A&A 383, 384-389 (2002)

DOI: 10.1051/0004-6361:20011751

**A. Mazure ^{1} -
H. V. Capelato^{2}**

1 -
LAM Traverse du Siphon, BP 8, 13376 Marseille Cedex 12, France

2 -
INPE/DAS CP 515, S. J. Dos Campos SP 12201-970, Brazil

Received 20 September 2001/ Accepted 23 November 2001

**Abstract**

Using the *Mathematica* package,
we find exact analytical expressions
for the so-called de-projected *de Vaucouleurs* and *Sérsic* laws as well as for
related spatial (3D) quantities - such the mass, gravitational potential, the total
energy and the central velocity dispersion - generally involved in astronomical
calculations expressed in terms of the Meijer G functions.

**Key words: **galaxies: elliptical and lenticular, cD - galaxies: fundamental
parameters - galaxies: clusters: general

Dynamical studies of astronomical systems like Elliptical Galaxies or Clusters of Galaxies involve the de-projection of observed (projected on the sky) quantities like surface brightness profiles, numerical density profiles, velocity dispersion profiles etc. The 3D profiles obtained are then used to derive e.g. the total luminosity (or mass) of the system or the gravitational potential and are used in the Jeans equation which is then resolved to get for instance the kinematics of the system.

The *de Vaucouleurs* profile (de Vaucouleurs 1948) and its generalization by the
*Sérsic* law (Sérsic 1968), is one of the most often used laws
particularly in the study of Elliptical Galaxies.

Unhappily these laws have so far lead to non-analytical de-projected (i.e. spatial) quantities. Efforts have been made in the last decades to provide either numerical tables (Poveda et al. 1960; Young 1976) or approximations and asymptotic expressions (Mellier & Mathez 1987; Ciotti 1991; Graham & Colless 1997; Ciotti & Bertin 1999; Marquez et al. 2001).

Here we give analytical exact expressions for 3D quantities
usually derived when using the *de Vaucouleurs* or *Sérsic* laws.

The classical *de Vaucouleurs* and *Sérsic* laws express the dependence on the
projected central
distance *R* of for instance the Luminosity Intensity *I*(*R*) of an
Elliptical Galaxy.

The *de Vaucouleurs* law relates the intensity *I*(*R*) to the central one, *I*(0),
by:

and the

The parameter

The requested 3D profiles are related to the derivative of
the
projected profiles by the usual Abel Integral written here for the 3D
density
profile *n*(*r*):

Except for some particular cases there is no known exact expression for theses integrals, in particular in the case of the

However, using *Mathematica * we succeeded in obtaining
exact analytical expressions for such integrals that involve the
* Meijer G functions*. Physical quantities such as the spatial
luminosity or mass profiles, the gravitational force, the gravitational
potential and energy, which are combinations or integrals of the above
functions, have also analytical expressions involving *Meijer G
functions*. Some of these expressions are given below and numerical
evaluations are compared to previous numerical calculations.

We first give the classical definition of the *Meijer G
functions* together with some of their properties which will be useful
in understanding the results given by *Mathematica*.

These functions are defined as integrals of products of
functions.
The *generalized* Meijer *G* function
is defined as:

(Gradshteyn & Ryzhik 1980; Wolfram 1991; for a collection of formulae related to the Meijer

The case *r* = 1 defines the *standard* Meijer function,
,
which is the form we
will be dealing
with in the rest of this work. In the *Mathematica* `StandardForm`

notation
it writes as:

For clarity we will keep both these notations, although suppressing the suffix "Meijer''. Notice that in these formulae, the empty case: means that the corresponding coefficients are not defined and thus do not exist. This may occur, for instance, when

The following identity may be easily obtained by a substitution of
the integration variable
,
where *c* is a constant, in Eq. (4):

The moments of the Meijer function are expressible in terms of the higher-order Meijer functions:

This may be straightforwardly demonstrated by inverting the order of the integrations.

In some particular cases the Meijer *G* functions may be expressed
in term of more classical special functions. As an example we give below the case
of the Meijer function
(cf. http://functions.wolfram.com/07.09.03.0330):

where is the modified Bessel function of order.

In the following, we give the analytical expressions using the
*Mathematica* formalism for the *G* Functions.

Let first start by introducing some useful dimensionless quantities.
The dimensionless 2D and 3D *x* and *s* radial distances are
expressed in terms
of
as:

The dimensionless 2D and 3D profiles are also defined as:

which lets Eqs. (1) and (2) transform in the following reduced form:

where the

The de-projection integral (3) also transforms to:

We give here some detailed results for the case of the *de Vaucouleurs* law as
an example and give more general results in the next section using
the *Sérsic* law. Integrating the preceding equations using
*Mathematica*, we first derive the expression for the 3D
profile *n*(*r*). We will then calculate the luminosity (or mass)
profiles as well as the gravitational potential and the gravitational
energy.

Only numerical estimations or asymptotic behaviors were given before (Poveda et al. 1960; Young 1976; Mellier & Mathez 1987) so we will compare our results with those provided by Young for the spatial density and for the luminosity (or mass).

s |
M(s) |
|||

4.4047 | 4.4047 | |||

1. | ||||

10. |

The basic ingredient to obtain the profile is the derivative of the

Integrating Eq. (11), the spatial density expressed in terms of the dimensionless 3D radial distance

One then obtains the luminosity (or mass) spatial profile defined by:

which with a new formal integration gives:

The gravitational potential is defined by:

which gives:

The values tabulated by Young (1976) are recovered defining the new potential:

such that

In Table 1, we compare the values given with the above expressions (normalized by the factor ) to the numerical values obtained by Young for

We give now more general expressions for the 3D profile derived from
the *Sérsic* law.
This law is parametrized by an index *m* and a parameter *b*(*m*).
The *Sérsic* law for *m*=1 corresponds to a 3-D *Exponential*
profile, already discussed by Fuchs & Materne (1982) and for *m*=4 to
the usual *de Vaucouleurs* law, treated in Sect. 2 above. In general,
*b*(*m*) is found as a solution of
Eq. (4) given in the Appendix. Values of *b*(*m*), (as well as )
have also been given before by Ciotti et al. and Ciotti & Bertin.
We give in the appendix, values of *b*(*m*) calculated using
*Mathematica* for the range *m*=1 to *m*=15 and compared to
those of Ciotti & Bertin.

The spatial density expressed in terms of the dimensionless 3D
distance *s* can be written as (see e.g. Ciotti 1991; Graham &
Colless 1997):

where

Again this integral is found in terms of Meijer *G* Functions
which
reduces to a* Bessel * function (*K*_{0}) for *m*=1. We
give below the various
expressions for various values of *m* and *b*(*m*). Defining
auxiliary constants:

and

we obtain, for

For

For

Notice that Eq. (23) is equivalent to Eq. (13) for the

The expressions above have the general form:

with denoting the 2

Related quantities as the mass, the gravitational potential, the (total)
potential energy and the
(central) velocity dispersion, can
then be formally calculated by other integrations, similar to what
has been done for the *de Vaucouleurs* profile. From Eq. (14) we find for the mass:

Notice that in the case

in which

For the gravitational potential, using Eqs. (16) and (26), we
find the following expression:

Notice that, as before with the equations for and for the same reasons, in the case

The gravitational potential energy is defined by:

Unfortunately there seems to be no formal solution for this integral in terms of

07.09.16.0025 one finds the following expression for the

where, besides the 2

The velocity dispersion of a spherical system in hydrostatic equilibrium is given by:

As for the gravitational potential energy, for systems endowed with a

with .

- 1.
- We obtain analytical solutions for the de-projected
*de Vaucouleurs*and*Sérsic*laws using formal integration with*Mathematica*as well as for other related quantities like the mass or the potential, total potential energy and the central velocity dispersion. - 2.
- Comparisons with existing numerical estimates show very few differences, however analytical expressions are always much more very convenient to deal with in many cases.

This work was supported by the Brazil-France CNRS-CNPq cooperation. H.V.C acknowledges partial support from FAPESP (Project No. 2000/06695-2).We warmly thank the referee, Dr. Mathews Colless, for very useful comments.

The projected luminosity profile is defined by:

In terms of the dimensionless quantities, and , this gives:

l(x) |
= | ||

(A.2) |

where is the incomplete gamma function and its complement (cf. Gradshteyn & Ryzhik 1980). For integer values of

(A.3) |

From this one may find *b*(*m*) as the solutions of the equation
,
where
is the total luminosity
integrated to infinity. We have (see also Ciotti 1991):

This is solved instantaneously using the following

`mm = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
b[m_] := FindRoot[Gamma[2m,b] == Gamma[2m]/2,{b,2m - 1/3},
WorkingPrecision -> 60, AccuracyGoal -> 30]
blist = Map[b, mm]; bb = N[b /. blist, 17]`

The derived values of *b*(*m*) are given in Table 2,
where we compare the *Mathematica* results with those from
the asymptotic expansions by Ciotti & Bertin (1999).
For *m*=4, we find of course
*b*(4) = 7.66925.

m |
b(m) |
||

1 | 1.67834699001666 | 1.67838865492157 | |

2 |
3.67206074885089 | 3.67206544591768 | |

3 |
5.67016118871207 | 5.67016250849902 | |

4 |
7.6692494425008 | 7.66924998466950 | |

5 | 9.66871461471413 | 9.66871488808778 | |

6 |
11.6683631530448 | 11.6683633097115 | |

7 |
13.6681145993449 | 13.6681146973462 | |

8 |
15.6679295443172 | 15.6679296096535 | |

9 |
17.6677864177885 | 17.6677864635090 | |

10 |
19.6676724233057 | 19.6676724565414 | |

11 |
21.6675794898319 | 21.6675795147457 | |

12 |
23.6675022752263 | 23.6675022943807 | |

13 |
25.6674371029624 | 25.6674371180047 | |

14 |
27.6673813599995 | 27.6673813720274 | |

15 |
29.6673331382212 | 29.6673331479896 |

The luminosity so writes:

for *m*=2, with *b*=*b*(2):

(A.5) |

for *m*=4, with *b*=*b*(4):

(A.6) |

and so on

- Ciotti, L. 1991, A&A, 249, 99 In the text NASA ADS
- Ciotti, L., & Bertin, G. 1999, A&A, 353, 447 In the text NASA ADS
- de Vaucouleurs, G. 1948, Ann. Astroph., 11, 247 In the text NASA ADS
- Fuchs, B., & Materne, J. 1982, A&A, 113, 85 In the text NASA ADS
- Gradshteyn, I. S., & Ryzhik, I. M. 1980, Table of Integrals, Series and Products (Academic Press) In the text
- Graham, A., & Colless, M. 1997, MNRAS, 287, 221 In the text NASA ADS
- Marquez, I., Lima Neto, G. B., Capelato, H. V., et al. 2001, A&A, 379, 767 In the text NASA ADS
- Mellier, Y., & Mathez, G. 1987, A&A, 175, 1 In the text NASA ADS
- Poveda, A., Iturriaga, R., & Orozco, I. 1960, Bol. Obs. Tonantzinla N. 20, 3 In the text
- Sérsic, J.-L. 1968, Atlas de Galaxias Australes, Observatorio Astronomico de Cordoba In the text
- Young, P. J. 1976, AJ, 81, 807 In the text NASA ADS
- Wolfram, S. 1991, The Mathematica Handbook (Addison-Wesley Publishing Company, Inc.) In the text

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