A&A 490, 477-486 (2008)
DOI: 10.1051/0004-6361:200809682
J.-M. Huré1,2 - F. Hersant2 - C. Carreau3 - J.-P. Busset1
1 - Université de Bordeaux, LAB, 351 cours de la Libération, Talence 33405, France
2 -
CNRS/INSU, UMR 5804/LAB, 2 rue de l'Observatoire, BP 89, 33271 Floirac Cedex, France
3 -
La Maurellerie, 37290 Bossay-sur-Claise, France
Received 29 February 2008 / Accepted 31 May 2008
Abstract
Aims. The first-order ordinary differential equation (ODE) that describes the mid-plane gravitational potential in flat finite size discs of surface density
(Huré & Hersant 2007, A&A, 467, 907) is solved exactly in terms of infinite series.
Methods. The formal solution of the ODE is derived and then converted into a series representation by expanding the elliptic integral of the first kind over its modulus before analytical integration.
Results. Inside the disc, the gravitational potential consists of three terms: a power law of radius R with index 1+s, and two infinite series of the variables R and 1/R. The convergence of the series can be accelerated, enabling the construction of reliable approximations. At the lowest-order, the potential inside large astrophysical discs (
)
is described by a very simple formula whose accuracy (a few percent typically) is easily increased by considering successive orders through a recurrence. A basic algorithm is given.
Conclusions. Applications concern all theoretical models and numerical simulations where the influence of disc gravity must be checked and/or reliably taken into account.
Key words: gravitation - methods: analytical - accretion, accretion disks
Gaseous discs in which the main physical quantities (density, pressure, temperature, thickness, velocity) scale with cylindrical radius as power laws, i.e. ``power-law discs'', represent an important class of theoretical systems. These are used customary to model accretion in evolved binaries (Shakura & Sunyaev 1973; Pringle 1981), circumstellar matter (Edgar 2007; Dubrulle 1992), the environment of massive black holes inside active galactic nuclei (Collin-Souffrin & Dumont 1990; Huré 1998; Semerák 2004) or even the stellar component of some galaxies (Evans 1994; Zhao et al. 1999). In most applications however, power-law discs are truncated either to avoid diverging values at the disc centre (such as density, mass) or in attempting to reproduce the properties of observed discs of finite extension and mass. Although self-similarity is not compatible with the presence of edges, it is generally considered that power laws offer a good description of disc properties in some regions (far from the edges). Note that the presence of sharp edges can be misleading when interpreting observational data (e.g. Hughes et al. 2008). In general, the surface density
in the outer parts of discs is a decreasing function of the cylindrical radius R. Depending on the models, hypotheses, and objects, we have, for instance,
in binaries (Shakura & Sunyaev 1973),
in active galactic nuclei (Collin-Souffrin & Dumont 1990),
for a Mestel disc (Mestel 1963), or
in circumstellar discs (Piétu et al. 2007). In the context of stationary viscous
-discs, a wide range of power-law exponents is allowed since the temperature T,
,
and R satisfy the condition (e.g. Pringle 1981):
The calculus of the gravitation potential of finite-size, power-law discs has received little attention yet. Several reasons can be put forward. Solving the Poisson equation or computing the integral of the potential is not a trivial procedure, especially in the presence of edges. It is generally believed that gravity due to low mass discs is unimportant compared with that of a central proto-star or black hole, and cannot be probed (see however Baruteau & Masset 2008). Many studies employ the multi-pole expansion which is known to converge too slowly inside sources to be efficient for the numerical applications (e.g. Stone & Norman 1992; Clement 1974). Huré & Hersant (2007) demonstrated that the mid-plane potential of flat power-law discs obeys an inhomogeneous first-order ordinary differential equation (ODE). In this second paper, we discuss the exact solutions of this ODE for the entire physical range (outside and inside the disc) in terms of infinite series. In particular, it is shown that the mid-plane potential is a combination of a power law for the radius R and two series of the variables R and 1/R. Since these series converge rapidly inside large discs, it is possible to derive reliable approximations by truncating the series at low orders.
This paper is organised as follows. The ODE for the potential is briefly recalled in Sect. 2 and its formal solution is derived in Sect. 3. In Sect. 4, we express the potential at the two disc edges and consider a few special cases. The inside and outside solutions in the form of series are presented in Sect. 5. In Sect. 6, we analyse the potential in the disc inside in detail, and in particular, the power-law contribution. Since all series involved converge rapidly, we are able to derive reliable approximations for the potential; this is done in Sect. 7. We discuss in Sect. 8 the case of discs with no inner and/or outer edge. The paper ends with a few concluding remarks.
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Figure 1: Configuration for a finite-size flat disc. |
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Following Huré & Hersant (2007) hereafter Paper I), the mid-plane potential
due to a flat power-law disc satisfies the ODE:
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(2) |
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(4) |
A formal solution of Eq. (1) is found by setting (e.g. Rybicki & Lightman 1979):
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(8) |
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(9) |
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(10) |
In the following, we shall analyse Eq. (11) analytically in terms of infinite series by considering in Eq. (3) the expansion of
over its modulus (e.g. Gradshteyn & Ryzhik 1965):
To determine
from Eq. (11) for
,
it is sufficient to calculate the potential at the two disc edges
, that is
and
.
For this purpose, we use Eq. (5) and define
.
After some algebra
we find:
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(17) |
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Figure 2:
Coefficient ![]() |
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At the outer edge, the potential is calculated in a similar manner, but using the variable
.
For
,
Eq. (5) writes1:
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(20) |
If the power law exponent s is such that:
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(22) |
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(24) |
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(27) |
The case
occurs i) when the disc has no inner hole (i.e.
)
but finite size, and/or ii) when the disc has an inner edge but is infinitely extended (i.e.
). In the first case,
(denoted
in Paper I) becomes the potential at the origin of coordinates and has infinite value as soon as
.
In the second case,
represents the gravitational potential at infinity. It diverges if
.
Figure 3 summarises the ranges of s where the edge surface density, edge potential, and total disc mass are finite. We note that only discs having either i)
,
with s > 0; or ii)
,
with
are physically meaningful since they are characterised by a finite surface density, a finite total mass and a finite potential. Mestel discs do not belong to these categories (e.g. Hunter et al. 1984; Mestel 1963).
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Figure 3: Ranges of s where the edge surface density, edge potential and total disc mass are infinite. |
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We see from Eqs. (3) and (13) that the function S can be easily expressed as a series. We have:
cn = -In -Jn. | (35) |
Figure 4 compares the total potential
with the power law contribution (i.e. the term
)
for three typical values of the exponent s in a disc of axis ratio
.
We clearly see that, in a finite size disc: i) the gravitational potential is not a power-law function of the radius; ii) a power-law contribution is present inside the disc only; and iii) the power law is not the dominant part of the potential. As expected, spatial self-similarity is broken due to edges. We note that, outside the disc (i.e. in regions I and III), the series coincides with the multi-pole expansions.
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Figure 4:
Potential ( plain lines) in a power-law disc with axis ratio
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The determination of the gravitational potential is usually straightforward outside the distribution where different kinds of expansions are efficient in practice (Kellogg 1929). In contrast, it is problematic inside matter where the classical multi-pole approach fails to converge rapidly (e.g. Stone & Norman 1992; Clement 1974).
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Figure 5:
Coefficients A ( open circles), ![]() ![]() ![]() ![]() |
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In region II, we have
.
If we set:
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(36) |
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(37) |
The coefficient A is plotted in Fig. 5. It is symmetric with respect to
since:
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(38) |
Once s is given, the coefficients A, ,
and A1 can be easily determined at the required accuracy. It is also possible to improve the convergence rate of the associated series. This accelerates the computation of the coefficients and makes their dependence with the exponent s more explicit. Convergence acceleration is performed by using the properties of the definite integrals of the complete elliptic integrals of the first and second kinds. The demonstration reported in the Appendix C yields, for
:
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(44) |
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(45) |
For
,
we then find:
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Figure 6: Relative error in A when computed approximately from Eq. (48). |
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Depending on the exponent s of interest, a good approximation for A,
or A1 can be obtained by considering only the largest terms in the sum, i.e. all terms up to the rank
or
.
For astrophysical discs, s is around -1 meaning that we retain only the first term in Eq. (43). We then find the following approximation:
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Figure 7: The exact potential for a power law disc with s=-1.5 ( thick line) compared to approximate values ( thin lines) found from Eq. (50) for N=0, 1, 2 and 3 (i.e. see Eqs. (52), (54), (55), etc.). The largest errors are found around edges. |
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Equations (33), (39) and (41) contain three rapidly converging series that can be truncated to derive reliable approximations for the potential. For
,
only a few terms can be considered (see Sect. 6). Although many truncations are possible, we have noticed that the most accurate approximations of
for discs
are obtained provided the coefficient A (or
or A1 depending on s) takes its converged value. Under these circumstances, the N-order approximation for the potential in region II becomes:
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(51) |
As argued in Sect. 6.3, a reliable formula for the potential in astrophysical discs (for which
)
is obtained by considering only the terms a0 and
,
in addition to the power law. At the lowest order, we thus have
:
It is worth noting that the accuracy remains of the same order if coefficient A is determined by Eq. (48). This is convenient when the explicit dependence of
on s is required. Under this hypothesis, the potential becomes (see note 4):
Figure 8 shows the accuracy of this formula in the
-plane. We see that the relative deviation of
10% observed previously for s=-1.5 holds globally for s roughly in the range
[-3,0]. This agrees with the fact that Eq. (48) produces values of A within a few percents for this range of exponents. The deviation can be reduced at the inner and outer edges provided additional terms are included (see below).
If necessary, more accurate expressions are obtained by accounting gradually for following terms (each acting as a smaller and smaller correction). For N=1, we have:
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Figure 8:
Contour map showing the decimal logarithm of the relative error in the potential in the
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If the disc has no inner edge but a finite size (i.e.
and
), then the ODE is (see Huré & Hersant 2007):
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(57) |
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(58) |
If the disc is infinite but has an inner edge (i.e.
and
), then
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(59) |
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(60) |
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(63) |
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(64) |
If the disc is infinite, the ODE become homogeneous:
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(65) |
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(66) |
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(67) |
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(68) |
In this paper, we have derived an exact expression for the gravitational potential in the plane of flat power-law discs as a solution of the ODE reported in Paper I. This expression is valid over the entire spatial domain and takes into account finite size effects. Inside the disc (the most difficult case to treat in general), it consists of three terms of comparable magnitude: a power law of the cylindrical radius R with index 1+s (where s is the exponent of the surface density) and two series of R and 1/R. In terms of convergence, our expression is by far superior to the multi-pole expansion method. Reliable approximations for the potential can be produced by performing fully controlled truncations. We have shown that the potential can be expressed by means of a simple function of R and s, which is valid to within a few percents in the range of exponents
.
This formula should be sufficiently accurate for most astrophysical applications. If necessary, more accurate formulae can be developped by including successive terms. These results should help in investigating various phenomena where disc gravity plays a significant role.
An interesting point concerns the case of discs for which the surface density is not a power-law function. As shown in Paper I, it is easy to reproduce numerically the potential when the profile
is a mixture of power laws. From an analytical point of view however, the construction of a reliable formula for
as compact as the one obtained here is not guaranteed at all. For instance, for an expansion of the form:
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(69) |
Acknowledgements
F. Hersant was supported by a CNRS fellowship which is gratefully acknowledged. We thank C. Baruteau. We thank the anonymous referee for valuable comments.
For
,
the coefficient is:
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(A.1) |
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= | ![]() |
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= | ![]() |
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= | ![]() |
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= | ![]() |
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= | ![]() |
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= | ![]() |
(A.2) |
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= | ![]() |
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= | ![]() |
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= | ![]() |
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= | ![]() |
(A.3) |
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= | ![]() |
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= | ![]() |
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= | ![]() |
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= | ![]() |
||
= | ![]() |
(A.4) |
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(A.5) |
Table B.1: Values of A (4th column) computed within the computer precision (double precision; about 16-digit) for different power law exponents s or s' (Cols 1 to 3) including those relevant for astrophysical applications (see also Fig. 5). Also given are the relative error on the coefficient (5th column) and the number of iterations (6th column) required.
The coefficient A is given by a series whose convergence can be accelerated by considering an interesting property of the integral of
(e.g. Gradshteyn & Ryzhik 1965), namely:
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= | ![]() |
|
= | ![]() |
(C.2) |
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(C.3) |
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(C.5) |
It follows that:
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= | ![]() |
|
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(C.6) |
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= | ![]() |
|
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= | ![]() |
(C.7) |
But, from Eq. (C.4), we have:
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(C.8) |
and then:
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(C.9) |
Finally, we have:
A | = | ![]() |
|
= | ![]() |
||
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|||
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(C.10) |
A possible algorithm is the following (see the text for cases where
):
compute dimensionless variables
and X (depending on the region I, II or III).
The loop ends after N steps; the accuracy of the potential value is then given by the next term (rank N+1).