Mon. Not. R. Astron. Soc. 000, 1–5 (yyyy) Printed 8 November 2018 (MN LATEX style file v2.2)
On spherically symmetrical accretion in fractal media
Nirupam Roy ⋆
NCRA-TIFR, Post Bag 3, Ganeshkhind, Pune 411 007, India
Accepted yyyy mmm dd. Received yyyy mmm dd; in original form yyyy mmm dd
ABSTRACT
We use fractional integrals to generalize the description of hydrodynamic accretion in
fractal media. The fractional continuous medium model allows the generalization of the equa-
tions of balance of mass density and momentum density. These make it possible to consider
the general case of spherical hydrodynamic accretion onto a gravitating mass embedded in
a fractal medium. The general nature of the solution is similar to the “Bondi solution”, but
the accretion rate may vary substantially and the dependence on central mass may change
significantly depending on dimensionality of the fractal medium. The theory shows consis-
tency with the observational data and numerical simulation results for the particular case of
accretion onto pre-main-sequence stars.
Key words: Accretion, accretion discs – hydrodynamics – turbulence – stars: pre-main-
sequence – ISM: clouds – ISM: structure.
1 INTRODUCTION
The interstellar medium (ISM) is believed to have a self-
similar hierarchical structure over several orders of magni-
tude in scale (Larson 1981; Falgarone, Puget & Perault 1992;
Heithausen et al. 1998). Direct H I absorption observations and
interstellar scintillation measurements suggest that the structure
extends down to a scale of 10 AU (Crovisier, Dickey & Kazès
1985; Langer et al. 1995; Faison et al. 1998) and possibly even
to a scale of sub-AU (Hill et al. 2005). However, the latter is
limited by the spatial resolution of the observations. Hence the
issue is far from being definite even after observational detec-
tion of lower limit of self-similarity scale in some ISM com-
ponents (Goodman, Barranco, Wilner & Heyer 1998). Numerous
theories have attempted to explain the origin, evolution and
mass distribution of these clouds (began with the hierarchi-
cal fragmentation picture, Hoyle 1953) and it has been estab-
lished, from both observations (Elmegreen & Falgarone 1996)
and numerical simulations (Burkert, Bate & Bodenheimer 1997;
Klessen, Burkert & Bate 1998; Semelin & Combes 2000), that the
interstellar medium has a clumpy hierarchical self-similar structure
with a fractal dimension 2.5 . D . 2.7 (Sánchez, Alfaro & Pérez
2005) in 3 dimensional space. The main reason for this is still not
properly understood but can result from the underlying fractal ge-
ometry that may arise due to turbulent processes in the medium.
Here we have investigated physical processes, in particular hy-
drodynamics, in such a medium with fractal dimension. We have
considered the simplest situation of hydrodynamic spherical steady
accretion of the medium onto a large gravitating mass embedded in
this medium. We have assumed the dimensionality of the medium
⋆ E-mail: nirupam@ncra.tifr.res.in
to be isotropic and homogeneous. It implies that the mass den-
sity (or, equivalently, dimensionality) is same for any surface inde-
pendent of the orientation. The famous solution of the continuous
medium case is the “Bondi Solution” for spherical hydrodynamic
accretion (Bondi 1952). Here we have extended that analysis for a
medium with fractal dimension D (= 3d) 6 3.
To describe physical processes in fractal medium, we have
used fractional integration and differentiation (Zaslavsky 2002, and
references therein). We replace the fractal medium by some contin-
uous medium and the integrals on the network of fractal medium
is approximated by fractional integrals (Ren et al. 2003). The inter-
pretation of fractional integration is connected with fractional mass
dimension (Mandelbrot 1983). Fractional integrals can be consid-
ered as integrals over fractional dimension space (up to a numerical
factor; Tarasov 2004). We have chosen the numerical factor prop-
erly to get the right dimension of any physical parameter and to de-
rive the standard expression in the limit d → 1. The isotropic and
homogeneous nature of dimensionality is also incorporated prop-
erly. This model allows us to describe the hydrodynamics in a self-
consistent way in a fractal medium.
In this paper we first derive, in §2, the steady state mass accre-
tion rate for fractal spherical hydrodynamic accretion using sim-
ple dimensional analysis. In §3 we derive the steady state hydro-
dynamic equations to describe the spherical accretion in a fractal
medium and the steady state accretion rate in terms of boundary
conditions at infinity is derived in §4. We discuss the actual astro-
physical implication of our analysis and compare our central re-
sult with observational data and numerical simulation results in §5.
Possible limitations of our analysis is discussed in §6. Finally, we
summarize and present our conclusions in §7.
c© yyyy RAS
http://arxiv.org/abs/0704.1110v2
2 N. Roy
2 DIMENSIONAL ANALYSIS OF FRACTAL
ACCRETION
We assume a medium that, on a range of length scale R, has a frac-
tal structure of dimensionality D = 3d < 3 embedded in 3 dimen-
sional space. Here D refers to the mass dimension and it implies
that in such a medium of constant density ρ, the mass enclosed in a
sphere of radius r will be
MD = kr
3(d−1)
3d (1)
where lc is a characteristic inner length of the medium and can take
arbitrary value in the limit d → 1. It is the scale below which the
medium will be continuous. We can define the modified “density”
for this fractal medium as ρ ≡ ρ/l3(d−1)c so that MD ∼ ρR3d.
For steady state hydrodynamic spherical accretion onto a cen-
tral mass M from its surroundings fractal medium with a mass di-
mension of D, the relevant physical parameters will be (1) sound
speed at a large distance away from the accretor (a∞), (2) modified
“density” of the fractal medium at a large distance away from the
accretor (ρ
= ρ∞/l
3(d−1)
c ) and (3) mass of the accretor scaled
by the gravitational constant (GM ). The dimensions of these three
parameters are
[a∞] = [M ]
1 (2)
] = [M ]
0 (3)
[GM ] = [M ]
. (4)
It is possible to uniquely construct, from these parameters, a quan-
tity Ṁ with dimension [M ]1[L]0[T ]−1. So, from simple dimen-
sional analysis we get the mass accretion rate
Ṁ ∼ ρ
3(d−1)
. (5)
Dimension analysis can not be used to fix the dimensionless con-
stant C in the above equation and does not give a detailed physical
picture. But we have used the fractional integrals to derived, in a
more detailed analysis in the following sections, the mass accretion
rate and found that to be consistent with the accretion rate derived
from the dimensional analysis.
3 FRACTIONAL INTEGRALS AND HYDRODYNAMIC
EQUATIONS
The integrals on network of fractals can be approximated by frac-
tional integrals (Ren et al. 2003) and the interpretation of the frac-
tional integration is connected with fractional mass dimension. We
consider the fractional integrals as integrals over fractional dimen-
sion space (up to a numerical factor) and the fractional infinitesimal
length for a medium with isotropic mass dimension will be given
ld−1c
dr, d = D/3 < 1 (6)
where the constant is chosen to derive the standard expression in the
limit d → 1. Note that the infinitesimal area and volume elements
in this “fractional continuous” medium of mass dimension D = 3d
will be
dAr =
2(d−1)
r2(d−1)
sin θdθdφ (7)
dAθ =
2(d−1)
r2(d−1)
rdr sin θdφ (8)
dAφ =
2(d−1)
r2(d−1)
rdrdθ (9)
3(d−1)
r3(d−1)
dr sin θdθdφ (10)
and hence the mass enclosed in a sphere of radius R for constant
density ρ will be
ρdV =
3(d−1)
∼ RD (11)
which will give the standard expression in the limit d → 1.
Below we consider the mass density and momentum density
balance in such a fractal medium of mass dimension 3d in the case
of accretion onto a central gravitating mass M for the case of our
interest (i.e. the steady state hydrodynamic spherical accretion).
We consider the infinitesimal volume dV in E3 and the bal-
ance of mass density or the conservation of mass in that volume in
the case of steady state spherical accretion will imply
[ρu(r/lc)
dAr] = 0 =⇒
) = 0. (12)
This is the generalized mass density balance equation or the gener-
alized continuity equation that can be integrated over a surface of
constant r to give a constant radial accretion rate
3(1−d)
= Constant (independent of r). (13)
In the infinitesimal volume dV in E3, the balance of momen-
tum density will imply
(ρVdV ) = F
S (14)
where V is the velocity vector, FM is the total gravitational force
acting on the mass contained in the infinitesimal volume dV and
S is the total surface force due to pressure acting on the surfaces
bounding the volume dV and are given by
r = −
dV (15)
r = ber.(r/lc)
[(pdArber)r − (pdArber)r+dr +
(pdAθbeθ)θ − (pdAθbeθ)θ+dθ + (pdAφbeφ)φ −
(pdAφbeφ)φ+dφ] (16)
and, for spherical accretion where p = p(r), it will be reduced to
r = −
dArdr(r/lc)
. (17)
The total change of radial momentum is given by
(ρudV ) =
(ρudV ) +
(r/lc)
dAr]dr. (18)
Combining this and equation (12), in case of steady state we get
(r/lc)
dArdr = −
dArdr(r/lc)
d−1 −
dV (19)
and this can be simplified to the generalized momentum density
balance equation
= 0. (20)
This equation, using the equation of state p = Kργ (1 6 γ 6 5
and the boundary condition at infinity, can also be integrated to give
c© yyyy RAS, MNRAS 000, 1–5
Accretion in fractal media 3
0.001
0.010
0.100
1.000
10.000
 0.01  0.1  1  10
log (r/r0)
λ=0.25λc
λ=1.00λc
λ=4.00λc
Figure 1. Velocity profile (u/a–r/r0, r0 = GM/a2∞) for D = 2.55 and
γ = 7
at λ = 0.25λc , 1.00λc , 4.00λc
γ − 1
γ − 1
∞ (21)
where a is the sound speed given by a ≡ (dp/dρ)1/2 and a∞ is
the sound speed at infinity. This is a local conservation law and,
as expected, has exactly the same form as that of the continuous
medium.
4 MASS ACCRETION RATE
We have solved equations (12) and (20) for a smooth, monotonic
solution without any singularities in the flow. Following the stan-
dard derivation of the “Bondi solution”, we found that there exist
unique eigen value solution of the problem for a given d and γ and
that the solution must pass through a critical point where
c = a
c = S(d, γ)a
∞, S(d, γ) =
(6d− 1)− 3γ(2d− 1)
. (22)
The mass accretion rate may not be steady in a small timescale
because of the “clumpy” structure of the medium but a time average
mass accretion rate can also be calculated following the standard
derivation and is given by
Ṁ = 4πλc(d, γ)
3(d−1)
”3d−1
= 4πλc(d, γ)ρ∞a∞(GM/a
−3(1−d) (23)
where F (M,a∞, lc) and λc(d, γ) are dimensionless parameters
F (M,a∞, lc) = GM/a
∞lc (24)
λc(d, γ) =
(3d− 1)(1−3d)
[S(d, γ)]
2(γ−1)
−(3d−1)
. (25)
This is consistent with the accretion rate derived from the dimen-
sional analysis (see Eqn.(5) in Sec.2). The solution also uniquely
corresponds to the physically likely situations properly matching
the boundary conditions (small velocity at large distance and high
velocity at small r in accretion flow and the opposite in “wind
flow”). The velocity profile is shown in figure (1) for a particu-
lar case where D = 3d = 2.55 and γ = 7
. The general nature
of the solution is similar to the “Bondi solution” (Bondi 1952).
Note that the smooth, physically selected solution exists only for
the transonic case with λ = λc. It also corresponds to the max-
imum accretion rate as there exists no meaningful solution either
for λ > λc or for λ < λc.
Note that value of λc is of order unity. In particular, for d = 1
and γ = 5
, λc =
(Bondi 1952) and for γ = 1, λc is given by
λc(d, γ = 1) =
(3d− 1)(1−3d)
3d− 3
2 (26)
which is the limit of equation (25) as γ → 1, so that λc is continu-
ous at γ = 1.
The most interesting consequence is the dependence of accre-
tion rate on the factor F (M,a∞, lc). In the interstellar medium
typical ambient sound speed (which goes as
T ) is ∼ 10 km s−1
for a temperature of ∼ 104 K. In this case the value of F is given
F (M,a∞, lc) = 8.907
10 km s
”2“1 AU
. (27)
This is simply the ratio of the length-scale of “sphere of influence”
of the accreting object to the scale length of the fractal structure. As
we are mainly interested in the medium inside the sphere of influ-
ence, our approach to the problem of accretion in fractal medium
will be valid only if F & 1.
In such situations, this factor can change the mass accretion
rate substantially and it will crucially depend on the fractional di-
mensionality d. More interestingly, for the cases where this fac-
tor is important, the accretion rate Ṁ will not be proportional
to M2 but will be significantly different from that. For example,
for 2.5 . D . 2.7 (Sánchez, Alfaro & Pérez 2005) or equiv-
alently 0.83 . d . 0.90, the accretion rate Ṁ ∼ Mα where
1.5 . α . 1.7.
In the actual astrophysical situation, however, the
Bondi-Hoyle accretion rate will probably be more rel-
evant (Bondi & Hoyle 1944; Bondi 1952; Edgar 2004,
and references therein). Bondi (1952) proposed the in-
terpolation formula (corrected upto a numerical factor by
Shima, Matsuda, Takeda & Sawada 1985) replacing a∞ by
(a2∞ + v
1/2 and Bondi radius rB = GM/a
∞ by Bondi-Hoyle
radius rBH = GM/(a
∞ + v
∞) where v∞ is the gas velocity
relative to the star at a large distance. For a medium with density
and velocity structure, it is not possible to define a unique v∞ and
the present analysis can not be extended to such a situation. But
it may be extended to a more idealized case where the medium
has a fractal structure and the accretor is moving in a velocity v∞
relative to the medium at a large distance. Following a similar
interpolation method, the accretion rate in this case will be
ṀBH =
4πλcG
(a2∞ + v
−3(1−d)
where F ∼ rBH/lc and λc is a factor of the order of unity. Here
we have assumed that the gas velocity relative to the star at a large
distance will modify the accretion rate in fractal medium in a simi-
lar way that of in the continuous medium. Though a more detailed
analysis is required to get the Bondi-Hoyle accretion rate and, in
particular, F in turbulent medium, it is most likely that it will not
change our main result that the accretion rate will be proportional
to M3d−1.
c© yyyy RAS, MNRAS 000, 1–5
4 N. Roy
-1.5 -1.0 -0.5 0.0 0.5
log M
M ~ M1.49
M ~ M2.00
Figure 2. Comparison of data and theory for accretion rates of PMS stars
and brown dwarfs. The stellar mass is in M⊙ and the accretion rate is in
M⊙ yr
5 ASTROPHYSICAL IMPLICATIONS
Our results have important implications for a number of astro-
physical problems. One of these, for example, is the problem of
pre-main-sequence (PMS) accretion. Numerical simulation shows
that approximating the PMS accretion process as Bondi-Hoyle ac-
cretion leads to agreement between simulation and observation
(Padoan et al. 2005). For a solar mass star and for typical sound
speed a∞ ∼ 0.2 km s−1 in ambient molecular cloud filaments
(Padoan et al. 2005), a fractal scale length of ∼ 10 AU will be
smaller than the size of the sphere of influence and hence our ap-
proach should be valid in this particular case. Here we compare our
central result with observational data and numerical simulation re-
sults. For that we have taken the accretion rates of PMS stars and
brown dwarfs compiled by Padoan et al. (2005) (from Natta et al.
2004; White & Hillenbrand 2004; Muzerolle et al. 2004, and refer-
ences therein) includes all detections but no upper limit. Though
the the data do not provide any strong support for one theory as op-
posed to another one due to scatter in the accretion rate attributed
to (i) an age dependence of the accretion rate, (ii) variation of ρ∞,
a∞ and v∞ and (iii) interaction of accretion flow with jets and out-
flows on smaller scale (Padoan et al. 2005), we find that our the-
ory is consistent with these data. For example, the best fit value of
α is 1.49 ± 0.13 for log (Ṁaccr/M⊙ yr−1) > −10. Even when
we include all the compiled data of Padoan et al. (2005), the best
fit value of α is consistent with our theory within 3σ error for
2.5 . D . 2.7. In figure (2) we have shown the data and the best
fit for log (Ṁaccr/M⊙ yr
−1) > −10. Data from Muzerolle et al.
(2004) and White & Hillenbrand (2004) are shown as filled squares
and empty squares respectively and the rest of the data are shown
as empty circles. We also found that for higher accretion rate and
higher central mass, the exponent is significantly different from that
of the Bondi-Hoyle accretion. For a smaller central mass, the self-
gravity that we have neglected in our analysis may change the ac-
cretion rate significantly.
It is possible to use a high resolution simulation (e.g.
Krumholz et al. 2006) to find out the fractional dimensionality and
the scale length lc by using equation (11) and counting the num-
ber of particle n(R) on boxes of different scale length (R). On the
other hand, comparing the result with existing numerical simula-
tion results is not very straightforward as in most of the cases it is
assumed that Ṁaccr = AM
2 and the coefficient A is computed
for the assumed M2 dependence on accretion rate. But with the
existing published simulation results, it is still possible to check if
the variation of this coefficient A within computational uncertainty
can consistently accommodate our result and we found that the best
fit of the observational data and our theoretical estimate are within
1.5σ scatter around the computed accretion rate of Padoan et al.
(2005). On the other hand, to get a normalized mean accretion rate
(see Krumholz et al. 2006, for details) of 0.1 for a Mach number
of 5 and D = 2.55, we get, from Eqn.(28), rB/lc ∼ 17740. For
a solar mass star and for a typical sound speed a∞ ∼ 0.2 km s−1,
this translates to lc ∼ 1.3 AU. Fractal structure at this scale is not
observationally ruled out. But one should take this as an order of
magnitude estimate of the scale length as factors like velocity struc-
ture or magnetic field in real complicated astrophysical situations
may alter the accretion rate significantly.
The other potential implication is the black hole accretion. The
model growth of galactic center black holes assume that the black
holes accrete at Bondi rate (e.g., Springel, Di Matteo & Hernquist
2005). If the gas around the black hole is fractal in nature, then one
should rather use the modified accretion rate for fractal medium.
Here we would like to mention the caveat that there is no defini-
tive observational evidence against or for the fractal nature of the
medium in this case. Even if the medium is fractal, the validity of
our approach, as mentioned in §3, crucially depends on the mass of
the accretor, the scale length of the fractal and the ambient sound
speed.
6 DISCUSSION
We have derived the steady state hydrodynamic equations to de-
scribe the spherical accretion in a fractal medium by replacing the
fractal medium by a “fractional continuous” model. We have de-
rived the steady state accretion rate in terms of boundary condi-
tions at infinity in this simplified situation without considering the
self-gravity of the material for a medium where the dimensionality
is independent of position and orientation. Magnetic fields, which
we have not included in these models, will certainly play a major
role in determining the dynamics. But even without the inclusion
of magnetic field we have got the following results
We have found that there exists a unique solution with max-
imum mass accretion rate and the general nature of the solution
is similar to the “Bondi solution” (Bondi 1952) even for a fractal
medium with D = 3d < 3. The mass accretion rate, in cases, may
differ substantially depending on the fractional dimensionality. In
particular, the accretion rate is likely to be significantly different
from the “Bondi accretion rate” and will be proportional to MD−1
in case of accretion in fractional medium with scale length lc very
different from GM/a2∞.
One limitation of our analysis is that we have not considered
the self-gravity of the medium. This is justified provided the central
mass M is very large. In cases where self-gravitation is not negligi-
ble, it can change the accretion rate significantly. A more important
limitation is that the present analysis does not account for turbu-
lent velocity structure. It will certainly play an important role to
determine the accretion rate but is probably unlikely to change the
c© yyyy RAS, MNRAS 000, 1–5
Accretion in fractal media 5
mass dependence. In a real astrophysical situation with both density
and velocity structure in the medium, the analytical mass accretion
rate will hence not be applicable. Neglecting the magnetic fields, as
mentioned earlier, is the other major limitation of the analysis. The
presence of magnetic field is more likely to suppress the accretion
rate and magnetohydrodynamic simulation in fractal medium will
be required to make a more definitive statement. But the central re-
sult, that the accretion rate will be proportional to MD−1 will not
be affected by the addition of magnetic fields.
Our assumption of the equation of state of the form p = Kργ
may seems to be a considerable limitation but that is not likely
to be the case. In ordinary Bondi-Hoyle accretion in continuous
medium, changing the equation of state changes changes the ac-
cretion rate by a numerical factor of the order unity (Ruffert 1994;
Ruffert & Arnett 1994) and hence will not change the result much.
A more detail justification of the assumption can be found in
Krumholz, McKee & Klein (2006).
7 CONCLUSION
We have shown that the accretion rate onto a mass embedded in
a fractal medium may differ, in some cases, significantly from the
Bondi accretion rate even in the simplest situation. We have used
the simple model of accretion onto a mass from a fractal medium
of mass dimensionality D 6 3 and derived a self-consistent so-
lution that matches the “Bondi solution” as D → 3. The primary
result of our investigation is that theoretically accretion rate will
be proportional to MD−1. The observational accretion rate data
and the numerical simulation for particular astrophysical problem
of accretion onto PMS stars is consistent with our result. Our find-
ings suggest that the fractal structure of the medium around the
accreting mass is playing a major role to determine the accretion
rate and its dependence on the central mass. The agreement of the
theoretical prediction with existing numerical simulation implies
the consistency of the approach. A number of previously published
numerical and analytical results have not considered this effect and
may need to be reconsidered. We leave a more detailed treatment of
the problem, including the effects discussed in this work, the effect
of self-gravity and stability of fractal accretion, to a future work.
8 ACKNOWLEDGMENTS
We thank Paolo Padoan and Vasily E. Tarasov for useful discus-
sions and Paolo Padoan for kindly providing us the compiled data
of the accretion rate of PMS stars and brown dwarfs. We are grate-
ful to Rajaram Nityananda for reading an earlier version and sug-
gesting the dimensional argument as a cross check of the calcula-
tions. Thanks to Kandaswamy Subramanian for his comments on
an earlier version of the paper. We also thank Jayaram N. Chen-
galur, Raghunathan Srianand and Ranjeev Misra for useful discus-
sion. We are grateful to the anonymous referee for useful com-
ments and for prompting us into substantially improving this paper.
This research was supported by the National Centre for Radio As-
trophysics (NCRA) of the Tata Institute of Fundamental Research
(TIFR).
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This paper has been typeset from a TEX/ LATEX file prepared by the
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c© yyyy RAS, MNRAS 000, 1–5
	Introduction
	Dimensional Analysis of Fractal Accretion
	Fractional Integrals and Hydrodynamic Equations
	Mass accretion rate
	Astrophysical Implications
	Discussion
	Conclusion
	acknowledgments

  We use fractional integrals to generalize the description of hydrodynamic
accretion in fractal media. The fractional continuous medium model allows the
generalization of the equations of balance of mass density and momentum
density. These make it possible to consider the general case of spherical
hydrodynamic accretion onto a gravitating mass embedded in a fractal medium.
The general nature of the solution is similar to the "Bondi solution", but the
accretion rate may vary substantially and the dependence on central mass may
change significantly depending on dimensionality of the fractal medium. The
theory shows consistency with the observational data and numerical simulation
results for the particular case of accretion onto pre-main-sequence stars.

Introduction
	Dimensional Analysis of Fractal Accretion
	Fractional Integrals and Hydrodynamic Equations
	Mass accretion rate
	Astrophysical Implications
	Discussion
	Conclusion
	acknowledgments
