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Mass loss and dynamical friction on the Fornax dSph galaxy in the Milky Way potential

Published online by Cambridge University Press:  15 October 2026

Pierfrancesco Di Cintio Open the ORCID record for Pierfrancesco Di Cintio [Opens in a new window] ,
Giuliano Iorio ,
Carlo Nipoti Open the ORCID record for Carlo Nipoti [Opens in a new window]  and
Francesco Calura Open the ORCID record for Francesco Calura [Opens in a new window]
Pierfrancesco Di Cintio*
Affiliation:
CNR-Istituto dei Sistemi Complessi, Via Madonna del piano 10, I-50019 Sesto Fiorentino, Italy INFN-Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy INAF-Arcetri, largo E. Fermi 5, I-50125, Firenze, Italy
Giuliano Iorio
Affiliation:
Institute of Cosmos Science, University of Barcellona, Martí i Franquès, 1 08028 Barcelona, Spain INAF-Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy
Carlo Nipoti
Affiliation:
Dipartimento di Fisica e Astronomia “Augusto Righi” - Alma Mater Studiorum - Università di Bologna, via Gobetti 93/2, I-40129 Bologna, Italy
Francesco Calura
Affiliation:
INAF-Bologna, via Gobetti 93/3, I-40129, Bologna, Italy
*
email: pdicintio@gmail.com
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Abstract

We study the interplay between mass-loss and dynamical friction (DF) on the orbital decay of the Fornax dwarf spheroidal galaxy in the potential of the Milky Way (MW). Using a simplified single particle approach combined with a mass-loss rate extrapolated by $N-$body simulations we find that the effect of a time-dependent mass partially compensates DF, and typically produces a much less evident decay of the perigalactic distance, thus confirming that $N-$body simulations in smooth MW potentials without DF can be taken as a good model of the dynamics of dwarf satellite galaxies.

Keywords

stellar dynamicsgalaxies: kinematics and dynamicsmethods: N-body simulationsdiffusion.

Information

Type
Poster Paper
Information
Proceedings of the International Astronomical Union , Volume 21 , Symposium S398: Compact Objects and Binaries in Dense Star Clusters , September 2025 , pp. 216 - 219
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Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of The International Astronomical Union

1. Introduction

The Fornax dwarf spheroidal galaxy (dSph) has a remarkably high fraction of stellar mass locked in its globular clusters (GCs) system, thus putting at stake some GC formation models. It has been conjectured that an even larger fraction of stellar mass has been lost via tidal stripping.

Aiming at constraining the amount of (stellar) mass lost by Fornax in Di Cintio et al. (Reference Di Cintio, Iorio, Calura, Nipoti and Cantari2024), using high resolution upper N minus $N-$ body simulations combined with an a-posteriori mass component (stellar or dark matter) assignment scheme, we evolved the progenitor of the Fornax dSph along two Gaia DR3-based orbits in a static Milky Way (MW) potential. The core result of this work is that, even though, in principle, a broad range of initial stellar mass distributions for Fornax would be consistent with the present observed stellar density profile, the stellar mass loss must have been in all cases rather low (i.e. of the order of less than or equivalent to 3 percent sign $\lesssim 3\%$ ). As a consequence, it appears rather unlikely that the mass loss only could explain the high fraction of stellar mass associated to Fornax’s GCs system. Therefore the relative amount of field and GC stars remains a puzzle (despite the significant uncertainty in the adopted GC mass-to-light ratio) for GC formation channels that assume originally much more massive progenitors.

2. Methods and models

In line with Di Cintio et al. (Reference Di Cintio, Iorio, Calura, Nipoti and Cantari2024) here we assume the Johnston et al. (Reference Johnston, Spergel and Hernquist1995) MW model consisting of three components (disk, bulge and halo) with total gravitational potential normal upper Phi Subscript normal upper M normal upper W comma normal t normal o normal t Baseline left parenthesis upper R comma z right parenthesis equals normal upper Phi Subscript disk Baseline plus normal upper Phi Subscript bulge Baseline plus normal upper Phi Subscript halo $\Phi_{\rm MW,tot}(R,z)=\Phi_{\text{disk}}+\Phi_{\text{bulge}}+\Phi_{\text{halo}}$ composed of a Miyamoto and Nagai (Reference Miyamoto and Nagai1975) disk

(1) normal upper Phi Subscript disk Baseline left parenthesis upper R comma z right parenthesis equals minus StartFraction upper G upper M Subscript disk Baseline Over StartRoot upper R squared plus left parenthesis a plus StartRoot z squared plus b squared EndRoot right parenthesis squared EndRoot EndFraction \begin{equation} \Phi_{\text{disk}}(R,z)= -\frac{GM_{\text{disk}}}{\sqrt{R^2 + \big(a+\sqrt{z^2 + b^2}\big)^{2}}} \end{equation}

and a spherical Hernquist (Reference Hernquist1990) bulge

(2) normal upper Phi Subscript bulge Baseline left parenthesis r Subscript normal upper G Baseline right parenthesis equals minus StartFraction upper G upper M Subscript bulge Baseline Over r Subscript normal upper G Baseline plus c EndFraction \begin{equation} \Phi_{\text{bulge}}(r_{\rm G})= -\frac{GM_{\text{bulge}}}{r_{\rm G}+c} \end{equation}

embedded in a spherical Binney (Reference Binney1981) logarithmic halo

(3) StartLayout 1st Row normal upper Phi Subscript halo Baseline left parenthesis r Subscript normal upper G Baseline right parenthesis equals v Subscript halo Superscript 2 Baseline ln left parenthesis r Subscript normal upper G Superscript 2 Baseline plus d squared right parenthesis period EndLayout \begin{align} \Phi_{\text{halo}}(r_{\rm G})= v^2_{\text{halo}} \ln{( r_{\rm G}^2 +d^2 )}. \end{align}

In the equations above upper R equals StartRoot x squared plus y squared EndRoot $R=\sqrt{x^2+y^2}$ and z are Galactocentric cylindrical coordinates, and r Subscript normal upper G Baseline equals StartRoot upper R squared plus z squared EndRoot $r_{\rm G}=\sqrt{R^2+z^2}$ is the Galactocentric distance, with a equals 6.5 $a=6.5$ kpc, b equals 0.26 $b=0.26$ kpc, c equals 0.7 $c=0.7$ kpc, d equals 12 $d=12$ kpc, upper M Subscript bulge Baseline equals 3.4 times 10 Superscript 10 Baseline upper M Subscript circled dot $M_{\text{bulge}}=3.4\times 10^{10}\text{ M}_{\odot}$ , upper M Subscript disk Baseline equals 10 Superscript 11 Baseline upper M Subscript circled dot $M_{\text{disk}}=10^{11}\text{ M}_{\odot}$ and v Subscript halo Baseline equals 128 $v_{\text{halo}}=128$ km/s.

To include the effect of Chandrasekhar (Reference Chandrasekhar1943) DF on the orbit of the Fornax dSph, here we perform single particle integrations, where a test mass upper M Subscript normal t normal o normal t $M_{\rm tot}$ orbits in the MW potential according to the equations of motion

(4) ModifyingAbove bold r With two dots equals minus nabla normal upper Phi Subscript normal upper M normal upper W comma normal t normal o normal t Baseline left parenthesis bold r right parenthesis plus bold a Subscript upper D upper F Baseline comma \begin{equation}\ddot{\mathbf{r}}=-\nabla\Phi_{\rm MW,tot}(\mathbf{r})+\mathbf{a}_{DF},\end{equation}

augmented by the deceleration produced by DF through

(5) bold a Subscript normal upper D normal upper F Baseline equals minus 2 pi upper G squared upper M Subscript normal t normal o normal t Baseline rho left parenthesis bold r right parenthesis ln left parenthesis 1 plus normal upper Lamda squared right parenthesis left bracket normal e normal r normal f left parenthesis upper X right parenthesis minus StartFraction 2 upper X squared Over StartRoot pi EndRoot EndFraction exp left parenthesis minus upper X squared right parenthesis right bracket StartFraction bold v Over v cubed EndFraction period \begin{equation}\mathbf{a}_{\rm DF}=-2\pi G^2 M_{\rm tot}\rho(\mathbf{r})\ln(1+\Lambda^2)\Bigg[{\rm erf}(X)-\frac{2X^2}{\sqrt{\pi}}\exp(-X^2)\Bigg]\frac{\mathbf{v}}{v^3}.\end{equation}

In the equation above rho left parenthesis bold r right parenthesis $\rho(\mathbf{r})$ is the total MW mass density at bold r $\mathbf{r}$ , bold v equals ModifyingAbove bold r With dot $\mathbf{v}=\dot{\mathbf{r}}$ and upper X equals v divided by left parenthesis StartRoot 2 EndRoot sigma right parenthesis $X=v/(\sqrt{2}\sigma)$ , where sigma left parenthesis bold r right parenthesis $\sigma(\mathbf{r})$ is the velocity dispersion at bold r ${\mathbf{r}}$ (i.e. we assume a local Maxwellian approximation ignoring the MW velocity distribution function) as obtained by solving the Jeans equations for the MW potential and density (e.g. see Dehnen et al. Reference Dehnen, McLaughlin and Sachania2006). In Equation (5), in order to take into account that the satellite is extended (e.g. see Petts et al. Reference Petts, Read and Gualandris2016), we define the ratio normal upper Lamda $\Lambda$ entering the Coulomb logarithm as

(6) normal upper Lamda equals r Subscript normal upper G Baseline divided by normal m normal a normal x left bracket r Subscript normal s normal a normal t Baseline comma StartFraction upper G upper M Subscript normal t normal o normal t Baseline Over v squared EndFraction right bracket comma \begin{equation}\Lambda={r_{\rm G}}/{{\rm max}\left [r_{\rm sat},\frac{G M_{\rm tot}}{v^2}\right]},\end{equation}

where r Subscript normal upper G $r_{\rm G}$ and r Subscript normal s normal a normal t $r_{\rm sat}$ are the scale radii of the Galaxy and the satellite, which we fix to 12 and 4.3 kpc, respectively. upper N minus $N-$ body simulations performed in a static MW potential revealed that the mass loss rate experienced by the Fornax progenitor over 12 Gyr is nearly constant in time for a broad range of putative orbits. In Fig. 1 we show the evolution of the mass within a fixed radius of 3 kpc around the centre of Fornax for the orbits O1 and O2 discussed in Di Cintio et al. (Reference Di Cintio, Iorio, Calura, Nipoti and Cantari2024). The dotted lines mark the best linear fits with the relation

(7) upper M left parenthesis t right parenthesis equals upper M 0 minus ModifyingAbove upper M With dot t \begin{equation}M(t)=M_0-\dot{M}t\end{equation}

yielding the two mass loss rates ModifyingAbove upper M With dot almost equals 3.4 times 10 Superscript negative 3 $\dot{M}\approx3.4\times 10^{-3}$ and 1.1 times 10 Superscript negative 2 Baseline left bracket 10 Superscript 10 Baseline upper M Subscript circled dot Baseline normal upper G normal y normal r Superscript negative 1 Baseline right bracket $1.1\times 10^{-2}[10^{10}M_{\odot}{\rm Gyr}^{-1}]$ .

Figure 1.

Evolution of the mass within a control radius of 3 kpc for two different upper N minus $N-$ body simulations of the Fornax dSph in a smooth MW potential (coloured solid lines) and their linear fit (black dotted lines).

Line graph showing mass evolution within a 3 kpc radius for two simulations of the Fornax dSph.

3. Numerical simulations and discussion

We integrated backward in time from t equals 12 $t=12$ Gyr to t equals 0 $t=0$ via Eq. (4) the orbit of a single particle representing Fornax using a modified leapfrog scheme (Mikkola and Merritt Reference Mikkola and Merritt2006) starting from the same two estimates of the present day phase-space coordinates used in our upper N minus $N-$ body simulations, with and without mass loss. In both cases we start by assuming upper M Subscript normal t normal o normal t Baseline equals 5.7 times 10 Superscript 9 Baseline upper M Subscript circled dot $M_{\rm tot}=5.7\times 10^9M_{\odot}$ at t equals 12 $t=12$ Gyr, that is, the present-day total mass of Fornax as estimated in our upper N minus $N-$ body simulations. Additionally, we have also propagated from the same sets of coordinates a particle of constant mass upper M Subscript normal t normal o normal t Baseline equals 1.13 times 10 Superscript 10 Baseline normal upper M Subscript circled dot $M_{\rm tot}=1.13\times 10^{10}{\rm M}_\odot$ , corresponding to the heaviest Fornax progenitor used in our upper N minus $N-$ body study. Figure 2 shows the Galactocentric distance r Subscript normal upper G $r_{\rm G}$ as a function of time for the two choices of present day coordinates and the different simulation set-ups. It appears evident that allows the mass to vary in time (purple lines, note that in backward integrations upper M Subscript normal t normal o normal t $M_{\rm tot}$ increases), in both simulations partially compensating the effect of DF, in particular with respect to the fixed mass runs with an overestimated present day mass (indicated by the blue dashed lines), thus implying a much lower apogalactic distance for the Fornax progenitor.

Figure 2.

Evolution of the Galactocentric distance r Subscript upper G $r_G$ of the Fornax dSph for two different models. The blue and cyan dashed lines refer to fixed mass single particle integrations with an artificially overestimated and a realistic present-day Fornax mass, respectively 1.13 times 10 Superscript 10 Baseline normal upper M Subscript circled dot $1.13\times 10^{10}{\rm M}_\odot$ and 5.7 times 10 Superscript 9 Baseline normal upper M Subscript circled dot $5.7\times 10^{9}{\rm M}_\odot$ . The purple solid lines refer to backward in time propagation where the mass loss has been accounted for. As a reference, the heavy black line marks the orbits propagated in the frozen MW potential without DF, and the yellow dots correspond to upper N minus $N-$ body simulations in the same frozen potential. Here t equals 12 $t=12$ Gyr is today.

Two line graphs depict the evolution of Galactocentric distance over time for different models.

For comparison, we have also reported the evolution of r Subscript normal upper G $r_{\rm G}$ in the upper N minus $N-$ body simulations (yellow dotted curves) where MW DF was not implemented. Remarkably, since the interplay between mass loss (nearly 10% for O1 and 40% for O2) and DF results in a much milder orbital decay, we obtained further confirmation that our upper N minus $N-$ body simulations neglecting the DF exerted by the Galaxy can be safely taken as a reasonable model of Fornax evolution along a family of Gaia-based reference orbits. We note that, as shown in Miller et al. (Reference Miller, van den Bosch, Green and Ogiya2020), under certain conditions the mass loss can sometimes enhance the orbital decay of a satellite system with respect to the simple point-particle with constant mass approximation. This is the case when extended tidal tails exert an additional torque on the satellite’s orbit inducing the dissipation of orbital kinetic energy. This so-called tidal self-friction is therefore partially compensating the mitigation of the standard DF caused by mass loss. However, for the specific choices of initial conditions used in Di Cintio et al. (Reference Di Cintio, Iorio, Calura, Nipoti and Cantari2024), the tidal self-friction appears to be rather mild as no appreciable decay of the pergalactic distance was detected. In order to disentangle the effect of mass stripping and host galaxy DF one should in principle run upper N minus $N-$ body simulations in a galactic model incorporating DF, in a similar fashion to what has been carried out in Giunchi et al. (Reference Giunchi, Marinacci, Nipoti, Claeyssens, Pascale, Calura and Ragagnin2025), where the sinking stellar system suffers the drag exerted by the host system on its center of mass. A numerical study of the strength of the DF force as a function of the satellite size and (possibly time dependent) mass is currently underway.

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Figure 0

Figure 1. Evolution of the mass within a control radius of 3 kpc for two different N−$N-$body simulations of the Fornax dSph in a smooth MW potential (coloured solid lines) and their linear fit (black dotted lines).

Figure 1

Figure 2. Evolution of the Galactocentric distance rG$r_G$ of the Fornax dSph for two different models. The blue and cyan dashed lines refer to fixed mass single particle integrations with an artificially overestimated and a realistic present-day Fornax mass, respectively 1.13×1010M⊙$1.13\times 10^{10}{\rm M}_\odot$ and 5.7×109M⊙$5.7\times 10^{9}{\rm M}_\odot$. The purple solid lines refer to backward in time propagation where the mass loss has been accounted for. As a reference, the heavy black line marks the orbits propagated in the frozen MW potential without DF, and the yellow dots correspond to N−$N-$body simulations in the same frozen potential. Here t=12$t=12$ Gyr is today.