Download Potential-Density pairs for galaxy models obtained by superposition

Galaxy models
Potential-Density pairs for galaxy models obtained by superposition of the Kalnajs
generalized disks and the Miyamoto-Nagai Potentials
Fernando Cortés Serrano & Guillermo A. González & Jerson I. Reina
Universidad Industrial de Santander, Escuela de Fı́sica
[email protected], [email protected], [email protected],
A.A 678, Bucaramanga 680002, Colombia
Abstract
We present an infinite family of potential-density pairs for galaxy models composed by thin disks and spheroidal
haloes. The models are obtained by expressing the gravitational potential as the superposition of two independent
components, one corresponding to the galactic disk and the other to the spheroidal halo. The component corresponding to the galactic disk is taken as the Kalnajs generalized disks, which are obtained by considering solutions
of the Laplace equation in oblate spheroidal coordinates, expressed in terms of products of Legendre polynomials
and Legendre functions of second class. The gravitational potential generated by the spheroidal halo of matter is
built considering a multipolar expansion of the potential and performing a transformation of the Miyamoto-Nagai
type in the variable z, so that the new potential is solution of the Poisson equation and represents the gravitational
potential of the matter halo.
Particular models for the gravitational potential of the disk and the halo of matter are proposed. Given the gravitational potential, we obtain analytical expressions for the surface density of the disk, the density of the halo of matter
and the corresponding expressions for the rotation curves. From these it is possible to know the changes in rotation
curves due to the variation of the ratios between the parameters of the disk and matter halo.
in which, Ãk = (Ak /A0)/ak . This
√ potential is written in cilindrical coordiantes (R, z) and we realize
the transformation z ∗ = ah + b2 + z 2 in order to show all the expressions in the same coordinate
system.
The disk’s models are given by the expression for the surface density
2m + 1
m−1/2
σm(R) =
M
[1
−
R/a]
,
(10)
2
2πa
and as the halo’s gravitational potential Φh(R, z; ah, b) satisfies the Poisson equation this allows to
obtain an expression for the density
Φh(R, z),R
1
(11)
Φh(R, z),RR +
+ Φh(R, z),zz .
ρ(R, z) =
4πG
R
Results
Introduction
In the following figures we show the results about the formulation of the galaxy model with two
components, a finite disk with zero thickness and a halo of infinite size, and the manipulation of the
constants of the model
Currently, one of the most important problems in dynamics of galaxies is the determination of its
mass distribution based on observations of the speed circular or rotation curve [4]. If a particular
model is assumed for the composition of a galaxy, it is possible to obtain an analytic expression for
the potential of the entire system. From the gravitational potential can be obtained the corresponding
expressions for the surface density of the galactic disk, halo density of matter and circular rotation
speed. The fit of the model with the rotation curve allows you to completely determine the mass distribution of the galaxy [1]. The most widely accepted description of the composition of spiral galaxies
is that a significant portion of its mass is concentrated in a thin disk, while other contributions to the
total mass of the galaxy come from a spheroidal halo of matter, the central bulb and a central black
hole [1].
Ž
z
2.5
2.5
2.0 0.2
2.0
Ž
Σ
1.5
1.0
0.5
0.5
0.0
0.6
0.9
0.8
0
0.4
2
0.1
Ž
R
3
0.0
1. To built models of spiral galaxies composed of a thin galactic disk and a spheroidal matter’s halo.
2. To determine the analytical expressions for the galactic surface disk’s mass density , the density of
the spheroidal halo and the rotational curve of the previous models.
3. To determine explicit values for the constants in the general solution of the previous models.
Ž
z
1.5
0.3
0.7
0.5
0.2
0.4
0
0.8
0.4
1
2
0.1
Ž
R
3
0.08
Ž
z
0.06
0.1
Materials and Methods
0.3
0.05
0.10
Ž
R
0.2
0.15
a)
Some of the models used to represent galaxies consist of continuous distributions of matter characterized by density ρ(r) and gravitational potential Φ(r). The two physical quantities are related through
the Poisson equation:
∇2Φ(r) = 4πGρ(r),
(1)
in matter-free regions, that is, when ρ = 0, the equation (1) reduces to Laplace’s equation:
2
0.4
0.20
3
4
5
Sin halo
0.15
2.5
Ž
V
1.0
0.20
2.0
0.40
1.5
1.0
0.5
0
1
2
3
Ž
R
0.0
4
0
2
4
Ž
R
6
8
10
c)
2.0
Sin halo
0.05
0.10
0.15
0.06
0.04
0.05
1.5
Ž
V
0.10
0.15
1.0
0.5
0.02
0.00
Ž
R
3.0
0.40
0.08
0.00
1
b)
0.10
0.5
0
c)
0.10
0.04
1.5
0.0
1.0
1.5
0.0
4
0.12
0.00
0.8
0.20
a)
Ž
z
Ž
R
0.6
0.5
0.6
0.9
0.02
0.00
0.05
Ž
R
b)
0.10
0.15
0.0
0.0
0.5
1.0
1.5
Ž
R
2.0
2.5
3.0
c)
Figure 1: Upper: (a) Density contours in the (R̃, z̃) plane corresponding to a halo with the following parameters:
ã = ah/a = 0.2, b/ah = 10. (b) Density profile of the disk model m = 2, through the coordinate R̃. (c) Rotation curve with the following parameters: disk mass Md and the halo mass Mh , where: ã = 0.2, b/ah = 10 and
M̃ = Mh /Md = {0, 1, 4, 8}. Center: (a) Density contours in the (R̃, z̃) plane corresponding to a halo with the following
parameters: ã = 0.2, b/ah = 10. (b) Density contour of three halos that satisfy ρ̃ = cte = 0.5ρ̃0 , with a variation in the
parameter ã: ã = {0.15, 0.20, 0.40}. (c) Rotation curve with variation over the parameters ah of the halo and the constant
a of the disk, the numerical values of the parameters are: M̃ = 7, b/ah = 10 and ã = {0.15, 0.20, 0.40}. Bottom: (a)
Density contours in the (R̃, z̃) plane corresponding to a halo with the following parameters: ã = 0.2, b/ah = 0.1. (b)
Density contour of three halos that satisfy ρ̃ = cte = 0.5ρ̃0 , with the parameters: ã = 0.2, b/ah = {0.05, 0.10, 0.15}. (c)
Rotation curve with variation over the b and ah halo parameters, where: M̃ = 1, ã = 0.2 and b/ah = {0.05, 0.10, 0.15}.
∇2Φ(r) = 0.
(2)
The general model adopted makes use of two symmetries: the axial symmetry and reflection symmetry. The axial symmetry of the potential implies independence with respect to the azimuthal coordinate φ, Φ = Φ(R, z); while the reflection symmetry states that the potential above and below the
plane z = 0 must be equal
Φ(R, z) = Φ(R, −z).
(3)
For a complete model of a galaxy, you should consider the contributions of all components, such
that the distribution of matter will be given by
Conclusions
• We perform well behaved models, that is, the distributions of matter both disk and halo don’t have
singularities.
• We could relate the rotation curves with the matter distribution of the halos, through the variation
of the parameters of the disk and the halo of matter.
• We obtain a great variety of distributions of matter to analize the effects of the size, form and the
mass of the halo over the form of the rotation curve.
(4)
such that ρbh, ρb, ρd and ρh are the densities of matter of the central black hole, the central bulge, the
thin galactic disk and the dark matter’s halo, respectively. So that the gravitational potential will have
the general form
Φ = Φbh + Φb + Φd + Φh + · · · .
(5)
Another physical quantity of interest is the circular speed, which is related to the gravitational potential by the expression:
∂R
0.0
2.0
2.0 0.2
1.0
8
0.5
0.15
0.5
4
2.0
b)
2.5
Main Objectives
∂Φ
2
V =R .
0.0
4
1
1.0
a)
Ž
z
ρ = ρbh + ρb + ρd + ρh + · · · ,
Ž
V
1.5
0.5
1
Sin halo
2.5
1.0
0.3
0.7
3.0
Forthcoming Research
In the present state of this investigation we are developing a research about the stability of the circular
orbits in this model.
(6)
Thin disk and halo model
References
In order to obtain galaxy models composed of a thin disc and the spheroidal halo we propose the total
gravitational potential as the superposition of two independent components:
Φ(R, z) = Φd(R, z) + Φh(R, z),
[1] James Binney and Scott Tremaine. Galactic dynamics. Princeton university press, 2011.
[2] Guillermo A González and Jerson I Reina. An infinite family of generalized kalnajs discs. Monthly
Notices of the Royal Astronomical Society, 371(4):1873–1876, 2006.
[3] Masanori Miyamoto and Ryuzaburo Nagai. Three-dimensional models for the distribution of mass
in galaxies. Publications of the Astronomical Society of Japan, 27:533–543, 1975.
[4] A. Pierens and J.-M. Huré. Rotation Curves of Galactic Disks for Arbitrary Surface Density
Profiles: A Simple and Efficient Recipe. ApJ, 605:179–182, April 2004.
[5] D. vogt and P. S. Letelier. On Multipolar Analytical Potentials for Galaxies. pasj, 57:871–875,
December 2005.
(7)
Φd and Φh are the gravitational potential of the disk and halo, respectively. The potential Φd is a
solution of the Laplace equation in oblate spheroidal coordinates [2]:
m
X
Φd(ξ, η) = −
C2l q2l (ξ)P2l (η),
(8)
l=0
q2l (ξ) are the Legendre functions of second kind and P2l (η) are the Legendre polinomials, additionaly,
the halo’s gravitational potential is proposed as an expansion on the way [3], [5]
n
X
Ãk Pk (cos θ)
Ψh(r, θ) = −A0
,
(9)
k+1
r
Acknowledgements
k=0
1
The authors want to give thanks for the financial support from the Industrial University of Santander
, Colombia.