
gests that it should be possible (at least as a first approx­
imation) to describe the initial pattern of motions of the
Gould Belt in such a form.
The time evolution of the orientation of the Belt and
of the Oort constants are described in the next Subsec­
tions for different possible initial patterns of motion. The
main aspect of interest here is in the early phases of this
evolution, this is, in ages of a few times 10 7 years which
are consistent with the observations. For illustrative pur­
poses, I extend the calculations to 3 \Theta 10 8 years, which
covers the first resonances with the epicyclic and galactic
rotation periods and their consequences on the quantities
whose evolution is studied here. Such an extension may
be in principle relevant to the study of other, older Gould
Belt­like structures which may be eventually discovered.
However, the study of such structures would be hampered
with increasing age as their more massive and brightest
members end up their lives, and the less massive ones di­
lute in phase space due to differential rotation and to the
dynamical heating mechanisms operating in the galactic
disk.
The evolution of the plane orientation and the Oort
constants are in general sensitive to the chosen initial
parameters. For this reason, I restrict the discussion to the
values that reproduce the present orientation of the Gould
Belt(\Omega = 285 ffi , i = 20 ffi ; see Section 2) and the small value
of G = jGj, which places the Gould Belt near its peak in­
clination at present. In general, it is always possible to find
a set of initial parameters fulfilling these constraints for
an age of the Belt compatible with the observational es­
timates. The consistency with these constraints thus sets
rather narrow limits on the possible age of the Belt, which
depend on the model adopted. As a convention, the inclin­
ation is defined here as a positive quantity,
and\Omega as the
galactic longitude of the ascending node of the Gould Belt
with respect to the galactic plane, so that the crossing of
the galactic plane by the Gould Belt results in a change
of 180 ffi in \Omega\Gamma rather than a reversal in the sign of i. In
this
way,\Omega = 90 ffi means that the Gould Belt reaches its
largest distance to the galactic plane in the direction to
the galactic anticenter,
while\Omega = 270 ffi implies that it
does so in the direction to the galactic center,
and\Omega = 0 ffi
places the direction of maximum inclination in the direc­
tion of the galactic rotation. For the sake of simplicity,
and in consistency with the observations, I will consider
x c = 0, —
x c = 0. As can be seen from Eq. (24), the Oort
constants are not affected by this choice.
The values adopted for the galactic constants A c and Ÿ
appearing in the matrices S and T are those corresponding
to a flat rotation curve at the position of the Sun with a
circular angular speed of rotation ! = 25:9 km s \Gamma1 kpc \Gamma1
(Kerr & Lynden­Bell 1986), namely A c = 1
2
! = 12:9 km
s \Gamma1 kpc \Gamma1 and Ÿ = 36:0 km s \Gamma1 kpc \Gamma1 . The vertical os­
cillation frequency is taken to be š = 99 km s \Gamma1 kpc \Gamma1 ,
corresponding to a vertical oscillation period of 6:2 \Theta 10 7
years (Binney & Tremaine 1987).

F. Comer'on: Vertical motion and expansion of the Gould Belt 7
Fig. 2. The evolution of the nodal
line\Omega\Gamma the axis of vertical
oscillation ff, the inclination i, the Oort constants A, B, C, K,
and the instantaneous angular velocity of oscillation G in the
case in which the stars of the Gould Belt follow circular orbits
around the galactic center while oscillating around the galactic
plane. In the upper panel, the solid curve represents the dir­
ection of the nodal line, and the dashed curve the direction of
the axis of vertical oscillation. The Oort constants are identi­
fied by the solid line (A), the dotted line (B), the dashed line
(C), and the dot­dashed line (K). Note that C = K = 0 in the
present case, so both curves overlap. The stars in the plane are
assumed to form simultaneously with a zero vertical motion;
the present orientation and peak inclination are obtained with
the initial
values\Omega 0 = 102 ffi 2, i 0 = 24 ffi 2, at an age of 3:4 \Theta 10 7
years.
4.1. Circular motions
The simplest case that can be considered, and that will be
used here as a reference, is that of a plane whose stars fol­
low circular orbits around the galactic center when their
positions are projected on the galactic plane. In the epicyc­
lic reference frame, the equations of motion are described
by Eqs. (4) with C = 0, and one obtains for U:
U =
0
@
0 0 0
2A c 0 0
\Gammak
sin\Omega 0 k
cos\Omega 0 0
1
A (29)
Fig. 3. Same as Figure 2, but now assuming that the stars are
formed in the galactic plane with a nonzero vertical motion.
The value of k in Eq. (28) is set to 43.2 km s \Gamma1 kpc \Gamma1 to
obtain a maximum inclination of 20 ffi at 5:2 \Theta 10 7 years. The
initial orientation in this case
is\Omega 0 = 101 ffi 1, i 0 = 0 ffi .
where the possibility of a nonzero vertical initial velocity is
taken into account by the terms in the last row of Eq. (29).
This includes also as a particular case one in which the
star­forming matter of the Gould Belt is suddenly knocked
away from the galactic plane towards opposite directions
(i 0 = 0, k 6= 0).
Figure 2 shows the time evolution of the position of the
nodal
line\Omega\Gamma the direction of the axis of vertical oscillation
ff, the inclination i, the value of the Oort constants A, B,
C, K, and the instantaneous angular velocity of oscillation
G. The values
of\Omega and ff are their galactic longitudes at
the corresponding time, i.e., they are expressed in the ro­
tating reference frame. The evolution of the orientation
can be described fairly simply: starting from the chosen
initial position, the direction of the nodal line rotates in
the sense of increasing galactic longitudes, assymptotic­
ally tending to align itself with the direction of galactic
rotation, as a consequence of the ''stretching'' of the plane
produced by galactic rotation. A point of the nodal line
at the initial position (x 0 ; y 0 ) will have moved, following
the galactic circular rotation, to (x 0 ; y 0 + 2A c x 0 t) after a

8 F. Comer'on: Vertical motion and expansion of the Gould Belt
time t, and the longitude of the nodal line will thus be
given (neglecting the possible 180 ffi difference due to the
actual inclination) by
tan\Omega = y=x = (y 0 + 2A c x 0 t)=x 0 ,
which increases indefinitely with time. The axis of ver­
tical oscillation is coincident with the nodal line, but its
180 ffi reversals take place at the times when the vertical
motions of stars change signs, rather than at the galactic
plane crossings as is the case for the nodal line.
The evolution of the inclination is also easy to under­
stand qualitatively, being dominated by the vertical oscil­
lation of the stars around the galactic plane. The initial
decrease of the amplitude of the inclination is due to the
stretching of the plane as the nodal line approaches the
galactocentric direction. Once the nodal line crosses it, the
plane continues to stretch, but a projection effect makes
the inclination amplitude grow again, and at sufficiently
large times (several times larger than the timespan shown
in Figure 2) it actually tends to 90 ffi . The projection effect
may be visualized as follows: let us assume,
when\Omega = 0 ffi , a
star lying at the coordinates (x; y; z), so that the distance
to the nodal line is y and the inclination of the plane is
simply tan i = z=y. As the nodal line rotates (i.e.,
as\Omega approaches the 90 ffi ! 270 ffi direction), the distance d of
the star to the nodal line decreases as d = y
cos\Omega\Gamma so that
the inclination amplitude grows as tan i max = z max =d,
with the vertical amplitude of the oscillation, z max , being
constant.
Since the motions of the stars as projected on the
galactic plane are circular around the galactic center in the
present case, the Oort constants have the values A = A c ,
B = A c \Gamma !, C = K = 0 characteristic of such motion.
The evolution of jGj is similar to that of jij, but anticor­
related with it, as expected from the stars reaching their
peak vertical velocity when they cross the galactic plane.
The projection effect discussed in the previous paragraph
applies to the gradients of the vertical velocity as well,
what causes the amplitude of the oscillation in G to vary
in a similar way to that of i.
The results remain essentially unchanged if the stars
are assumed to be born in the galactic plane and expelled
from there in a coherent way, with vertical velocities pro­
portional to the distance to the nodal line. This is shown in
Figure 3, which displays the same behaviour as Figure 2,
the differences being due to the somewhat different initial
parameters that are necessary to match the constraints set
by the presently observed orientation and state of vertical
motion of the Belt.
4.2. Radial expansion
Let us assume now that the stars in the Gould Belt were
born simultaneously in a volume much smaller than the
one they occupy at present, with their initial motions con­
tained in a plane and directed radially away from the cen­
ter of such volume. This is one of the most widely con­
sidered kinematical models for the Gould Belt, motivated
Fig. 4. The evolution of the nodal
line\Omega\Gamma the axis of vertical
oscillation ff, the inclination i, the Oort constants A, B, C,
K, and the instantaneous angular velocity of oscillation G in
the case in which the stars of the Gould Belt are born simul­
taneously and expand initially from a negligibly small volume
with random velocities (Ü0 ! 0 in Eq. (30)). In the upper panel,
the solid curve represents the direction of the nodal line, and
the dashed curve the direction of the axis of vertical oscilla­
tion. The Oort constants are identified by the solid line (A),
the dotted line (B), the dashed line (C), and the dot­dashed
line (K). The present orientation and peak inclination are ob­
tained with the initial
values\Omega 0 = 40 ffi 7, i 0 = 63 ffi 5, at an age
of 4:9 \Theta 10 7 years.
by the long­recognized expansion term in the velocities of
nearby young stars. The early work of Blaauw 1952 de­
scribing the expansion of an unbound group of stars mov­
ing in the galactic potential was extended by Lesh 1968,
Lindblad 1980, and Westin 1985. Lindblad et al. 1973 and
Olano 1982 developed the models to account for the ob­
served radial velocities of the gas associated to the Belt.
Let us assume an initial velocity modulus of the stars
proportional to the distance to the center. At very early
times, it is possible to define an expansion age Ü 0 such
that the pattern of motions can be described, using the ¸ 0
system defined above, as follows:

F. Comer'on: Vertical motion and expansion of the Gould Belt 9
M = 1
Ü 0
0
@
1 0 0
0 1 0
0 0 0
1
A (30)
If the initial volume is negligible, then Ü 0 ! 0, and the
pattern described by Eq. (30) is equivalent to that of a
system of stars expelled from a single point with random
velocities. The evolution of the orientation of the plane
is then as given in Figure 4. The main characteristic is
the fast decrease of the amplitude of the tilt with time
at early ages, as a consequence of the expansion of its
stars away from the center while maintaining constant the
amplitude of their vertical motions. An important feature
of this model is the need for a large initial tilt, i = 63 ffi 5,
to still obtain a maximum tilt of 20 ffi at the age of 4:9 \Theta 10 7
years, after the first galactic plane crossing.
The initial expansion along the plane from a very small
volume causes a large initial gradient in the vertical com­
ponent of the velocity resulting in a large value of G, which
rapidly decreases as the distance of the stars to the center
of the expansion grows. Afterwards, the evolution of G and
i are approximately anticorrelated like in the case of circu­
lar motions, although the expansion of the system causes
a small lag between the peak in i and the minimum in
G: since the distance to the center of expansion increases,
the inclination can start decreasing while the stars are still
moving away from the galactic plane. A common feature
to the expansion models, including those to be considered
in the next Section, is the predicted permanent alignment
between the nodal line and the vertical oscillation axis.
The evolution of the Oort constants plotted in Figure
4 reproduces the results of Lindblad 1980, giving low val­
ues at early times for all the Oort constants (including a
permanent null value of B) with the exception of K. Sin­
gularities in some Oort constants and in the orientation of
the plane appear around the times of the epicyclic period
and the galactic rotation period, due to the alignment of
the stars of the plane along a single line, what gives in­
finite values for some spatial derivatives of some velocity
components. This is a feature common to the evolution of
all the models in which the stellar orbits projected on the
galactic plane are not circular.
A similar, but less marked behaviour, is found when
Ü 0 is finite. This may be regarded as an approximation
to the case in which the stars formed out of an extended
molecular cloud become an unbound system shortly after
their formation, as a consequence of the dispersal of the
gas remaining in the system. Assuming that the stars loc­
ated at the outskirts of the cloud expand initially with a
velocity of 1 km s \Gamma1 , and that the star forming cloud had
an initial radius of 50 pc, one obtains Ü 0 ' 5 \Theta 10 7 years.
The evolution is given in Figure 5, and is nearly identical
to that depicted in Figure 4 except for the existence of a
negative value of B. The initial nonzero extent of the Belt
allows to start with an inclination of its plane somewhat
smaller than in the case Ü 0 ! 0 considered before, but still
Fig. 5. The evolution of the nodal
line\Omega\Gamma the axis of vertical
oscillation ff, the inclination i, the Oort constants A, B, C, K,
and the instantaneous angular velocity of oscillation G in the
case in which the stars of the Gould Belt are born simultan­
eously over a circular region, expanding with initial velocities
proportional to the distance to the center. The expansion age,
which relates the size of the initial volume and the expansion
rate, is taken to be 5 \Theta 10 7 years, as explained in the text.
In the upper panel, the solid curve represents the direction of
the nodal line, and the dashed curve the direction of the axis
of vertical oscillation. The Oort constants are identified by the
solid line (A), the dotted line (B), the dashed line (C), and the
dot­dashed line (K). The present orientation and peak inclina­
tion are obtained with the initial
values\Omega 0 = 63 ffi 0, i 0 = 58 ffi 4,
at an age of 4:1 \Theta 10 7 years.
very considerable despite of the small expansion velocity
at the outskirts of the cloud.
4.3. Expansion from a line
In this Secton I consider the case in which the initial ex­
pansion takes place along a preferential direction, rather
than being isotropic as it has been assumed so far. Such an
expansion pattern was proposed by Comer'on et al. 1994
on the basis of the distribution of residual velocities of
Gould Belt stars when a purely circular rotation is sub­
tracted from the observed velocities. It was suggested in

10 F. Comer'on: Vertical motion and expansion of the Gould Belt
Fig. 6. The evolution of the nodal
line\Omega\Gamma the axis of vertical
oscillation ff, the inclination i, the Oort constants A, B, C,
K, and the instantaneous angular velocity of oscillation G in
the case in which the stars of the Gould Belt are born sim­
ultaneously on a very narrow strip, and expand initially with
random velocities directed prependicular to it. The strip is as­
sumed to coincide with the apsidal line, so that the initial ex­
pansion takes place parallel to the direction of the nodal line.
In the upper panel, the solid curve represents the direction of
the nodal line, and the dashed curve the direction of the axis
of vertical oscillation. The Oort constants are identified by the
solid line (A), the dotted line (B), the dashed line (C), and the
dot­dashed line (K). The present orientation and peak inclina­
tion are obtained with the initial
values\Omega 0 = 58 ffi 3, i 0 = 25 ffi 9,
at an age of 3:3 \Theta 10 7 years.
that paper that such a pattern may have arisen from the
sudden compression of a gas layer precursor to the Gould
Belt, threaded by a magnetic field aligned with the dir­
ection of galactic rotation. The subsequent expansion of
the gas would have taken place preferentially along the
magnetic field lines, and would be reflected now in the
motions of the stars formed out of that gas. Rather than
a radiant point marking the center of expansion, it is pos­
sible in this case to define a radiant line so that stars move
initially away from it following trajectories perpendicular
to it.
Fig. 7. The evolution of the nodal
line\Omega\Gamma the axis of vertical
oscillation ff, the inclination i, the Oort constants A, B, C, K,
and the instantaneous angular velocity of oscillation G in the
case in which the stars of the Gould Belt are born simultan­
eously and expand initially on a plane, with velocities that are
parallel to the nodal line and proportional to their distance to
it. The expansion age of the system is set to 5 \Theta 10 7 years.
In the upper panel, the solid curve represents the direction of
the nodal line, and the dashed curve the direction of the axis
of vertical oscillation. The Oort constants are identified by the
solid line (A), the dotted line (B), the dashed line (C), and the
dot­dashed line (K). The present orientation and peak inclina­
tion are obtained with the initial
values\Omega 0 = 87 ffi 3, i 0 = 32 ffi 0,
at an age of 3:4 \Theta 10 7 years.
Assuming that the initial velocity of any given star has
a modulus proportional to the distance to the radiant line,
one obtains an initial expansion law that can be expressed
in the general form (9), with
M = 1
Ü 0
0
@
cos 2 fi cos fi sin fi 0
cos fi sin fi sin 2 fi 0
0 0 0
1
A (31)
where fi is the angle between the direction of expansion
and that of the nodal line, and Ü 0 is again the expansion
age, now defined as the ratio between the initial distance of
a star to the radiant line and its initial velocity. Two cases
similar to those presented in Sect. 3.2 are shown in Figures

F. Comer'on: Vertical motion and expansion of the Gould Belt 11
6 and 7, corresponding to Ü 0 ! 0 and Ü 0 = 5 \Theta 10 7 years.
The angle fi is taken to be 0 ffi for both cases, implying that
the radiant line is coincident with the apsidal line. The
foundation for this choice lies in the physical motivation of
this expansion law, as outlined above. The present position
of the nodal line implies an initial position roughly aligned
with the direction of galactic rotation, especially in the
case Ü 0 = 5 \Theta 10 7 (Figures 6 and 7) and this is also the
approximate orientation of the systemic component of the
galactic magnetic field.
The Ü 0 ! 0 case has in common with the correspond­
ing one in the radial expansion scenario the existence of
a null constant B and the large positive initial K term,
which decreases with time and eventually becomes negat­
ive when the epicyclic orbits reverse the initial expansion.
However, the present case is characterized by large abso­
lute values of A and C, unlike in the radial expansion case.
The evolution of the orientation is similar between both
cases, but now, due to the fact that the expansion takes
place initially perpendicular to the direction of maximum
inclination, the decrease in the amplitude of the inclina­
tion is much slower; this is, the initial tilt of the plane is
not very different from the presently observed one, unlike
in both of the radial expansion scenarios discussed before.
The early evolution when Ü 0 = 5 \Theta 10 7 years is qual­
itatively very similar to that of the radial expansion case
with the same expansion age, with the largest differences,
such as the initial C ! 0 term, appearing only in the first
10 7 years of evolution.
As to the evolution of ff and G, a behaviour similar to
the radial expansion case (alignment of the axis of vertical
oscillation with the nodal line, anticorrelation between i
and G) is found for both cases, with the exception of the
very early evolution in the case of G that now starts from
a null value.
4.4. Rotation
The last model considered here concerns a tilted plane
formed by stars which initially rotate with an angular ve­
locity w around a perpendicular axis. The velocity v of a
star on the plane is thus given by v = w \Theta r, with r being
the position vector of the star with respect to the center
of rotation. This allows one to express the initial pattern
of motion in terms of
M = w
0
@
0 \Gamma1 0
1 0 0
0 0 0
1
A (32)
with w being the modulus of w. To illustrate the resulting
pattern of motions and choose a value best matching the
actual observations, w is set to ­6.5 km s \Gamma1 kpc \Gamma1 . The
positive value
of\Omega + w corresponds to a prograde rota­
tion, i.e., in the same direction as the galactic rotation
(although with lower angular speed than the latter) in a
fixed reference frame. The results are shown in Figure 8.
Fig. 8. The evolution of the nodal
line\Omega\Gamma the axis of vertical
oscillation ff, the inclination i, the Oort constants A, B, C,
K, and the instantaneous angular velocity of oscillation G in
the case in which the stars of the Gould Belt are born simul­
taneously in a plane which rotates like a solid body and move
independently afterwards. The initial angular velocity of the
plane is set to ­6.5 km s \Gamma1 kpc \Gamma1 , i.e., the sense of rotation in
the epicyclic reference frame has a direction opposite to that of
the galactic rotation. In the upper panel, the solid curve rep­
resents the direction of the nodal line, and the dashed curve
the direction of the axis of vertical oscillation. The Oort con­
stants are identified by the solid line (A), the dotted line (B),
the dashed line (C), and the dot­dashed line (K). The present
orientation and peak inclination are obtained with the initial
values\Omega 0 = 103 ffi 3, i 0 = 36 ffi 0, at an age of 3:4 \Theta 10 7 years.
The evolution of the orientation of the plane, the Oort
constants, and the parameters defining the oscillation of
the stars around the galactic plane is clearly different from
that in the cases studied so far. The most remarkable dif­
ference is the introduction of an offset between the nodal
line and the axis of vertical oscillation, which is initially
90 ffi as it would correspond to a rigid body rotation. The
initial value of G is small but not zero, as the stars are
moving across the plane and have their largest vertical ve­
locity component at the nodal line. The offset
between\Omega and ff is maintained with time, although its value varies
as the initial circular rotation pattern is distorted by the

12 F. Comer'on: Vertical motion and expansion of the Gould Belt
differential galactic rotation. The combination of initial
rigid body rotation and subsequent independent epicyclic
orbits results in an anisotropic expanding motion, char­
acterized by the positive values of C and K. The initial
rotation also yields a large negative value of B (which is
the rotational of the velocity field) at the earliest stages.
5. Discussion
Kinematical studies of the system of young stars in the
solar neighbourhood over the last decades have revealed
some clear features associated to the Gould Belt, which
have been reinforced by the accurate picture of the stellar
kinematics in the solar neighbourhood provided by Hip­
parcos (Lindblad et al. 1997, PalouŸs 1998, Torra et al.
1999). These features can be summarized as follows:
-- The A constant has a value smaller than that corres­
ponding to pure galactic differential rotation.
-- The B constant is negative, and its absolute value is at
least similar, and probably somewhat larger, than that
corresponding to pure galactic differential rotation.
-- The K constant, which should be null for pure galactic
differential rotation, is clearly positive. The same
seems to be true for the C constant.
To these features, one can add the ones related to the
vertical motion that have been described in Section 2 of
this paper:
-- There is a small gradient of 6:5 \Sigma 1:8 km s \Gamma1 kpc \Gamma1 in
the vertical component of the velocity.
-- The stars of the Gould Belt oscillate around the
galactic plane around an axis that is misaligned by
¸ 52 ffi with respect to the nodal line marking the inter­
section between the Gould Belt and the galactic plane.
These are the features, together with the observed ori­
entation of the Gould Belt and its persistence over a time
span of 3 \Theta 10 7 years or more, that any kinematic model
should aim at accounting for. Is any of the scenarios de­
scribed in Section 4 favoured by these observed features?
Expansion models in which the stars of the Gould Belt
formed in a small volume successfully account for the ob­
served behaviour of the A, C, and K constants. Neverthe­
less, they seem to be excluded by the large negative value
of B, as noted by Lindblad 1980 and PalouŸs 1998. Models
that assume that the stars formed in a narrow strip and
expand initially along a preferential direction also account
for the values of A, C, and K, although they do so only
after an early evolution characterized by large positive val­
ues of A and K, and a large negative value of C. However,
the previous objection concerning B applies in this case
too, as a permanent null value of B is also predicted in
this scenario.
The main objection to those expansion models, namely
their inability to reproduce a value of B significantly dif­
ferent from zero, weakens if the Gould Belt started its
expansion from a plane that had already a considerable
extent. In this case, the observed features in A, C, and K
are still successfully reproduced, but now with a clearly
negative value of B, in better agreement with the obser­
vations. As far as the Oort constants and the orientation
of the Gould Belt are concerned, the predictions of both
the model of radial expansion and the model of expansion
along a line are essentially the same. However, none of the
expansion models is able to account for the main feature
discussed in this paper concerning the vertical motion of
stars, namely the large offset between the axis of vertical
oscillation and the nodal line. Moreover, radial expansion
models require a large initial tilt of the plane of the Belt,
as can be seen in Figures 4 and 5, and it is difficult to ima­
gine a formation mechanism that could account for such
a large value.
The rotation model is the only one among the models
studied here that predicts the offset between the axis of
vertical oscillation and the nodal line. Most interestingly,
the choice of an initial rigid body rotation with w = \Gamma6:5
km s \Gamma1 kpc \Gamma1 , and of the initial orientation parameters of
the Gould Belt, is able to produce a remarkably good fit,
at least qualitatively, to all the features observed in the
Gould Belt, including its present orientation, the peculi­
arities of the Oort constants, and the characteristics of its
vertical motion, if the age of the Gould Belt is 3:4 \Theta 10 7
years. The agreement can be appreciated by comparing
the measured values of the different parameters with those
predicted by the model at that age, given in Table 1. In
such comparison, we have adopted the values of Table 4
of Torra 1999 for their sample of stars at a distance of less
than 600 pc from the Sun and an age below 6 \Theta 10 7 years.
As to the results of Lindblad et al. 1997, the values listed
correspond to their sample of 144 stars within the rectan­
gular area limited by \Gamma450 pc \Gamma y ! \Gammax ! 600 pc \Gamma y,
\Gamma450 pc + y ! \Gammax ! 990 pc + y. The differences between
both sets of results may provide an idea of the subsisting
uncertainties in the derived value of the Oort constants.
The dating of the Belt based on its kinematical be­
haviour at present is strongly constrained by the mild
gradient observed in the vertical component of the ve­
locity: a difference of only 5 \Theta 10 6 years would result in
G ? 10 km s \Gamma1 kpc \Gamma1 , a value that would start conflicting
with the observations. The accurate dating based on the
small value of G is a common feature for all the models,
but unfortunately different models place the age at which
G is sufficiently small at different times. Therefore, it is
not possible to use the observed value of G for a model­
independent dating of the Gould Belt. Nevertheless, the
fact that the pattern based on the evolution of a disk ini­
tially in rotation is by far the one providing the best fit
to the data leads one to strongly favour 3:4 \Theta 10 7 years
as the true age of the Gould Belt. Remarkably, as can be
seen from Figure 8, the direction of the axis of vertical os­
cillation changes rapidly with time when the Belt is near
its greatest tilt with respect to the galactic plane. For this

F. Comer'on: Vertical motion and expansion of the Gould Belt 13
reason, even the large uncertainty in the value of ff is a
more stringent constraint on the present age of the Belt,
as a difference of only 3 \Theta 10 6 years brings the predicted
value of ff outside the range allowed by the observations.
An age of 3:4 \Theta 10 7 years is in good agreement with the
one adopted by PalouŸs 1998, although the Keplerian rota­
tion pattern proposed in that paper is different from the
solid body rotation studied here. It should be noted that
the non­linear dependence between positions and velocit­
ies implied by the Keplerian rotation pattern makes the
formulism developed in Section 3 not applicable to that
case, which is why such a pattern has not been considered
in the present study. Moreover, an initially Keplerian ro­
tation lacks the property of keeping the stars distributed
on a plane as time passes.
The initial rotation of the Gould Belt system has im­
portant implications concerning possible hypotheses for its
origin. The failure of pure expansion models in explaining
basic features of the present day kinematical behaviour of
the Belt rules out models invoking a very energetic event,
or a chain of them, in a small volume as the cause for
the velocities of the stars. This does not mean that such
explosive events have not taken place at all: on the oppos­
ite, they must have been relatively frequent as the oldest
most massive members of the Gould Belt have exploded
as supernovae, or as the stellar winds from its O and B
stars have injected large amounts of energy in their sur­
roundings. Many features in the interstellar medium of
the Gould Belt, such as the Local Bubble (Cox & Reyn­
olds 1987, the Lindblad Ring of HI (Lindblad et al. 1973),
or the shells around the Scorpius­Centaurus­Lupus associ­
ation (de Geus 1992) testify to the importance of the past
and present interaction of the Gould Belt stars with the
interstellar medium. However, the mechanism that gave
origin to these stars in the first place has most probably
to be searched elsewhere.
A simplified, straightforward interpretation of the kin­
ematical history of the Gould Belt may be to assume the
existence of a giant, rotating molecular cloud that started
to form stars about 3:4 \Theta 10 7 years ago, becoming unbound
in the process, probably because of the loss of the gas that
was not employed in the formation of stars. Although such
a scenario seems in principle plausible, it would require an
initial tilt of the molecular cloud of ' 36 ffi with respect to
the direction perpendicular to the galactic plane, whereas
actual giant molecular clouds are seen to have their rota­
tion axes well aligned with that direction (Blitz 1993).
An alternative model for the origin of the Gould Belt
was proposed by Comer'on & Torra 1992, 1994 based on
the consequences of the impact of a high velocity cloud
from the galactic halo on the galactic disk. The model
was proposed mainly to account for the initial tilt of the
Belt, which appeared as a natural consequence of an im­
pact along a direction not perpendicular to the galactic
disk. The rotation pattern of the resulting layer of dense
shocked gas, where star formation would then proceed,
could not be considered in the two­dimensional hydro­
dynamical simulations presented in those works. However,
it is conceivable that some rotation of the resulting struc­
ture may well result as a consequence of the collision, due
to the transfer of the angular momentum of the impinging
cloud to the shocked layer. A detailed examination of this
aspect of the collision is beyond the scope of this paper,
and should be studied by means of fully three­dimensional
hydrodynamical simulations, which should ultimately de­
cide whether the observed kinematical patterns are com­
patible or not with this hypothesis.
6. Summary
The main results of the present work can be summarized
as follows:
-- There is a systematic gradient in the vertical com­
ponent of the velocity of the stars belonging to the
Gould Belt along the galactic plane, subtle but detect­
able in the Hipparcos astrometric data. Such a gradient
amounts of 6:5 \Sigma 1:8 km s \Gamma1 kpc \Gamma1 , a rather small value
that suggests that the Gould Belt is at present near its
maximum tilt.
-- The pattern of such vertical motions implies an in­
stantaneous axis of oscillation around the galactic
plane oriented along a line forming an angle of 52 ffi \Sigma20 ffi
with the direction of the nodal line in which the Gould
Belt intersects the galactic plane.
-- The maintenance of the distribution of the Gould Belt
stars on a plane for a time that is comparable to the
vertical oscillation period around the galactic plane
can be achieved if the initial pattern of motions has
a linear dependence with their initial positions.
-- Analytical expressions can be found for the evolution
of the orientation of the Gould Belt, the properties of
the vertical velocities of their stars, and the Oort con­
stants as a function of time under the assumption of a
linear dependence between the initial positions and ve­
locities and the validity of the epicyclic approximation
to galactic orbits.
-- Such evolution has been considered for different cases,
whose initial parameters have been chosen so as to fit
the presently observed orientation of the Gould Belt
and its nearly maximum tilt.
-- Comparison between model results and measured
parameters seem to rule out kinematical models of the
Gould Belt in which the stars initially expand away
either from a point or from a line.
-- A model in which the stars of the Gould Belt are ini­
tially rotating around an axis perpendicular to the
plane of the Belt, and then move independently fol­
lowing their epicyclic orbits, achieves the best match
to the observations, simultaneosly fitting the orienta­
tion of the Gould Belt, the Oort constants A, B, C,
and K, the gradient in the vertical component of the

14 F. Comer'on: Vertical motion and expansion of the Gould Belt
Table 1. Comparison between observed characteristics of the Gould Belt and those predicted by the best fitting rotation model.
observed predicted
(Torra et al. 1999) (Lindblad et al. 1997)
A (km s \Gamma1 kpc \Gamma1 ) 7:2 \Sigma 0:9 \Gamma6:1 \Sigma 4:1 9.5
B (km s \Gamma1 kpc \Gamma1 ) \Gamma18:8 \Sigma 0:9 \Gamma20:6 \Sigma 5:2 ­15.3
C (km s \Gamma1 kpc \Gamma1 ) 6:0 \Sigma 1:0 2:9 \Sigma 3:7 9.9
K (km s \Gamma1 kpc \Gamma1 ) 4:9 \Sigma 1:0 11:0 \Sigma 3:5 13.8
G (km s \Gamma1 kpc \Gamma1 ) 6:5 \Sigma 1:8 (this work) 4.5
ff( ffi ) 337 \Sigma 20 (this work) 337
age (yr) ? 3 \Theta 10 7 3:4 \Theta 10 7
velocity, and the offset between the axis of vertical os­
cillation and the direction of the nodal line. Such a
best fit implies an age of (3:4 \Sigma 0:3) \Theta 10 7 years for the
Gould Belt.
-- It seems unlikely, in view of actual observations of gi­
ant molecular clouds, that the formation of the Gould
Belt can be simply explained by the dissolution of such
a complex, due to the requirement of a significant mis­
alignment between its rotation axis and the direction
perpendicular to the galactic plane. The impact of a
high velocity cloud with the galactic disk as the form­
ation mechanism of the Belt may provide a more ad­
equate explanation.
Acknowledgements. Discussions with the Hipparcos group
at the University of Barcelona, in particular Jordi Torra,
Francesca Figueras, David Fern'andez, and Montserrat Mestres,
contributed to shape the work presented here. I am pleased to
thank the careful reading and helpful comments by Prof. Per
Olof Lindblad and by the referee, Prof. Adriaan Blaauw.
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