c ESO 2010

Astronomy & Astrophysics manuscript no. 15330
October 1, 2010

An analytical approach to the dwarf galaxies cusp problem
R. S. de Souza1,2 and E. E. O. Ishida1,2
1IAG, Universidade de São Paulo, Rua do Matão 1226, Cidade Universitária,
CEP 05508-900, São Paulo, SP, Brazil
2IPMU, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
Released Xxxxx XX
ABSTRACTAims. An analytical solution for the discrepancy between observed core-like profiles and predicted cusp profiles in dark
matter halos is studied.
Methods. We calculate the distribution function for Navarro-Frenk-White halos and extract energy from the distribution, taking into account the effects of baryonic physics processes.
Results. We show with a simple argument that we can reproduce the evolution of a cusp to a flat density profile by adecrease of the initial potential energy.
Key words. galaxies: formation, dwarf, cosmology: large scale structure of Universe

1. Introduction
There are many dwarf galaxies in the Universe (e.g. Marzke& da Costa (1997); Sawala et al. (2010)), and the Local
Group is an excellent place to study these systems and theirproperties (Mateo 1998; Pasetto et al. 2010).
The nearly solid body rotation curve observed in dwarf
galaxies indicates a central core in the dark matter distribution (e.g., Burkert (1995); de Blok & Bosma (2002); Gentileet al. (2005); de Blok (2005)). These results disagree withpredictions from numerical simulations, which require density profiles with cusps (e.g. Navarro et al. (1997); Mooreet al. (1999); Navarro et al. (2004)).
Navarro et al. (1996) were one of the first to point out
the possibility that feedback mechanisms can turn the central dark-matter cusp into a cored one. Following this line ofinvestigation, Gnedin & Zhao (2002) analyzed the influenceof winds. Read & Gilmore (2005) discussed how external
impulsive mass-loss events can successfully act as a flattener for the central density cusps, and Mashchenko et al.
(2006) showed that random bulk motions of gas in small primordial galaxies could flatten the central dark matter cuspin ∼ 108 years. Beyond that, several authors have suggested
that the interstellar medium (ISM) of dwarf galaxies systems could be entirely removed as a result of star formation(Dekel & Silk 1986; Mori et al. 1997; Murakami & Babul
1999; Mac Low & Ferrara 1999; Mori et al. 2002; Hensler
et al. 2004; Mori et al. 2004).
In the present work, we construct a simple model under the hypothesis that astrophysical processes are able toremove baryonic gas from dwarf galaxies. We show for the
first time a clear and analytical argument to solve the cusp
problem. We argue that baryonic feedback could changethe distribution function on the phase space of energy andangular momentum, flattening the density profile.
Send offprint requests to: Rafael S. de Souza e-mail:rafael@astro.iag.usp.br

This paper is organized as follows. In Sect. 2, we briefly
discuss the removal of baryonic gas from dark matter halos as a consequence of different astrophysical processes. InSect. 3, we review the formalism of distribution functions.
In Sect. 4, we discuss the changes in density profile caused
by gas removal and show our results. Finally we present ourconclusions in Sect. 5.

2. Expulsion of baryonic gas
Our main goal is to give an analytical explanation to theflat density profile observed in dwarf galaxies. Our starting point will be the results obtained from ΛCDM numerical simulations (Moore et al. 1999; Navarro et al. 2004).
In this context, we begin with a dwarf galaxy that has aNavarro-Frenk-White profile (NFW) (Navarro et al. 1997)
after the formation of its first stellar population, and study
the signatures left in the dark matter profile when part ofits baryons are removed. The first natural question is: whyare the baryons removed?
Before answering that question, it is important to emphasize that a cusp profile is not only characteristic, butnecessary. In the initial phase of galaxy formation, when
only a small part of the stellar population is formed, a deep
gravitational potential well is needed in order to trap thebaryons (Mori et al. 1997; Bradamante et al. 1998). Beyond
that, baryon accretion in the center of the halo will suffer
adiabatic contraction and drag the dark matter profile into
an even steeper one (Blumenthal et al. 1986).
On the other hand, the baryonic characteristic of thegas causes it to radiate energy, increasing the entropy ofthe dark matter background and leaving strong signaturesin the density profile (El-Zant et al. 2001; Romano-Dı́azet al. 2008). In this context, also the effects of gas bulk
motions need to be considered, as shown in Mo & Mao(2002) and Mashchenko et al. (2006).
After this initial collapse, the galaxy will go throughone (or probably more) star-formation bursts. Mac Low
1Article published by EDP Sciences and available at http://www.aanda.org
To be cited as: A&A preprint doi http://dx.doi.org/10.1051/0004-6361/201015330

de Souza & Ishida: The dwarf galaxies cusp problem& Ferrara (1999) and Pasetto et al. (2010) showed that SF
alone can remove almost all the interstellar baryonic gas inside dwarf galaxies, for Mhalo ≤ 107 M⊙ . Gelato & SommerLarsen (1999) explored how different mass ejection fractions
and disk density distribution could fit the observed rotation
curve of the dwarf galaxy DDO 154. Hayashi et al. (2003)
demonstrated that the mass loss can vary from 5% to 95%
as a result of tidal effects caused by a host larger galaxy.
Below we will consider only astrophysical effects thatare able to remove from a small percentage up to all thebaryonic gas initially trapped by the dark matter halo.
Following Pasetto et al. (2010), we estimate that 1/50 of
the total mass in dwarf galaxies is visible. Moreover, 15%of this visible mass is in form of stars and 85% is in formof interstellar gas. We consider that the astrophysical processes cited before are able to remove from 5% to 100% ofthe gas in the galaxy halo. This means that for halos withMhalo ∼ 107 M⊙ , from 0.085% to 1.7% of the total halo
mass can be removed.
Note that the crucial point in our analysis is the percentage of total halo mass lost. All the other astrophysical mechanism cited above that do not necessarily remove
baryons are not considered here. Our results are valid forany mass loss mechanisms, as long as they expel the same
percentage of baryons out of the dark matter halo potentialwell.

3. Dark matter density profile
At a given time, t, a full description of any collisionless system can be obtained by specifying the number of particles,f (x, v, t)d3 xd3 v, that have positions in the small volume
d3 x centered in x and small range of velocities d3 v centeredin v. f (x, v, t) is called the distribution function, hereafter
DF (Binney & Tremaine 2008). The DF of a mass distribution in a steady state is related to its density profile throughZρ(r) = f (r, v)d3 v .
(1)
For any general spherical system, we can define a DF,
f (E, L), which depends on the energy, E, and the angularmomentum absolute value, L. Osipkov (1979) and Merritt(1985) found that in anisotropic spherical systems, f canbe expressed as:f (Q, L2 ) = F (Q) L2α ,(2)where α is a real number andL2,(3)
2ra2
for a radial parameter ra , known as the anisotropy radius.
It is useful to define the relative potential, Ψ = −φ, where
φ is the gravitational potential.
For distribution functions that can be written in theform of Eq. (2), the fundamental integral equation can bewritten as (Cuddeford 1991; Binney & Tremaine 2008)
Q=E−Z2π Ψρ(r) = 2F (Q) dQ(4)r 0
Z 2r2 (Ψ−Q)/(1+ r22 )raL2α dL2q.
×202(Ψ − Q) − (L2 /r2 )(1 + rr2 )α2After performing the integration over L, we obtain a single
integral equation for F (Q),Z Ψ(Ψ − Q)α+1/2 F (Q)dQ ,(5)ρ(Ψ) = Θ(α)
0whereΘ(α) = 2

α+3/2 3/2π1−αr2
Γ(α + 1) 2α
1+ 2r.
Γ(α + 3/2)rα(6)For a general α, Eq. (5) can be inverted, resulting inZ Qsin(n − 1 − α)π dn+1ρ(Ψ)
F (Q) =dΨ,πΘ(α)(α + 1/2)! dQn+1 0 (Q − Ψ)α+ 32 −n(7)with n defined byn ≡ [α + 1/2] + 1,(8)and [α + 1/2] representing the largest integer less than orequal to (α + 1/2).
From the above expressions, it is clear that any perturbation in E and/or L will change the DF (Eq. (2)) and,
consequently, leave a signature in the galaxy density profile (Eq. (1)). Our strategy in this work is to start with aNFW profile in Eq. (7), find it a suitable DF and analyze
the change in the density profile caused by perturbations inthe DF. Although, as we initially proposed an entirely analytical calculation, we must express the NFW profile in a
way that makes a completely analytical approach possible.
This is done in the next section.

4. Erasing the cusp
4.1. Spherical density profiles
In order to evaluate the distribution function for a cusp-likeprofile, we need to consider an initial density profile witha central cusp and place it into Eq. (7). For simplicity, weadopt the following family of spherical potentials (RindlerDaller 2009),
φ0φ(r) = −(9)η γ ,
1 + rbwhere φ0 is the value of potential inside the halo radius, η, γ,and b are positive parameters. Using the Poisson equation,
we found the dimensionless density profile corresponding tothis potential,
ρe(r) =bηγ+2 (1 + η)bη + (1 − ηγ)rη
.
1+ηr2−η (bη + rη )γ+2(10)e ≡ −φ(r)/φ0 ,Defining Ψ
we can also write the density as function of the relative
potential,
nhie = 1e γ1 +1 + η(γ + 1)Ψ
e γ2 +1 ×ρe(Ψ)(1 − ηγ)Ψ
η+1
1− η2 −1/γe,(11)−1× Ψ
where the tilde denotes dimensionless quantities. The Eq.
(11) is general enough to be applied to any spherical density profile, given an appropriate choice of parameters. The
advantage of this expression is that it permit us to writee which is not always analytically possible.
ρe in terms of Ψ,However, we can always find a fit of Eq. (11) corresponding
to a specific density profile.

de Souza & Ishida: The dwarf galaxies cusp problem

1001001010ΡΡs1000

ΡΡs

1000

10.1
0.01
0.001

10.10.01

0.10.01
0.001

1rrs
Fig. 1. Reconstructed density profile (red dashed line) comparedwith original NFW profile (blue solid line).

0.01

0.1
rrs

1

Fig. 2. New equilibrium configuration of the perturbed halo removing 0.1% (red dot-dashed line) and 1.7% (dashed green line)of total energy Q, compared to the NFW profile (blue solid line).

4.2. Connection to the Navarro-Frenk-White profile
One of the strongest predictions from cosmological simulations is the universal density profile present in dark matter
halos (Navarro et al. 1997),
ρ(r) =ρs,r/rs(1 + r/rs)2(12)where ρs is de central density and rs is the scale radius ofthe halo.
Using η = 1, b = 1 and γ = 1/2, Eq. (10) becomesnumerically similar to NFW profile, as we can see in Fig.
e Eqs. (10) and (11)
1. Given this choice of parameters, Ψ,
become1eΨ(r)= √,r+1

and

ρe(r) =(13)r+4,4r(r + 1)5/2

e =ρe(Ψ)(14)

e7 + Ψe53Ψ.
e24 − 4Ψ(15)In order to test our calculations, we first found a DFfor the NFW profile and reconstructed the density profile
using the Eq. (15) in Eq. (7). We found that for α ∼ 1/2
and rα ∼ 1, Eq. (7) agrees well with the DF for NFWprofile. With these parameter values we have

F (Q) = −Q3 15Q6 − 39Q4 + 27Q2 + 532π 2 (Q2 − 1)Θ(1/2)−1 . (16)The above expression placed into Eq. (5) recovers Eq. (15).
We suppose that the macroscopic variables, like radial density profile, will change if the halo experimentsa physical process that perturbs its DF. We consideredthat the removal of baryonic gas will shift the energy ofthe system by Q → Q ± ∆Q. According to the arguments in Sect. 2, SF and external tidal effects can remove(0.085 − 1.7)% of the halo gravitational potential energy. Inorder to find a new DF configuration, we have to evaluate
F (Q−∆Q) ≈ F(Q)−∆Q ∂F∂Q . Which is just a solution of the∂Q ∂F
Boltzmann equation, ∂F∂t + ∂t ∂Q = 0, if we consider thatthe gas removal occurs in a short time interval, ∆t ≈ dt.
The results are presented in Fig. 2, where we show thenew configuration of the density profile after remotion of(0.085 − 1.7)% from gravitational potential energy. It is evident from this figure that a little change in the total energyof halo can flatter the profile in the inner region. Note thatthe core-like behavior we were looking for is already evidentfor ∆Q ≈ 0.001. As a consequence, any feedback capable ofremoving ≈ 10% of interstellar gas would be able to smooththe matter density profile.

5. Conclusions
The N-body simulations based on collisionless cold dark
matter do not show the core-like behavior observed in local
dwarf galaxies. Instead, they are better described by a steep
power-law mass density distribution called cusp. Several
attempts were made with numerical simulations in order totake into account the gas dynamics in the evolution of dark
matter profiles.
Considering that the total energy of the halos willchange because of baryonic feedback, we showed that it ispossible to model the evolution of an initial cusp-like profile
into a core-like one. We used an analytical analysis basedon the density profile DF.
The advantage of this approach is that we obtain results similar to those extracted from numerical simulations,which take into account the gas dynamics (Pasetto et al.
2010; Governato et al. 2010) in a very simple way. On top of
that, considering the baryon loss as a mechanism of flattening the dark matter density profile of dwarf galaxies allows
us to support the low amount of baryons observed in dwarf
spheroidal galaxies (Gilmore et al. 2007).
However, as discussed in the introduction, there are twocompetitive processes occurring during the formation ofgalaxies, baryon accretion in the center of the halo will suffer adiabatic contraction and drag the dark matter profile
into an even steeper one, and on the other hand, the baryonic characteristic of the gas causes it to radiate energy,
increasing the entropy of the dark matter background andpreventing the contraction. As our knowledge about these3

de Souza & Ishida: The dwarf galaxies cusp problem

two processes is still incomplete, it is important to take into
account these effects in future works.
Acknowledgements. R.S.S. thanks the Brazilian agencies FAPESP(2009/05176-4) and CNPq (200297/2010-4) for financial support.
E.E.O.I. thanks the Brazilian agency CAPES (1313-10-0) for financial support. We also thank the anonymous referee for fruitful comments and suggestions. This work was supported by World PremierInternational Research Center Initiative (WPI Initiative), MEXT,
Japan.

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