D. Yar-Mukhamedov*

Cavendish Laboratory, University of Cambridge CBS OHE. Cambridge. United Kingdom

Received July 8, 2014

An analytic model for evolution of galactic magnetic fields in hierarchical galaxy formation frameworks is introduced. Its major innovative components include explicit and detailed treatment of the physics of merger events, mass gains and losses, gravitational energy sources and delays associated with formation of large-scale magnetic fields. This paper describes the model, its implementation, and core results obtained by its means.

DOI: 10.7868/S0044451015040047


1.1. Initial assumptions

By creating this model, we aim to develop a semi-analytic approach for modeling galactic magnetic fields, which solves several known problems, including:

• the problem of overprediction of magnetic field strengths for high -rilclSS galaxies, as pointed out in [1];

• problems arising as a result of treating evolution of magnetic fields completely independently of mass evolution;

• computational efficiency problems.

We proceed first to formulate assumptions that define the galaxy formation and evolution framework.

• Each galaxy consists of the central supermassive black hole; the bulge (or spheroid); the disk; the halo of hot gas; the dark matter halo. The spheroid and the disk contain cold gas and stars.

• All components can grow through mergers and accretion. However, mergers can also trigger mass transfer from the disk to the spheroid through the disk instability mechanism, leading to a decrease in the disk


• Both star formation and supernovae reduce the amount of cold gas. Star formation, however, returns some gas into the system, while supernovae input some energy.

* E-mail: danial.su'fflgmail.com

We next consider assumptions on the structure and behavior of galactic magnetic fields.

• Magnetic fields in a disk of a galaxy are represented by ordered magnetic fields, which exhibit large-scale structure, and chaotic (also called random or turbulent) fields, which have no explicit structure on the galactic scales, but can show ordered behavior on smaller scales.

• The ffctS 111 ct galactic disk can be treated as a magnetohydrodynamic fluid, which results in equipart it ion of the total energy S-z of the system between the energy of the turbulent motions Si of the disk gas and its magnetic fields Sm fl]

¿■i = ¿m- (1)

• In the assumed approximation, magnetic fields are tied to the components where they formed and their evolution in each component proceeds independently.

• We further assume that the ordered magnetic fields form merely in galactic disks; we therefore consider disks only. Nevertheless, the same reasoning with minor corrections can be used to derive various properties of magnetic fields for other components of a galaxy if needed.

We next consider the assumptions that are new to this model and, to the best of our knowledge, have not been implemented in other semi-analytic models.

• All energy components of a system are tied to a cold gas, and therefore any decrease in the mass in a gas container results in the corresponding loss of energy

£ =




where £ is some energy component (St, Si, Sm, etc.), M is the mass of the cold gas, and M- is the negative part of the gas mass rate.

• The timescale of the process of formation of ordered magnetic fields from chaotic magnetic fields in this model is parameterized, and is therefore proportional to the period of rotation of the galaxy instead of being equal to it. The latter simpler approach was used in [1].

• Mass gains from mergers are treated explicitly, thus allowing for a detailed investigation of mcrger-re-lated effects on the evolution of energies of the ordered and random components of galactic magnetic fields.

1.2. Equations of energy balance

The rate of change of the total energy of a system with time depends merely on the rate of energy inputs and outputs £io, which, after taking (2) into account, yields

£T = ¿-in ~ ¿i

M: M


£0 =



For the rate of change of the energy of ordered magnetic fields £0, everything is somewhat more complicated. When the ordered energy is less than a half of the total energy, it would draw energy from the chaotic magnetic field £c and lose it only due to mass losses (2). However, in the case where the total energy decreases and the ordered magnetic field energy is half the total energy, it follows that due to energy equipartition (1), the energy of the ordered magnetic field should decrease along with the total energy without any delays. Both considerations together give

£o =


£c c M-t L'° M

Sa = and St, < 0,




where r is the ordered magnetic field formation timescale, which is proportional to a period of rotation of the considered galaxy, which hence depends indirectly on time.

As a result of the merger, the disk structure and laminar motion of interstellar gas can be disrupted, leading to a partial destruction of large-scale magnetic fields, which can be accounted for by adding more sum-mands to (4), leading to

M- + kdgMmg + kdsMr M

S0 = and St, < 0,



where kdg and kd8 are the efficiencies of the respective mechanisms associated with infall of gas and stars, and Mmg and Mms are the corresponding IIlclSS infall rates. This, however, does not decrease the total energy of the system.

Finally, all other energies i.e., the turbulent energy, the total energy of magnetic fields, and the chaotic energy, can be obtained from the energy balance equations

St = Si, + Sm, (6)

— L'c c'o' 0)

and equipartition assumption (1).

1.3. Sources and sinks

In the recent work fl] on semi-analytic modeling of magnetic field formation and evolution, its authors assumed that the total energy rate consists merely of


• the gravitational energy rate corresponding to the energy brought into the system by accretion;

• a positive energy rate caused by various supernovae feedback mechanisms;

• a negative energy rate due to removal of energy by star formation.

In this model, in addition to those sources, we account for

• energy changes due to mergers including the gravitational energy of infalling matter and in the case where merger causes a disk instability, the negative energy changes due to the transfer of IIlclSS from the disk to the bulge;

• supernovae expulsive feedback, which causes all energies of the system to decrease as a result of incurring mass losses.

All the onunioratod energy losses are caused by the corresponding HiaSS losses, and hence it is possible to account for all of them just by explicitly defining all the mass rates as

M = M+-M-, M+ = Ma+Mwi+Msfi,

M = >!,/„ + Msn + M mo ■


(8) 0) (10)

where M+ is the positive part of the total mass rate, Ma is the positive mass rate due to accretion of matter to the disk, Mm¿ is the positive mass rate due to acquisition of additional HiaSS through mergers, Ms/i is the mass input due to gas recycling, Ms/0 is a negative mass rate due to the effects of star formation, Msn is a negative mass rate due to expulsive supernovae feedback, Mmo is a negative mass rate due to mergers, and Mdi a negative mass rate due to disk instability.

The rest of energy rates should be accounted explicitly,

£■ — £

C'Q [ C-JÎ


where £gn is the energy input rate due to supernovae feedback, £a is the energy input rate due to accretion of mass into the galactic disk, and £,n is the energy input rate due to mergers.

We now define model parameters that determine efficiencies of various energy sources and sinks. We account for the efficiency of supernovae with kgn, accretion with ka- and mergers with k,ng for the gaseous component and k,ng for the stellar one; finally, kT defines the relation between a characteristic timescale r and the period of rotation of a galaxy.

We now consider the energy sources individually. We begin with accretion, where the energy rate is


£a = kaGMa

= KGMa

M + AM

■ dr =


1 1



■ dr

■I'd rgh J J r*



where G is the universal gravitatinal constant, M is the total rilciSS inside the disk radius, AM is the fraction of mass between the current infall distance and the disk radius, rd is the radius of the galactic disk, and r9/¡ is the radius of the hot gas halo. This result can be simplified by first assuming

M > AM


and then

which leads to

I'd -C rgh,

Sa « kaGMa AI (---

I'd rgh




M a AI




The energy rate of the source associated with mergers is

— hm.gG./\/img

M + AM



M + AM

-dr =

= G (kmgMmg + kmsMms I X


1 1

I'd r„




where rg is the distance between merging galaxies, and Mmg and Mms are the respective rates of gas and star infall. To simplify the obtained result, in addition to (13), we can assume that

■I'd < rg


which leads to

G [IrngMmg + kmsMms ) AI

1 1

I'd r„


kmgMmg + kmsMms ) AI I'd


Further possible simplifications of both these sources include

• equivalence of the merger and accretion efficiency coefficients

^ a — ^ mg *


which can be assumed to be true because accretion of gas from a hot halo shares many similarities with the accretion of gas from a satellite galaxy in course of the merger event;

• equivalence of the total negative rate to the overall negative rate


0 -M M


where 9 is the Heaviside step function; this assumption holds when









• oquivalonco of the total positive rate to the overall positive rate

Mm» + MaKB[M\M,

M m, s « 0 [Mls)Mls,

(22) (23)

where Mts is the total stellar IIlclSS change in the disk; approximation (22) works only if (19) is applicable, (21) is assumed, and


Ma + M,

i. e., the recycled gas is negligible, and (23) works if (21) is assumed for stellar masses and


ÍS +


i.e., the amount of forming stars is much smaller than the amount of stars incorporated

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