# Ratio of kinetic luminosity of the jet to bolometric luminosity of the disk
at the “cold” accretion onto a supermassive black hole

S.V.Bogovalov  National Research Nuclear University (MEPHI), Kashirskoje
shosse, 31, Moscow, Russia

(25.06.2018)

###### Abstract

Disk accretion onto black holes is accompanied by collimated outflows (jets).
In active galactic nuclei (AGN), the kinetic energy flux of the jet may exceed
the bolometric luminosity of the disk a few orders of magnitude. This
phenomena can be explained in frameworks of so called “cold” disk accretion
when the only source of energy of AGN is the energy released at the accretion.
The radiation from the disk is suppressed because the wind from the disk
carries out almost all the angular momentum and the gravitational energy of
the accreted material. In this paper, we calculate an unavoidable radiation
from the “cold” disk and the ratio of the kinetic energy luminosity of the
outflow to the bolometric luminosity of the accretion disk around a super
massive black hole in framework of the paradigm of the optically thick
$\alpha$-disk of Shakura & Sunyaev. The ratio of the luminosities is defined
predominantly by the ratio of the magnetic field pressure inside the disk to
the magnetic field pressure at the base of the wind. The obtained equations
applied to the jet of M87 demonstrate good agreement with observations. In the
case of Sgr A*, these equations allow us to predict the kinetic energy flux
from the disk around Galactic SMBH.

###### pacs:

98.62.Js;98.62.Mw;98.62.Nx

##  I Introduction

In the standard theory of disk accretion onto a black hole Shakura and Sunyaev
(1973) is assumed that the gravitational energy of the accreted material is
entirely carried out by photons radiated from the surface of the optically
thick disk. Measurements of the bolometric luminosity of the disk gives total
energy released at the accretion. However, the disk accretion can be
accompanied by formation of collimated outflows (jets). Moreover, the
observations of recent years show that the power of kinetic energy of the jets
in some AGN exceeds the bolometric luminosity of the disk.

The famous galaxy M87 is a striking example of an AGN with a very large
kinetic-to-bolometric luminosity. According to Biretta  _et al._ (1991), the
bolometric luminosity of the disk does not exceed $10^{42}~{}\rm erg/s$ while
the kinetic luminosity of the jets is of the order of $10^{44}~{}\rm erg/s$
Bicknell and Begelman (1996); Reynolds  _et al._ (1996).

Starting with the paper Rawlings and Saunders (1991), the energetics and
lifetimes of extended double radio sources have been used for calculation of
the jet power in radio galaxies and quasars. The ratio of kinetic-to-
bolometric luminosity can be estimated from radio and X-ray data. Heinz and
Sunyaev (2003); Merloni  _et al._ (2003); Falcke  _et al._ (2004); Körding
_et al._ (2006); Merloni and Heinz (2007); de Gasperin  _et al._ (2011);
Saikia  _et al._ (2015) argued that the radio and X-ray luminosities are
likely to be related to the kinetic and bolometric luminosities, respectively.
It is established that in large fraction of AGNs the jet kinetic luminosity
exceeds the bolometric luminosity Körding  _et al._ (2008); Ma  _et al._
(2008); López-Corredoira and Perucho (2012); Daly (2016); Daly  _et al._
(2016); Fernandes  _et al._ (2011); Punsly (2011).

The jet power in 191 quasars detected by the Fermi Large Area Telescope (LAT)
in gamma rays, calculated within the frameworks of an one-zone model show that
in this sample of blazars $L_{kin}$ systematically exceeds the bolometric
luminosity Ghisellini  _et al._ (2014).

An indirect evidence of high kinetic luminosity of an outflow exceeding the
bolometric luminosity is provided by observations of the Galactic Center in
TeV gamma-rays HESS Collaboration  _et al._ (2016). To explain the observed
diffuse flux of the VHE gamma-rays from the Galactic Center region, the
production rate of protons accelerated up to 1 PeV, should be $\sim
10^{38}~{}\rm erg/s$. Assuming that the accelerator of protons is powered by
the kinetic energy of the outflow (a wind or jets) from the super massive
Black Hole in the the Galactic Center (Sgr A*), even in the case of 100 %
conversion of the bulk kinetic energy to non thermal particles, the kinetic
luminosity of the outflow would two orders of magnitude exceed the bolometric
luminosity of Sgr A* which is estimated close to $10^{36}~{}\rm erg/s$ Genzel
_et al._ (2010).

Observations of the very powerful and bright in gamma-rays AGN 3C 454.3 give
even more interesting information. During the outbursts of this object, its
apparent luminosity in GeV gamma-rays could exceed $10^{50}~{}\rm erg/s$
Striani  _et al._ (2010); Ackermann  _et al._ (2010); Vercellone  _et al._
(2011); Abdo  _et al._ (2011). The mass of the black hole in this AGN is
estimated in the region $(0.5-4)\cdot 10^{9}~{}\rm M\odot$. Thus the Eddington
luminosity is in the range of $(0.6-5)\cdot 10^{47}~{}\rm erg/s$. Because of
the Doppler boosting effect, the intrinsic gamma-ray luminosity of this source
is much smaller, by several orders of magnitude, than the apparent luminosity.
Yet, the estimates of the jet kinetic luminosity in an any realistic scenario
give a value exceeding the Eddington luminosity Khangulyan  _et al._ (2013).

In general, the estimates of the bolometric and kinetic luminosities are model
dependent Pjanka  _et al._ (2017). Nevertheless, it is difficult to avoid a
conclusion that at least some AGN demonstrate extremely high kinetic
luminosities of jets which are not only above the bolometric luminosity, but
in some cases can exceed the Eddington luminosity of the central super massive
black hole (SMBH). In this case the question about the source of the energy of
the jets becomes the central one for understanding of the processes at the
accretion onto SMBH McNamara  _et al._ (2011).

Presently, the rotational energy of a black hole is considered as the most
likely source of energy which is transformed in to the energy of jets due to
the so-called mechanism of Blandford and Znajek Blandford and Znajek (1977).
In this scenario, SMBH provides an additional (to the accretion) source of
energy which, in fact, could be the dominant source of energy of the system.
According to numerical simulations Tchekhovskoy  _et al._ (2011); Barkov and
Khangulyan (2012) this mechanism can provide the energy flux in the jet
$\approx 3\dot{M}c^{2}$ , where $\dot{M}$ is the accretion rate. But the black
hole has to rotate with maximal possible angular momentum. In this case the
radiation luminosity of the optically thick disk can achieve
$\eta\dot{M}c^{2}$ with $\eta=0.3$. Thus, the maximal ratio of the kinetic-to-
bolometric luminosities is close to 10. This value can be increased if the
accretion occurs in radiatively inefficient regime Shapiro  _et al._ (1976);
Rees  _et al._ (1982); Narayan and Yi (1995). In this regime the luminosity of
the disk is suppressed because the energy released at the accretion is
predominantly advected in to the black hole. But the alternative model is also
possible in which the only source of the energy of AGN is the energy released
at the accretion .

It was firstly pointed out in Pelletier and Pudritz (1992) that the magnetized
wind from the disk can carry out a significant fraction of the angular
momentum of the accreted matter rather than the viscous stresses. This
splendid idea has been explored in a range of papers by Grenoble group
Ferreira and Pelletier (1993, 1995); Ferreira (1997) which called this type of
flow around black holes as Magnetized Accretion-Ejection structures (MAES). A
range of other authors last years explored similar approach Cao and Spruit
(2013); Li and Begelman (2014); Li (2014) in different physical context. In
the last works of the Grenoble group Marcel  _et al._ (2018a, b) the radiation
from the Jet Emitting Disks (JED) is discussed. But the discussion is limited
by the case when the jet carries out a moderate fraction (20%-80%) of the
energy released at the accretion.

Starting with our first work Bogovalov and Kelner (2010) devoted to this
problem we stressed that the idea of Pelletier and Pudritz (1992) can be
applied to the solution of the problem of high ratio of the kinetic luminosity
of the jets to the bolometric luminosity of the disks. Actually Ferreira and
Pelletier (1995); Ferreira  _et al._ (2006) specially stressed much before us
that at the conventional conditions the jet can carry out almost all the
angular momentum and energy from the accretion disk. The radiation of the disk
can carry out only a minor fraction of the accretion energy. Therefore, the
ratio of the kinetic luminosity of the jets on a few orders of magnitude can
exceed the bolometric luminosity of the disks in this regime of accretion. We
called this type of accretion as ”dissipationless“ or ”cold“ accretion because
the dissipative processes like turbulent viscosity play no role in the
dynamics of the accretion. Two questions arise in this regard:

  * •

Do solutions describing the regime of ”cold“ accretion exist?

  * •

If yes then what is the expected ratio of the kinetic-to-bolometric
luminosity? Is it really possible to have very high ratio of the kinetic-to
bolometric luminosity exceeding 100?

Presently, we have a positive answer to the first question. The selfconsistent
solution of the problem of accretion and outflow has been obtained in Ferreira
and Pelletier (1995) in the self similar approximation. This allowed to the
authors to calculate the structure of the accretion disk assuming that the
magnetic field is formed due to advection and diffusion Ferreira (1997); Casse
and Ferreira (2000).

In the work by Bogovalov and Kelner (2010) a self similar solution of the
problem of ”cold“ accretion has been obtained neglecting the structure of the
accretion disk. From one side this simplified the model but from other side
this approach allows us to develop a method of numerical solution of the
problem without self similarity assumptions and obtain more general numerical
solutions Bogovalov and Tronin (2017, 2018). All these works remain no doubts
that the ”cold“ accretion due to magnetized wind is really possible.

In this paper we try to answer to the second question. ”Cold“ or Jet Emitting
Disks Ferreira  _et al._ (2006) can formally exist without any dissipation
processes (viscosity and finite electric conductivity) and therefore formally
can remain cold. But in reality the heating of the disk is unavoidable because
the ”cold“ disk accretion is possible only in presence of magnetic field at
the base of the wind. Therefore, the magnetic field has to be present also
inside the disk producing magnetic and turbulent viscosity. The objective of
the work is to estimate the heating of the disk due to turbulence and
determine the expected level of the ratio of the kinetic-to-bolometric
luminosity of the ”cold“ disks.

##  II Physics of the ”cold“ accretion

In the work of the Grenoble group the magnetic field in the disk and outflow
has formed under advection and diffusion of the external regular field
Bisnovatyi-Kogan and Ruzmaikin (1976); Bisnovatyi-Kogan and Lovelace (2012).
The magnetic field lines predominantly vertically crosses the plane of the
disk. But this is apparently strongly simplified case. Following to Shakura
and Sunyaev (1973), we assume that the magnetic field of the accretion disk is
generated by the turbulent motion of plasma due to a dynamo mechanism Meier
(2001); Livio  _et al._ (2003); King  _et al._ (2004); Begelman and Armitage
(2014); Begelman  _et al._ (2015); Salvesen  _et al._ (2016); Riols  _et al._
(2017). The formation of regular vertical component of the magnetic field
occurs due to emerging, reconnection Galeev  _et al._ (1979) and opening of
magnetic loops by the disk differential rotation Romanova  _et al._ (1998).
The magnetic field structure inside the disk is shown schematically in fig.1.
The wind is formed quite high, in the corona where all the field lines are
open, and the magnetic pressure $B^{2}/8\pi$ exceeds the thermal pressure $p$.
The outflow of plasma from the disk occurs due to centrifugal Blandford and
Payne (1982) mechanism.

Figure 1: Schematic structure of the magnetic field inside the disk and in the
wind. The wind is formed in the corona where all field lines are open. Inside
the disk, the chaotic magnetic field is generated due to turbulent motion of
plasma in the disk.

The equation of conservation of the angular momentum can be obtained directly
from MHD equations integrated over the central volume shown in the form of
rectangular box in fig.1 (for technical details see Bogovalov and Kelner
(2010)). This equation in framework of the $\alpha$-disks paradigm Shakura and
Sunyaev (1973) has the following form:

$\dot{M}{\partial rV_{k}\over\partial r}+{\partial\over\partial r}4\pi
r^{2}t_{r\varphi}h+r^{2}<B_{\varphi}B_{z}>|_{wind}=0,$ (1)

where $t_{r\varphi}$ is the tangential stress, $V_{k}$ is the Keplerian
velocity of rotation, $\dot{M}$ is the accretion rate, $r$ is the cylindrical
radius and $h$ is the semi-height of the disk. $B_{\varphi}$ and $B_{z}$ are
toroidal and z-components of the magnetic field at the base of the wind. The
brackets $<>$ mean averaging over time. The term ${\partial\over\partial
r}4\pi r^{2}t_{r\varphi}h$ is responsible for the viscous loss of the angular
momentum by the disk. The last term in the left hand side of the equation is
responsible for the angular momentum loss due to the magnetized wind. Symbol
$|_{wind}$ means that the value is taken at the base of the wind at the
surface of the disk.

The tangential stress in the disk $t_{r\varphi}$ is defined by the turbulent
motion and by the magnetic field. According to Shakura and Sunyaev (1973)

$t_{r\varphi}=-\alpha\rho v_{s}^{2}.$ (2)

where $v_{s}$ is the sound velocity, $\rho$ is the matter density and $\alpha$
is the parameter connecting the pressure and tangential stress in the disk.

As it was pointed out by Pelletier and Pudritz (1992), the momentum loss due
to the wind will dominate the losses caused by the viscous stresses provided
that

$4\pi rt_{r\varphi}h\ll r^{2}<B_{\varphi}B_{z}>|_{wind}.$ (3)

In the opposite case we have the standard Shakura and Sunyaev (1973) version
of the disk accretion.

The integration of the energy conservation equation over the control volume
gives

${\partial\over\partial r}\dot{M}{V_{k}^{2}\over 2}-{\partial\over\partial
r}4\pi\Omega r^{2}t_{r\varphi}h+4\pi r\rho V_{z}E_{t}|_{wind}+4\pi rQ=0,$ (4)

where

$E_{tot}={v^{2}\over 2}-{B_{z}B_{\varphi}\over 4\pi\rho V_{z}}\Omega
r-{GM\over r}$ (5)

is the total energy per particle in the wind, $Q$ is the energy radiated from
unit square of one side of the disk surface. We neglect by the contribution
from the turbulent Ohmic heat production and advection of the energy in the
optically thick disk.

Taking into account the mass conservation

${\partial\dot{M}\over\partial r}-4\pi r\rho v_{z}|_{wind}=0,$ (6)

a simple algebra with eqs. (1) and (4) gives like in Shakura and Sunyaev
(1973), that the heat production equals

$Q=t_{r\varphi}hr{\partial\Omega\over\partial r}.$ (7)

Below we consider the case when the inequality in eq.(3) is fulfilled.
Dissipative terms can be neglected in equations (1 and 4). The equation for
the angular momentum conservation takes a form

$\dot{M}{\partial rV_{k}\over\partial r}+r^{2}<B_{\varphi}B_{z}>|_{wind}=0.$
(8)

Energy conservation equation is reduced to

${1\over 2}{\partial V_{k}^{2}\dot{M}\over\partial r}+4\pi r\rho
v_{z}E_{tot}|_{wind}=0.$ (9)

The luminosity of the disk is calculated by eq. (7).

This system of equations together with the system of MHD equations describing
the flow of the wind outside the disk totally defines the dynamics of the disk
and the wind in the regime of ”cold“ accretion. These equations do not contain
dissipative terms connected with viscosity or finite conductivity. According
to eq. (8),

$<B_{\varphi}B_{z}>|_{wind}={\dot{M}V_{k}\over 2r^{2}}.$ (10)

This allows us to estimate the minimum magnetic field at the base of the wind
which can support the regime of the ”cold“ accretion:

$B_{min}=\sqrt{{\dot{M}V_{k}\over 2r^{2}}}.$ (11)

It is easy to demonstrate that this value is small compared to the field in
the Shakura-Sunyaev regime of accretion. Nevertheless, the presence of the
magnetic field at the base of the wind means that the magnetic field is
present inside the disk and this field generates turbulence which makes
inevitable heating and radiation from the disk. The main question for us is
how strong is this radiation?

To connect the magnetic field at the base of the wind with $t_{r\varphi}$ we
introduce a parameter

$\theta={4\pi t_{r\varphi}\over<B_{\varphi}B_{z}>|_{wind}}.$ (12)

Physical sense of this parameter becomes clear if to pay attention that
according to the assumptions of Shakura and Sunyaev (1973) and recent
numerical simulations of the magnetic field generation Salvesen  _et al._
(2016)

$-t_{r\varphi}\approx 0.4\cdot{B^{2}\over 4\pi},$ (13)

where the magnetic field $B$ is taken at the midplane of the disk. Roughly,
$\theta$ equals to the ratio of the magnetic pressure inside the disk to the
magnetic pressure at the base where the wind starts to flow. The ratio
$\theta$ depends on variables like the mass flow rate $\dot{M}$, mass of the
central object $M$, radius, and others. This is one of the key difference of
our work from other works. We distinguish the magnetic field inside the disk
from the magnetic field at the base of the wind. In the works of the Grenoble
group $\theta\sim 1$ Ferreira and Pelletier (1995); Ferreira (1997) and other
authors Cao and Spruit (2013); Li and Begelman (2014); Li (2014) $\theta\sim
1$ because the magnetic field vertically crosses the disk. In reality, the
magnetic field inside the disk can essentially exceed the field at the base of
the wind Salvesen  _et al._ (2016). Therefore, below we assume that
$\theta\geq 1$ but should satisfy to the condition of ”cold“accretion
following from eq.(3)

${\theta h\over r}\ll 1.$ (14)

For the geometrically thin disks with $h/r\ll 1$ the ”cold“ disk accretion can
take place even for $\theta>1$.

##  III Thermal radiation from the disk at the ”cold“ accretion

In the work Shakura and Sunyaev (1973), three regimes of the disk accretion
have been considered: a) when the radiation pressure exceeds the gas pressure
and the Thomson scattering dominates over the free-free absorption; b) when
the gas pressure dominates over the radiation pressure but the Thomson
scattering dominates the free-free absorption; c) when the gas pressure
dominates over the radiation pressure and the opacity of the matter is defined
by the free-free absorption. We consider only the case when the gas pressure
dominates over the radiation pressure. These are the regimes b) and c). Below
we will see that when the radiation dominates, the accretion proceeds in the
Shakura-Sunayev regime.

###  III.1 Scattering dominates over free-free absorption

Firstly, we consider the case when the Thomson scattering dominates over the
free-free absorption. Hereafter we call this regime as Thomson regime. The
pressure of radiation $P_{rad}$ equals to ${\varepsilon/3}$, where
$\varepsilon=bT^{4}$. The sound velocity is defined as $v_{s}^{2}={kT/m_{p}}$,
where $m_{p}$ is the proton mass. According to Shakura and Sunyaev (1973) the
heat conductivity of the disk is defined by the transport of radiation. Then

$\varepsilon={3\over 4}{Q\sigma u_{0}\over c},$ (15)

where $\sigma=0.4~{}~{}\rm cm^{2}/g$ is the Thomson opacity and $u_{0}=2\rho
h$. Then the rate of heating of the disk equals

$Q={3\theta\dot{M}V_{k}v_{s}\over 16\pi r^{2}}.$ (16)

We used here that $h=v_{s}/\Omega$. The solution of this system of equations
gives

$T={\sqrt{3}\over 4\sqrt{\pi}}\left({\theta^{2}\dot{M}^{2}V_{k}\sigma\over
b\alpha cr^{3}}\right)^{{1\over 4}}.$ (17)

The sound velocity equals

$v_{s}={6^{1/4}\over
2(2\pi)^{1/4}}{k^{1/2}V_{k}^{1/8}(\theta\dot{M})^{1/4}\sigma^{1/8}\over
m_{p}^{1/2}b^{1/8}\alpha^{1/8}c^{1/8}r^{3/8}},$ (18)

and the density flux of radiation from one side of the disk is expressed as

$Q={(3\pi)^{5/4}6^{1/4}\over
32}{(\theta\dot{M})^{5/4}V_{k}^{9/8}k^{1/2}\sigma^{1/8}\over
r^{19/8}m_{p}^{1/2}b^{1/8}\alpha^{1/8}c^{1/8}}.$ (19)

Let us express $\dot{M}=\dot{m}\dot{M}_{crit}$ , the radius $r$ in
$r=(3r_{g})x$ and the mass $M$ in the solar masses $M=mM_{\odot}$. In these
variables we obtain

$Q=0.75\cdot 10^{23}{(\theta\dot{m})^{5/4}\over
m^{9/8}x^{47/16}\alpha^{1/8}},~{}~{}\rm{erg/s/cm^{2}}.$ (20)

The integration of this expression over the disk gives the bolometric
luminosity of the disk

$L_{bol}=0.84\cdot
10^{36}{(\theta\dot{m})^{5/4}m^{7/8}\over\alpha^{1/8}},~{}~{}\rm{erg/s}.$ (21)

The kinetic luminosity of the jets equals to the total energy release at
accretion. Therefore

$L_{kin}={\dot{M}c^{2}\over 12}=1.4\cdot 10^{38}m\dot{m},~{}~{}\rm{erg/s}.$
(22)

Then, the ratio of the kinetic luminosity over the bolometric luminosity
equals

${L_{kin}\over L_{bol}}=168{(m\alpha)^{1/8}\over\dot{m}^{1/4}\theta^{5/4}}.$
(23)

The bolometric luminosity can be expressed in conventional variables:

$L_{bol}={4\over 5}\theta\dot{M}V_{k0}v_{s0},$ (24)

where $V_{k0}$ and $v_{s0}$ are the Keplerian and sound velocities at the
inner edge of the disk. Taking into account that the kinetic luminosity

$L_{kin}={\dot{M}V_{k0}^{2}\over 2},$ (25)

the condition $L_{bol}/L_{kin}\ll 1$ becomes

${8\over 5}{\theta v_{s0}\over V_{k0}}={8\over 5}{\theta h\over r}\ll 1,$ (26)

which practically coincides with the condition of applicability of the ”cold“
disk accretion approximation defined by eq. (14). Similar condition has been
obtained earlier in Ferreira  _et al._ (2006). The condition $L_{kin}\gg
L_{bol}$ indicates that the accretion occurs in the ”cold“ regime.

The temperature in the disk

$T=2.5\cdot
10^{7}{\sqrt{\theta\dot{m}}\over\alpha^{1/4}x^{7/8}m^{1/4}},~{}~{}\rm K$ (27)

can be less than the temperature in the Shakura and Sunyaev (1973) disk
provided that $\theta<100$.

Let us calculate the ratio of the radiation pressure over the gas pressure in
the disk,

${P_{rad}\over P_{gas}}={3\over 32\pi}{\theta\dot{M}\sigma\over
rc}=0.85{\theta\dot{m}\over x}.$ (28)

This means that all the estimates are valid at $0.85\theta\dot{m}<1$.

Other parameters are estimated in the disk as follows. Density equals

$\rho={1\over
2\sqrt{3\pi}}{\sqrt{\theta\dot{M}}V_{k}^{3/4}m_{p}b^{1/4}c^{1/4}\over
r^{5/4}k\alpha^{3/4}\sigma^{1/4}}=0.6{\sqrt{\theta\dot{m}}\over
m^{3/4}x^{13/8}\alpha^{3/4}},~{}~{}\rm g/cm^{3}.$ (29)

The aspect ratio of the disk is

${h\over r}=3.7\cdot 10^{-3}{(\dot{m}\theta)^{1/4}x^{1/16}\over(\alpha
m)^{1/8}}.$ (30)

The true optical depth $\tau^{*}=\sqrt{\sigma\cdot\sigma_{ff}}\cdot u_{0}$ of
the disk is expressed as

$\tau^{*}=51{(\theta\dot{m})^{1/8}m^{3/16}x^{5/32}\alpha^{-13/16}},$ (31)

where

$\sigma_{ff}=0.11\cdot T^{-7/2}n,~{}~{}\rm cm^{2}/g$ (32)

is the free-free opacity of the disk. The surface temperature of the disk
$T_{S}$ is defined from the equation $bcT_{s}^{4}/4=Q$ and has a form

$T_{s}=6\cdot 10^{6}{(\theta\dot{m})^{5/16}\over
m^{9/32}x^{47/64}\alpha^{1/32}}~{}~{}\rm K.$ (33)

###  III.2 Free-free absorption dominates over scattering

At the condition

$4.6\cdot 10^{-3}{(\alpha m)^{1/10}x^{23/20}\over(\theta\dot{m})}>1$ (34)

the Thomson scattering opacity becomes small compared with the free-free
absorption. Hereafter we call this regime as free-free. Similar calculations
give the following temperature inside the disk

$T=10^{7}{(\theta\dot{m})^{6/17}\over x^{12/17}(\alpha\cdot m)^{4/17}}~{}\rm
K.$ (35)

The bolometric luminosity of the disk is

$L_{bol}=0.6\cdot
10^{36}(\theta\dot{m})^{20/17}m^{15/17}\alpha^{-2/17}~{}~{}\rm erg/s.$ (36)

The ratio of the kinetic luminosity over the bolometric luminosity equals

${L_{kin}\over L_{bol}}={228(\alpha
m)^{2/17}\over\dot{m}^{3/17}\theta^{20/17}}$ (37)

The full optical depth, the density of plasma and the aspect ratio of the disk
are

$\tau=93(\theta\dot{m})^{4/17}m^{3/17}x^{1/34}\alpha^{-14/17},$ (38)
$\rho={1.2(\theta\dot{m})^{11/17}\over(\alpha m)^{13/17}\cdot x^{61/34}}~{}\rm
g/cm^{3},$ (39) ${h\over r}=2.5\cdot
10^{-3}x^{5/34}(\theta\dot{m})^{3/17}(\alpha m)^{-2/17}.$ (40)

Finally, the surface temperature is

$T_{s}=5.5\cdot 10^{6}{(\theta\dot{m})^{5/17}\over
m^{19/68}x^{97/136}\alpha^{1/34}}~{}~{}\rm K.$ (41)

##  IV Comparison with the fundamental plane of the black holes

The fundamental plane encapsulates the relationship between the compact radio
luminosity, X-ray luminosity, and the black hole mass and provides a good
description of the data over a very large range of black hole mass. There are
reasons to believe that the Fundamental Plane (FP) of the black holes
reproduces the actual relationship between the kinetic luminosity of jets and
bolometric luminosity of the disks. In the work Daly  _et al._ (2016), the
position of objects of different masses in the coordinates $L_{kin}/L_{bol}$
and $L_{bol}/L_{Edd}$ has been collected in one FP. If this is true, the FP
can be used to extract information about the dependence of $\theta$ on
$\dot{m}$ and $m$. All data at the FP can be approximated by a power law
function of the form

$\log{L_{kin}\over L_{bol}}=(A-1)\log({L_{bol}\over L_{Edd}})+B$ (42)

with $A$ in the range (0.43 - 0.47) and $B$ in the range from -0.94 to -1.37.
For estimates the values $A=0.457$ and $B=-1.1$ have been chosen which are
close to the average. The ratio of $L_{bol}/L_{Edd}$ in the Thomson regime is

${L_{bol}\over L_{Edd}}=6\cdot 10^{-3}{(\theta\dot{m})^{5/4}\over(\alpha
m)^{1/8}},$ (43)

while in free-free regime this ratio equals to

${L_{bol}\over L_{Edd}}=4.4\cdot 10^{-3}{(\theta\dot{m})^{20/17}\over(\alpha
m)^{2/17}},$ (44)

Obviously, at the constant $\theta$ the theoretical predictions are not
consistent with observations. The reason is that $\theta$ must depend on
$\dot{m}$ and $m$. The most natural option is to assume that $\theta$ depends
on $\dot{m}$ as a power law

$\theta=D\dot{m}^{\gamma}.$ (45)

In the Thomson regime

$X={L_{bol}\over L_{Edd}}=6\cdot
10^{-3}{\dot{m}^{5(\gamma+1)/4}D^{5/4}\over(\alpha m)^{1/8}},$ (46)

and

$Y={L_{kin}\over L_{bol}}={168(\alpha m)^{1/8}\over
D^{5/4}\dot{m}^{(5\gamma+1)/4}}$ (47)

After simple algebraic calculations, we obtain that

$A={4\over 5(\gamma+1)},$ (48)

For $A=0.457$ the value $\gamma=3/4=0.75$. Then $D=5\cdot 10^{3}(\alpha
m)^{1/10}$. Thus, in the Thomson regime

$\theta=5\cdot 10^{3}\dot{m}^{3/4}(\alpha m)^{1/10}$ (49)

Similar calculations in the free-free regime give

$\theta=11.4\cdot 10^{3}\dot{m}^{0.86}(\alpha m)^{1/10}.$ (50)

The power at $\dot{m}$ is chosen to provide uniform dependence of $Y$ on $X$
of the form (42) with constant $A$ in both regimes.

The dependencies (49) and (50) seem physically reasonable. They show that the
smaller the accretion rate, more uniform is the magnetic field across the disk
and, therefore, the $\theta$ is close to 1.

The plot of $\theta/(\alpha m)^{1/10}$ is presented in fig. 2. $\theta$
corresponding to FP agree with the assumption of the ”cold“ accretion because
this curve is located well below the curve separating the regime of the cold
accretion from the Shakura-Sunyaev regime. In fig. 2 we present in dashed-
dotted the line which separates regions of domination of the gas pressure and
radiation pressure defined by equation (28). Thin solid line separates the
Thomson regime from the free-free regime.

Figure 2: Dependence of $\theta/(\alpha m)^{1/10}$ on $\dot{m}$. Shakura-
Sunyaev regime of accretion takes place above thick dashed line. Below this
line the regime of ”cold“ accretion takes place. The Thomson scattering
dominates above the thin solid line, while below this line the free-free
absorption gives the major contribution into the opacity of the medium.
Dashed-dotted line (calculated at $m=10^{8}$) divides the plane in parts where
the radiation pressure (above this line) or the gas pressure (below) dominate.

##  V Comparison with the specific sources

It is interesting to apply the estimated dependencies to the specific sources.
Below we consider M87 and the SMBH in galactic center, Sgr A*. We will see
later that both sources are in the free-free regime. Therefore eq. (50) has
been used at the estimations. For definiteness we accept $\alpha=0.1$.

###  V.1 M 87

For this object $\dot{m}$ and $m$ can be easily estimated. The kinetic
luminosity of this object is $L_{kin}=10^{44}~{}\rm erg/s$ which we assume is
equal to the total rate of the gravitational energy released at the accretion.
Mass of the central black hole $m=3.5\cdot 10^{9}$ Walsh  _et al._ (2013).
With the Eddington luminosity equal to $L_{Edd}=1.4\cdot 10^{38}m~{}\rm
erg/s$, we find $\dot{m}=L_{kin}/L_{Edd}=2\cdot 10^{-4}$. From eq. (37) we
obtain that

${L_{kin}\over L_{bol}}=10^{4}\theta^{-20/17}.$ (51)

Eq. (50) gives $\theta=54$. Then ${L_{kin}/L_{bol}}\approx 95$ and $L_{bol}$
from eq. (36) equal to $10^{42}~{}\rm erg/s$ in accordance with observations.
Optical depth of the disk exceeds $\tau>10^{4}$.

###  V.2 Sgr A*

The kinetic luminosity of the outflow from the disk around SMBH in Sgr A* is
not known. The flux of TeV gamma-rays from the Galactic Center can be
explained by very high energy accelerated protons with a luminosity close to
$10^{38}~{}\rm erg/s$. The kinetic luminosity of the wind has to be higher. We
consider a reversed problem. Assuming that the dependence (50) is valid for
the disk in Sgr A* we can predict what is the kinetic luminosity of wind from
the disk in the Galactic Center. Let us to estimate $\dot{m}$ from the
bolometric luminosity of the disk (see eq. (36)). In this case

$\dot{m}=\left(({L_{bol}\over 3.6\cdot 10^{40}~{}({\rm
erg/s})})^{17/20}{1\over m}\right)^{1\over 1.86}=8\cdot 10^{-6},$ (52)

$\theta=1.7$ and from eq. (22) we obtain that

$L_{kin}=4.4\cdot 10^{39},~{}\rm erg/s,$ (53)

at $L_{bol}=10^{36}~{}\rm erg/s$. The kinetic luminosity of the wind from the
Galactic accretion disk $4.4\cdot 10^{3}$ times exceeds the bolometric
luminosity of the disk. Remarkably, this power is sufficient to explain the
flux of the PeV protons in the Galactic Center.

##  VI Discussion

The main result of this work is the conclusion that the main source of energy
in AGN is the energy released at the accretion even in the case of high ratio
of the kinetic-to-bolometric luminosity. Observation of such objects can point
out that the accretion occurs in the ”cold“regime. Outflow in the form of a
wind from the accretion disk carry out the largest fraction of the angular
momentum of the accreted material. This results into suppression of radiation
from the disk. Accretion onto SMBH appear very efficient process which
transforms practically all the gravitational energy into the kinetic energy of
jets. This strongly contrasts with also radiationally inefficient ADAF models
Esin  _et al._ (1997). In the last case AGN appears very inefficient machine
which transforms almost all the gravitational energy into mass and rotational
energy of the SMBH. This model needs an additional source of energy which is
believed can be the rotational energy of SMBH.

The accretion in the ”cold“ regime occurs even when the magnetic field inside
the accretion disk essentially exceeds the magnetic field at the base of the
wind. This is explained by the geometrical reason. The angular momentum
transport due to viscosity is proportional to the magnetic pressure in the
disk times the thickness of the disk $h$ while the flux of the angular
momentum from the disk is proportional to the magnetic pressure at the base of
the wind times the radius. The ratio of the viscous losses to the losses due
to wind is $\sim\theta\cdot({h\over r})$, where $\theta$ roughly equals to the
ratio of the magnetic pressures inside and at the surface of the disk.
Therefore, the Shakura-Sunyaev regime of accretion Shakura and Sunyaev (1973)
is realized when $\theta\gg({r\over h})$. The assumption that $\theta$ is
constant contradicts to the observations if to assume that FP correctly
reproduces the ratio of the kinetic-to bolometric luminosities. $\theta$
increases on 2 orders of magnitude with $\dot{m}$. This value of $\theta$
agrees with the results of modeling of MRI dynamo Salvesen  _et al._ (2016).
This estimate should be considered as only a rough approximation.
Nevertheless, it allows us to make certain conclusions about realization of
the regime of ”cold“ disk accretion. It appears that at small accretion rates
$\dot{m}<10^{-2}$, the estimated value of $\theta$ is located in the region
well below the line were the Shakura-Sunyaev model is valid. The magnetic
pressure inside the disk appears less than the magnetic pressure estimated in
the model Shakura and Sunyaev (1973). Therefore, it is quite reasonable to
assume that at relatively low rates of accretion, $\dot{m}<10^{-2}$, the
accretion occurs predominantly in the regime of the ”cold“ accretion. At
higher values of $\dot{m}>0.1$ the accretion occurs in the regime of Shakura &
Sunyaev. The transition between the regimes takes place at the value of
$\dot{m}$ between $0.01$ and $0.1$ which well agree with location of the
transition from very bright to very dim disks around SMBH with powerful
outflow deduced in Churazov  _et al._ (2005). Remarkably, the rough estimate
of the dependence of $\theta$ on $\dot{m}$ gives good agreements with
observations of two SMBHs, M87 and Sgr A*. Hardly, such an agreement could be
accidental.

In this paper, we have considered only the case when the gas pressure
dominates over the radiation pressure. But according to fig. 2, the radiation
pressure dominates basically in the Shakura-Sunyaev accretion regime. The
”cold“ accretion at the domination of the radiation pressure apparently may be
realized in rather narrow range of parameters. This case is planned to be
considered separately.

##  VII Acknowledgments

###### Acknowledgements.

The work was supported by Russian science foundation, project N 16-12-10443.
Author is grateful to Felix Aharonian for stimulation of this work and
comprehensive discussions. Grateful also follow to N.I.Shakura for the
discussion of realization of different accretion regimes, and to M.R.Gilfanov
who called my attention to the information that can be extracted from
Fundamental Plane of black holes.

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# The updated BaSTI stellar evolution models and isochrones. IV.
$\alpha$-depleted calculations

Adriano Pietrinferni,1 Maurizio Salaris,2,1 Santi Cassisi,1,3 Alessandro
Savino,4 Alessio Mucciarelli,5,6 E-mail<EMAIL_ADDRESS>David
Hyder,2 Sebastian Hidalgo7,8
1INAF – Osservatorio Astronomico di Abruzzo, Via M. Maggini, s/n, I-64100,
Teramo, Italy
2Astrophysics Research Institute, Liverpool John Moores University, 146
Brownlow Hill, Liverpool, L3 5RF, UK
3INFN – Sezione di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy
4 Department of Astronomy, University of California, Berkeley, Berkeley, CA,
94720, USA
5 Dipartimento di Fisica e Astronomia “Augusto Righi”, Alma Mater Studiorum,
Università di Bologna, Via Gobetti 93/2, I-40129 Bologna, Italy
6INAF – Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via
Gobetti 93/3, I-40129, Bologna, Italy
7Instituto de Astrofisica de Canarias, Via Lactea s/n, La Laguna, Tenerife,
Spain
8Department of Astrophysics, University of La Laguna, Via Lactea s/n, La
Laguna, Tenerife, Spain

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

This is the fourth paper of our new release of the BaSTI (a Bag of Stellar
Tracks and Isochrones) stellar model and isochrone library. Following the
updated solar-scaled, $\alpha$-enhanced, and white dwarf model libraries, we
present here $\alpha$-depleted ([$\alpha$/Fe] = $-$0.2) evolutionary tracks
and isochrones, suitable to study the $\alpha$-depleted stars discovered in
Local Group dwarf galaxies and in the Milky Way. These calculations include
all improvements and updates of the solar-scaled and $\alpha$-enhanced models,
and span a mass range between 0.1 and 15 $M_{\odot}$, 21 metallicities between
[Fe/H] = -3.20 and +0.45 with a helium-to-metal enrichment ratio $\Delta
Y$/$\Delta Z$ = 1.31, homogeneous with the solar-scaled and $\alpha$-enhanced
models. The isochrones –available in several photometric filters– cover an age
range between $\sim$ 20 Myr and 14.5 Gyr, including the pre-main-sequence
phase. We have compared our isochrones with independent calculations of
$\alpha$-depleted stellar models, available for the same $\alpha$-element
depletion adopted in present investigation. We have also discussed the effect
of an $\alpha$-depleted heavy element distribution on the bolometric
corrections in different wavelength regimes. Our $\alpha$-depleted
evolutionary tracks and isochrones are publicly available at the BaSTI
website.

###### keywords:

Hertzsprung–Russell and colour–magnitude diagrams – stars: evolution – stars:
horizontal branch – stars interiors

††pubyear: 2015††pagerange: The updated BaSTI stellar evolution models and
isochrones. IV. $\alpha$-depleted calculations–The updated BaSTI stellar
evolution models and isochrones. IV. $\alpha$-depleted calculations

##  1 Introduction

Extended (in terms of mass and chemical composition ranges) and accurate sets
of stellar model calculations are fundamental inputs to a wide variety of
numerical tools employed to interpret spectroscopic and photometric
observations of individual stars, star clusters, and galaxies, both resolved
and unresolved.

In the last years we have undertaken a massive update –in terms of physics
inputs and parameter space coverage– of our BaSTI (a Bag of Stellar Tracks and
Isochrones) stellar models and isochrones library, and published in Paper I
(Hidalgo et al., 2018) a new grid of solar-scaled models and isochrones, in
Paper II its $\alpha$-enhanced counterpart (Pietrinferni et al., 2021),
followed in Paper III by the extension of the library to the white dwarf
regime (Salaris et al., 2022)111All these libraries are publicly available at
the new official BaSTI website http://basti-iac.oa-abruzzo.inaf.it..

In this paper (Paper IV of this series) we present a further extension of the
new BaSTI library to cover chemical compositions with an $\alpha$-depleted
([$\alpha$/Fe]=$-$0.20) metal distribution. To the best of our knowledge,
there are only two other stellar model and isochrone libraries for
$\alpha$-depleted chemical mixtures in the literature, published by Dotter et
al. (2007) and VandenBerg et al. (2014).

The presence of $\alpha$-depleted stars has been revealed by spectroscopic
observations of Local Group dwarf galaxies, as discussed in the classic review
by Tolstoy et al. (2009) – see also the results by Vargas et al. (2013) for a
sample of ultra-faint dwarfs, and Hill, V. et al. (2019) for the Sculptor
dwarf galaxy – and in the Milky Way (Nissen et al., 2020; Hayes et al., 2022;
Wang et al., 2023). An accurate characterization of these populations of stars
can indeed greatly benefit from stellar models and isochrones with the
appropriate metal abundance distribution, given the sensitivity of the
bolometric corrections at optical and shorter wavelength to the specific metal
abundance pattern (see Cassisi et al., 2004, and the next sections).

This paper is organized as follows. Section 2 summarizes the physics inputs
and the heavy element distribution adopted in the calculations, followed by
Sect. 3 which first presents the stellar model grid and its mass, chemical
composition, evolutionary phases coverage, and then shows comparisons with
previous calculations available in the literature. Section 4 discusses the
importance of using bolometric corrections calculated for the appropriate
$\alpha$-depleted metal mixture when comparing models and isochrones to data
in colour-magnitude diagrams, and a summary follow in Section 5.

##  2 Metal distribution and physics inputs

For these calculations, we have employed the same stellar evolution code and
physics inputs of Paper I and II. The adopted $\alpha$-depleted metal
distribution listed in Table 1 has been calculated with the $\alpha$-elements
O, Ne, Mg, Si, S, Ca, and Ti uniformly depleted with respect to Fe in
comparison to the corresponding solar ratios (the reference solar-scaled metal
distribution is from Caffau et al., 2011, as in Paper I and II) by 0.2 dex
([$\alpha$/Fe]=$-$0.2).

This metal distribution has been employed consistently for the calculation of
the nuclear energy generation, radiative and electron conduction opacities, as
well as in the equation of state (and bolometric corrections, see Sect. 4).
The sources for all these physics inputs are given in Paper I, together with
details about the treatment of the neutrino energy loss rates, superadiabatic
convection (we employ a mixing length $\alpha_{\rm ML}$ = 2.006 obtained from
a solar model calibration), overshooting from the convective cores, and atomic
diffusion (without radiative levitation). As for the outer boundary
conditions, in the case of models with initial mass larger than about
$0.5~{}M_{\odot}$ we used the integration of a $T(\tau)$ relation as in Paper
I and II. At lower masses, i.e. in the range of very low-mass stars, for both
solar-scaled and $\alpha$-enhanced models we used outer boundary conditions
provided by model atmosphere computations (see Paper I and II for details).
However, for our $\alpha$-depleted calculations we have not been able to find
publicly available outer boundary conditions from model atmospheres with the
appropriate $\alpha$-depleted metal mixture, consistent with those adopted for
the other sets of models in the BaSTI-IAC library. We were therefore forced to
use also for very low-mass stars the integration of the same $T(\tau)$
relationship adopted for larger masses.

Mass loss has been included by means of the Reimers (1975) formula, with the
free parameter $\eta$ set to 0.3 (see Paper I and II for details).

Table 1:  The adopted $\alpha$-depleted heavy element mixture. Element | mass fraction | number fraction
---|---|---
C | 0.238566 | 0.34146
N | 0.063738 | 0.07823
O | 0.364857 | 0.39204
Ne | 0.093950 | 0.08004
Na | 0.002955 | 0.00221
Mg | 0.034963 | 0.02473
Al | 0.005117 | 0.00326
Si | 0.039486 | 0.02417
P | 0.000559 | 0.00031
S | 0.017940 | 0.00962
Cl | 0.000412 | 0.00020
Ar | 0.008319 | 0.00358
K | 0.000318 | 0.00014
Ca | 0.003404 | 0.00146
Ti | 0.000167 | 0.00006
Cr | 0.001543 | 0.00051
Mn | 0.001118 | 0.00035
Fe | 0.116168 | 0.03576
Ni | 0.006419 | 0.00188

##  3 The model library

To allow easy comparisons and interpolations with our solar-scaled and the
$\alpha$-enhanced libraries, the $\alpha$-depleted models have been calculated
for the same [Fe/H] values of the other two metal mixtures. For each assumed
value of [Fe/H] we have derived the corresponding metallicity Z, and the
associated He mass fraction (Y) adopting $dY/dZ$ the helium-enrichment ratio
equal to 1.31 (see Paper I). Our model grid includes 21 metallicity values,
listed in Table 2.

Table 2:  Grid of initial chemical abundances. [Fe/H] | [M/H] | Z | Y
---|---|---|---
0.45 | 0.33 | 0.03037 | 0.2867
0.30 | 0.18 | 0.02210 | 0.2759
0.15 | 0.03 | 0.01596 | 0.2679
0.06 |  $-$0.06 | 0.01309 | 0.2641
$-$0.08 |  $-$0.20 | 0.00959 | 0.2595
$-$0.20 |  $-$0.32 | 0.00733 | 0.2566
$-$0.30 |  $-$0.42 | 0.00585 | 0.2546
$-$0.40 |  $-$0.52 | 0.00466 | 0.2531
$-$0.60 |  $-$0.72 | 0.00296 | 0.2509
$-$0.70 |  $-$0.82 | 0.00235 | 0.2501
$-$0.90 |  $-$1.02 | 0.00149 | 0.2489
$-$1.05 |  $-$1.17 | 0.00106 | 0.2484
$-$1.20 |  $-$1.32 | 0.00075 | 0.2480
$-$1.30 |  $-$1.42 | 0.00059 | 0.2478
$-$1.40 |  $-$1.52 | 0.00047 | 0.2476
$-$1.55 |  $-$1.67 | 0.00033 | 0.2474
$-$1.70 |  $-$1.82 | 0.00024 | 0.2473
$-$1.90 |  $-$2.02 | 0.00015 | 0.2472
$-$2.20 |  $-$2.32 | 0.00007 | 0.2471
$-$2.50 |  $-$2.62 | 0.00004 | 0.2471
$-$3.20 |  $-$3.32 | 0.00001 | 0.2470

For each initial chemical composition, evolutionary sequences for 56 different
values of the initial stellar mass have been computed: the minimum initial
mass is $0.1M_{\odot}$, while the maximum value is 15$M_{\odot}$. For initial
masses below $0.2M_{\odot}$, we computed evolutionary tracks for masses equal
to 0.10, 0.12, 0.15, and $0.18M_{\odot}$. In the range between 0.2 and
$0.7M_{\odot}$ a mass step equal to 0.05$M_{\odot}$ has been adopted. Mass
steps equal to $0.1M_{\odot}$, $0.2M_{\odot}$, $0.5M_{\odot}$, and
$1M_{\odot}$ have been adopted for the mass ranges $0.7-2.6M_{\odot}$,
$2.6-3M_{\odot}$, and $3-10M_{\odot}$, and masses larger than $10M_{\odot}$,
respectively.

Models less massive than $4M_{\odot}$ have been computed from the pre-MS,
whereas more massive models have been computed starting from the zero age main
sequence configuration.

All stellar models –but those corresponding to very low-mass stars whose core
H-burning lifetime is much longer than the Hubble time– have been calculated
until the start of the thermal pulses on the asymptotic giant branch, or
C-ignition for the more massive ones. For the very low-mass stars, the
calculations have been stopped when the central H mass fraction is $\approx
0.3$ (corresponding to ages already much older than the Hubble time). For each
metallicity, an extended set of low-mass core He-burning models suitable for
the study of the horizontal branch (HB) in old stellar populations has been
also computed, as done in Paper I and II.

All the evolutionary sequences have been normalized, i.e. reduced to the same
number of lines using the technique of ‘equivalent points’ as described in
Paper I and II.222Although the normalization procedure is exactly the same
adopted in Hidalgo et al. (2018), we have slightly modified the definition of
one of the selected key points to better account for the extremely long
evolutionary timescales of very low-mass stars. More in detail, the fourth key
point for these models now corresponds to the evolutionary stage when
H-burning has decreased the central hydrogen mass fraction by 0.01 with
respect to the initial value. For consistency, the same criterion has been
applied to the solar-scaled and $\alpha$-enhanced models in the library and
used for computing isochrones in a wide age range.

Having evolutionary tracks and isochrones computed this way, with the same
number of points together with the homogeneous [Fe/H] grid for solar-scaled,
$\alpha$-enhanced and $\alpha$-depleted calculations, makes it easy to
calculate evolutionary tracks and/or isochrones at fixed [Fe/H] for any
arbitrary value of [$\alpha$/Fe] between -0.2 and +0.4 dex. One could use for
example a parabola to interpolate in [$\alpha$/Fe] point-by-point along the
tracks and isochrones. The same is true also for interpolations in [Fe/H] at a
given [$\alpha$/Fe].

Both evolutionary tracks and isochrones are provided in the theoretical HR
diagram (HRD) and in various photometric systems as described in Paper I and
II, by applying bolometric corrections obtained from theoretical spectra
(consistent with those employed in Paper I and II) calculated with the
appropriate $\alpha$-depleted metal distribution. However, due to the lack of
suitable $\alpha$-depleted synthetic spectra for VLM stars, in this mass
regime we have been forced to adopt a different approach. For each value of
[Fe/H] we have employed the solar-scaled bolometric corrections of Paper I
after applying - for each photometric filter - a shift in order to match the
corresponding bolometric correction at $T_{\rm eff}\approx 5000~{}K$ and
$\log{g}=4.5$ provided by the $\alpha$-depleted synthetic spectra obtained
with the ATLAS9 code (see the discussion in Paper I).

Several photometric systems have been added (or updated) since the publication
of Paper I, and Table 3 shows an updated list of the systems available in our
model library for $\alpha$-enhanced, solar-scaled and, $\alpha$-depleted
calculations.

Figure 1: Comparison between Dotter et al. (2007) and our [$\alpha$/Fe]=$-$0.2
isochrones from the MS to the tip of the RGB (red dashed and black solid
lines, respectively) with the labelled [Fe/H] and ages.

Figure 1 shows the HRD of our 1, 5 and 12 Gyr isochrones for two values of
[Fe/H], compared to the corresponding Dotter et al. (2007) $\alpha$-depleted
isochrones (we could not compare with the calculations by VandenBerg et al.,
2014, because they are for a 0.4 dex $\alpha$ depletion), from the main
sequence (MS) to the tip of the red giant branch (RGB)333These isochrones have
been computed using the online interpolation tool at
http://stellar.dartmouth.edu/models/isolf_new.html. Dotter et al. (2007)
calculations employ a metal distribution with [$\alpha$/Fe]=$-$0.20 as in our
models, although the reference solar mixture (the Grevesse & Sauval, 1998,
solar metal composition is used by these authors) is not the same, and there
are differences also in some physics inputs (see Dotter et al., 2007, for
details).

Apart from the lower MS, the two sets of isochrones are in close agreement.
Differences in $T_{\rm eff}$ at fixed luminosity are at most by a few tens of
K, and the TO luminosities are always practically the same, apart from the 1
and 5 Gyr, [Fe/H]=$-$1.9 cases, with the Dotter et al. (2007) isochrones
showing a slightly brighter TO than our calculations, corresponding to an age
difference by about 100 Myr.

The disagreement between the two sets of models along the fainter portion of
the MS is due to the use of different approaches for the outer boundary
conditions, an input that significantly affects the $T_{\rm eff}$ scale of
very low-mass stars, as well as to differences in the adopted chemical
mixtures (see below for more details).

##  4 The role of the bolometric corrections

As shown by Cassisi et al. (2004) and confirmed by our more recent
calculations (Pietrinferni et al., 2021), $\alpha$-enhanced isochrones with
ages above $\sim$1 Gyr can be well mimicked by solar-scaled ones with the same
total metallicity [M/H] not only in the HRD, but also in $VI$ and infrared
colour magnitude diagrams (CMDs), whilst the agreement is less satisfactory in
optical CMDs and worsens at shorter wavelengths. The good match in the HRD
confirms the results of the pioneering study by Salaris et al. (1993) who at
the time did not have available bolometric corrections (BCs) calculated with a
consistent $\alpha$-enhanced metal distribution. Indeed, the poorer agreement
in CMDs at optical and shorter wavelengths is due to differences in the BCs
induced by the effect of the change in the metal abundance distribution on the
stellar spectra.

We expect that similar results should hold also when comparing
$\alpha$-depleted with solar-scaled isochrones at the same [M/H], and indeed
this is the case, as shown in Fig. 2.

Figure 2: Comparison in the HRD between our $\alpha$-depleted (black solid
lines) and solar-scaled (Hidalgo et al., 2018) isochrones (red dashed lines)
with the labelled total metallicities [M/H] and ages.

Here we compare our $\alpha$-depleted 1, 5, and 12 Gyr isochrones with the
solar-scaled ones from Hidalgo et al. (2018) at two different common values of
[M/H]. There is a perfect match between the two pairs of isochrones along MS,
TO, and the core He-burning phase after the He-flash at the tip of the RGB.
Differences appear only on the lower MS for log$(L/L_{\odot})$ below
$\sim-$1.0 corresponding to masses below $\sim$ 0.5$M_{\odot}$. Similar
differences for the very low-mass regime were found in Paper II when comparing
solar-scaled (from Paper I) with $\alpha-$enhanced models. To verify the role
played by the outer boundary conditions in causing these differences, we
computed solar-scaled stellar models in this mass regime for [M/H]=$-$0.52, by
adopting the same $T(\tau)$ relationship used for our $\alpha$-depleted
computations. Figure 3 shows the resulting 12 Gyr isochrone compared to our
new $alpha$-depleted calculations, and a solar-scaled one from Paper I models.
When comparing the two solar-scaled isochrones it is easy to notice that the
use of a $T(\tau)$ relationship instead of the more appropriate boundary
conditions provided by model atmospheres employed in Paper I calculations,
makes the very low-mass models hotter (see also Baraffe et al., 1997; Cassisi
et al., 2000, and references therein). However, the impact is not able to
explain the full difference between our solar-scaled calculations and the
$\alpha-$depleted ones at the same [M/H] (the same is true for the differences
with the $\alpha-$enhanced models shown in Paper II). Indeed the main source
of these $T_{\rm eff}$ differences is not so much the way the boundary
conditions are computed, rather the metal mixture used in their calculations.
As already discussed by VandenBerg et al. (2022) and VandenBerg (2023), the
boundary conditions are much more sensitive to the adopted specific metal
abundance distribution than for more massive models. In particular, VandenBerg
(2023) has demonstrated the crucial role played by oxygen – whose abundance
changes substantially when moving from the solar-scaled mixture to the
$\alpha-$depleted/enhanced ones – in determining the atmospheric thermal
stratification of very low-mass stars. The differences in the boundary
conditions for very low-mass models at a given total metallicity are the
primary reason for the $T_{\rm eff}$ differences between solar-scaled and
$\alpha$-depleted isochrones shown in Fig. 3.

Figure 3: Comparison in the HRD between $\alpha$-depleted (black solid lines)
and solar-scaled (Hidalgo et al., 2018) isochrones (red dashed lines) for the
labelled values of the total metallicity and age. The open dots denote solar-
scaled models with mass between $0.1~{}M_{\odot}$ and $0.5~{}M_{\odot}$ for
the same values of [M/H] and age, but computed employing for the outer
boundary conditions the same $T(\tau)$ relationship adopted for the
$\alpha-$depleted calculations. Figure 4: Comparison in various CMDs of a
self-consistent 12 Gyr, [Fe/H]=$-$0.7 $\alpha$-depleted isochrone (black solid
lines), with isochrones for the same age and chemical composition but
calculated using solar-scaled bolometric corrections with either the same
[Fe/H] (red dashed lines) or with the same [M/H] (blue dash-dotted lines) of
the self-consistent isochrone, respectively. We didn’t display the portion of
the isochrones populated by masses below $\sim$0.5$M_{\odot}$(see text for
details).

Figure 4 highlights the effect of the $\alpha$-depleted heavy element
distribution of the model BCs, the same way it was shown by Cassisi et al.
(2004) for the $\alpha$-enhanced case. We display here 12 Gyr, [Fe/H]=$-$0.7
$\alpha$-depleted isochrones in different CMDs, calculated using the
appropriate BCs for a consistent $\alpha$-depleted metal mixture (we denote
them as self-consistent isochrones), BCs calculated for a solar-scaled metal
mixture and the same [Fe/H] of the isochrone (denoted as [Fe/H] isochrones)
and BCs computed for a solar-scaled metal mixture and the same [M/H] of the
isochrone (denoted as [M/H] isochrones), respectively. We do not show the
lower MS, for which we do not have models computed using boundary conditions
from $\alpha$-depleted model atmospheres.

As in Cassisi et al. (2004) we find that the perfect agreement between the
three isochrones in the infrared CMD progressively worsens, especially at low
$T_{\rm eff}$, when moving to CMDs at increasingly shorter wavelenghts.
Moreover, these differences quantitatively decrease with decreasing [Fe/H] of
the self-consistent $\alpha$-depleted isochrone

Just for reference, in the $UB$ CMD the RGB at $M_{U}$=3 for the [M/H]
isochrone is 0.08 mag bluer than the self-consistent one, while the [Fe/H]
solar-scaled isochrone is 0.03 mag bluer. The absolute $U$ magnitude and
$(U-B)$ colour of the TO are the same for the self-consistent and the [Fe/H]
isochrone, while $M_{U}$ is $\sim$0.03 mag fainter and $(U-B)$ is 0.02 mag
bluer for the [M/H] isochrone. On the MS at $M_{U}$=9 the [M/H] isochrone is
0.07 mag bluer than the self-consistent one, while the [Fe/H] solar-scaled
isochrone is 0.04 mag bluer. These differences are substantially larger when
considering magnitudes and colours in the UV wavelength range.

Moving to the $BV$ CMD the differences decrease. On the RGB at $M_{V}$=1 the
[M/H] isochrone is now 0.04 mag bluer than the self-consistent one, while the
[Fe/H] isochrone is 0.02 mag bluer. The TO colours and magnitudes are
basically identical among the three isochrones, while on the MS at $M_{V}$=7.5
the [M/H] isochrone is 0.06 mag bluer than the self-consistent one and the
[Fe/H] solar-scaled isochrone is 0.05 mag bluer. In the $VI$ CMD differences
appear only when approaching the tip of the RGB, which is redder in both
[Fe/H] and [M/H] isochrones by about 0.12 mag.

On average, the [Fe/H] isochrone is a better approximation of the self-
consistent one, in agreement with the results by Cassisi et al. (2004) for the
$\alpha$-enhanced regime. Not surprisingly, the sign of the colour differences
is the opposite of the $\alpha$-enhanced comparison, whereby the [Fe/H] and
[M/H] isochrones are redder along the RGB and the MS. As discussed briefly in
Cassisi et al. (2004), the different abundances of mainly O, Si, Mg, and Ca
between $\alpha$-depleted and solar-scaled compositions that affect both the
line opacity and the continuum opacity in the atmospheres, causing a
redistribution of the flux among the various wavelengths, and a change of the
values of the BCs at fixed $T_{\rm eff}$ and gravity.

##  5 Summary

We have presented our new BaSTI $\alpha$-depleted ([$\alpha$/Fe]=$-$0.20)
stellar models and isochrones, suitable to investigate the evolutionary
properties of the $\alpha$-depleted populations discovered in the Milky Way
and Local Group dwarfs. The models have been calculated with the same physics
inputs and reference solar metal mixture employed for our updated solar-scaled
and $\alpha$-enhanced ([$\alpha$/Fe]=0.4) calculations published in Paper I
and II, and cover the same mass and [Fe/H] ranges.

We have compared the HRDs of selected $\alpha$-depleted isochrones with the
Dotter et al. (2007) [$\alpha$/Fe]=$-$0.20 counterparts, from the MS to the
tip of the RGB, finding an overall good agreement despite differences in the
reference solar mixture and some physics inputs. We have also discussed the
importance of employing bolometric corrections calculated for the appropriate
$\alpha$-depleted heavy element mixture when comparing isochrones (and
evolutionary tracks) to data in optical CMDs and at shorter wavelengths.

Like for the solar-scaled and $\alpha$-enhanced libraries, these new
$\alpha$-depleted evolutionary tracks and isochrones are available in several
photometric systems at http://basti-iac.oa-abruzzo.inaf.it. In the near
future, we will make available on the BaSTI website an online tool to
interpolate among our sets of calculations, to obtain tracks ad isochrones in
all available photometric filters for any value of [$\alpha$/Fe] between
$-$0.2 and +0.4.

## Acknowledgements

We warmly thank our anonymous referee for pointing out the origin of the
differences between our sets of very low-mass models computed with different
heavy element mixtures. SC and AP acknowledge support from PRIN-MIUR2022 (PI:
S. Cassisi), from INAF Theory grant "Lasting" (Responsabile: S. Cassisi), INFN
(Iniziativa specifica TAsP), and from PLATO ASI-INAF agreement
n.2015-019-R.1-2018. MS acknowledges support from STFC Consolidated Grant
ST/V00087X/1.

## Data Availability

The $\alpha$-depleted evolutionary tracks and isochrones in several
photometric systems are available at the new official BaSTI website
http://basti-iac.oa-abruzzo.inaf.it.

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Table 3: Available photometric systems. For each system, information about the
source from the response curves and reference zero-points is provided.

Photometric system | Calibration | Passbands | Zero-points
---|---|---|---
2MASS | Vegamag | Cohen et al. (2003) | Cohen et al. (2003)
CFHT (MegaCam) | ABmag | CFHT Documentation1 | 0
Euclid (VIS + NISP) | ABmag | Euclid mission database2 | 0
Gaia DR1 | Vegamag |  Jordi et al. (2010)3 | Jordi et al. (2010)
Gaia DR2 | Vegamag |  Maíz Apellániz & Weiler (2018)4 | Maíz Apellániz & Weiler (2018)
Gaia DR3 | Vegamag | Riello et al. (2021) | Riello et al. (2021)
GALEX | ABmag | NASA5 | 0
Hipparcos + Thyco | ABmag | Bessell & Murphy (2012) | Bessell & Murphy (2012)
HST (WFPC2) | Vegamag | SYNPHOT | SYNPHOT
HST ( WFC3) | Vegamag | SYNPHOT | SYNPHOT
HST (ACS) | Vegamag | SYNPHOT | SYNPHOT
Johnson, Bessel & Brett | Vegamag | Bessell & Brett (1988); Bessell (1990) | Bessell et al. (1998)
J-PLUS | ABmag | J-PLUS collaboration 6 | 0
JWST (NIRCam - Post Launch) | Vegamag | JWST User Documentation 7 | SYNPHOT
JWST (NIRCam - Sirius) 7a | Vegamag | JWST User Documentation 7 | SYNPHOT
JWST (NIRISS - Post Launch) | Vegamag | JWST User Documentation 7 | SYNPHOT
Kepler | ABmag | Kepler collaboration8 | 0
Nancy Grace Roman Telescope (WFI) | Vegamag | Roman Documentation 9 | SYNPHOT
PanSTARSS1 | ABmag | Tonry et al. (2012) | 0
SAGE | ABmag | SAGE collaboration | 0
SDSS | ABmag | Doi et al. (2010) | Dotter et al. (2008)
Skymapper | ABmag | Bessell (2011) | 0
Spitzer (IRAC) | Vegamag | NASA10 | Groenewegen (2006)
Strömgren | Vegamag | Maíz Apellániz (2006) | Maíz Apellániz (2006)
Subaru (HSC) | ABmag | HSC collaboration11 | 0
SWIFT (UVOT) | Vegamag | HEASARC12 | Poole et al. (2008)
TESS | ABmag | TESS collaboration13 | 0
UVIT (FUV + NUV + VIS) | ABmag | UVIT collaboration 14 | Tandon et al. (2017)
Vera C. Rubin Observatory | ABmag | LSST collaboration 15 | 0
Victor M. Blanco Telescope (DECam) | ABmag | CTIO 16 | 0
VISTA (VIRCAM) | Vegamag | ESO17 | Rubele et al. (2012)
WISE | Vegamag | WISE collaboration18 | Wright et al. (2010)

  * 1

https://www.cfht.hawaii.edu/Instruments/Filters/megaprimenew.html

  * 2

http://euclid.esac.esa.int/epdb/db/SPV02/SPV02/EUC_MDB_MISSIONCONFIGURATION-
SPV02_2018-06-16T140000.00Z_01.01.xml.html

  * 3

The nominal G passband curve has been corrected following the post-DR1
correction provided by Maíz Apellániz (2017)

  * 4

Two different $G_{BP}$ passbands are provided for sources brighter and fainter
than G=10.87, respectively.

  * 5

https://asd.gsfc.nasa.gov/archive/galex/Documents/PostLaunchResponseCurveData.html

  * 6

http://www.j-plus.es/survey/instrumentation

  * 7

https://jwst-docs.stsci.edu/

  * 7a

This photometric system adopts the "post launch passbands but Vega zeropoints
obtained by using Sirius as the color reference (see JWST User Documentation
for more details).

  * 8

https://nexsci.caltech.edu/workshop/2012/keplergo/CalibrationResponse.shtml

  * 9

https://roman.gsfc.nasa.gov/science/WFI_technical.html

  * 10

https://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/calibrationfiles/spectralresponse/

  * 11

https://hsc-release.mtk.nao.ac.jp/doc/index.php/survey/

  * 12

https://heasarc.gsfc.nasa.gov/docs/heasarc/caldb/data/swift/uvota/index.html

  * 13

https://heasarc.gsfc.nasa.gov/docs/tess/data/tess-response-function-v1.0.csv

  * 14

https://uvit.iiap.res.in/Instrument/Filters

  * 15

https://github.com/lsst/throughputs/tree/master/baseline

  * 16

http://www.ctio.noirlab.edu/noao/content/decam-filter-information

  * 17

http://www.eso.org/sci/facilities/paranal/instruments/vircam/inst/

  * 18

https://wise2.ipac.caltech.edu/docs/release/prelim/expsup/sec4_3g.html#WISEZMA