A closer look at supernovae as seeds for galactic magnetization

Explaining the currently observed magnetic fields in galaxies requires relatively strong seeding in the early Universe. One theory proposes that magnetic fields of the order of $\mu$G were expelled by supernova (SN) explosions after primordial, nG or weaker fields were amplified in stellar interiors. In this work, we calculate the maximum magnetic energy that can be injected in the interstellar medium by a stellar cluster of mass $M_{cl}$ based on what is currently known about stellar magnetism. We consider early-type stars and adopt either a Salpeter or a top-heavy IMF. For their magnetic fields, we adopt either a Gaussian or a bimodal distribution. The Gaussian model assumes that all massive stars are magnetized with $10^3<\langle B_* \rangle<10^4$ G, while the bimodal, consistent with observations of Milky Way stars, assumes only 5-10 per cent of OB stars have $10^3<\langle B_* \rangle<10^4$ G, while the rest have $10<\langle B_* \rangle<10^2$ G. We find that the maximum magnetic energy that can be injected by a stellar population is between $10^{-10}-10^{-7}$ times the total SN energy. The highest end of these estimates is about five orders of magnitude lower than what is usually employed in cosmological simulations, where about $10^{-2}$ of the SN energy is injected as magnetic. Pure advection of the stellar magnetic field by SN explosions is a good candidate for seeding a dynamo, but not enough to magnetize galaxies. Assuming SNe as main mechanism for galactic magnetization, the magnetic field cannot exceed an intensity of $10^{-7}$ G in the best-case scenario for a population of $10^{5}$ solar masses in a superbubble of 300 pc radius, while more typical values are between $10^{-10}-10^{-9}$~G. Therefore, other scenarios for galactic magnetization at high redshift need to be explored.


Introduction
The origin of magnetic fields is one of the most compelling problems in astrophysics and cosmology.While some constraints of the primordial magnetic field strength exist, the mechanism that can generate them are very diverse (e.g., Gnedin et al. 2000;Subramanian 2019).What we do know is that the magnetic fields currently observed in spiral galaxies are too strong to be primordial.Therefore, they were either seeded by a primordial field in the early Universe and then amplified through a dynamo, or directly generated by an astrophysical process in galaxies (e.g., Brandenburg & Ntormousi 2022).
One theory for magnetizing early galaxies involves stars (Rees 1987).The idea is that a dynamo process in stellar interiors amplifies the tiny magnetic seeds coming from a Biermann battery or similar mechanism.When these stars explode as supernovae (SNe), the amplified fields are injected in the surrounding interstellar medium (ISM).
A number of numerical experiments adopt this mechanism as a subgrid model in galactic dynamo (Hanasz et al. 2009;Kulpa-Dybeł et al. 2011, 2015;Butsky et al. 2017) or cosmological structure formation simulations (Beck et al. 2013, Vazza et al. 2017;Katz et al. 2019;Garaldi et al. 2021;Martin-Alvarez et al. 2021), injecting a magnetic field together with thermal and kinetic energy as feedback from stellar populations.However, the assumptions about the relative ratio of magnetic to thermal SN energy differ strongly between models.For instance, Kulpa-Dybeł et al. (2011, 2015) assume that 10% of the SNe introduce a field on the order of 10 −5 µG in dense regions of the ISM.A similar assumption on the incidence of SN magnetization is made by Hanasz et al. (2009); although, they do not specify the value of the magnetic field.Butsky et al. (2017) seed each SN event with 10 −7 of the SN energy as magnetic energy.Cosmological models typically assume that the contribution from SN is much larger, and inject an energy E mag,inj = E SN with typically ranging from 0.01 to 0.03.This implies field strengths of about 10 −4 G on the typical scale of a remnant, which is about 5−10 pc.While these approaches provide a useful starting point for structure formation and cosmological simulations, they are not based on detailed models or observations of stellar magnetism.
In this Letter, we use our current understanding of stellar magnetism to provide a simple estimate of the maximum amount of magnetic energy that can be injected by a primordial L6, page 1 of 5 stellar population.To our knowledge, this estimate has not been performed previously, and it can be very useful for studies of cosmological magnetic field evolution.We show that the typical magnetic seeding by SN explosions used so far in cosmological simulations is much larger than what can be produced by a typical stellar population.Instead, the smaller values assumed by some galactic dynamo models are closer to our estimates.

Stellar magnetic fields
Equipartition between magnetic energy and gravitational binding energy sets the maximum-allowed magnetic field B max ∝ M/R 2 in a star.In early-type stars, this maximum surface magnetic field is about 10 7 G.On the other hand, the largest observed fields at the surface of stars seem to be much smaller, with B max,obs ∼ 10 −3 −10 −4 B max (Braithwaite & Spruit 2017).This means that magnetic fields usually provide a very small perturbation to the stellar hydrostatic equilibrium.Also, in earlytype stars, the maximum observed amplitude of surface magnetic fields is 10 3 −10 4 G (Petit et al. 2019).
The origin and evolution of stellar magnetism is not fully understood.It is clear that dynamo action can amplify a seed magnetic field in stars, with the dynamo tapping into the energy reservoir provided by turbulent convection and/or differential rotation (Brandenburg et al. 2012).This is the case of the Sun and of low-mass stars in general.On the other hand, the largescale, high amplitude fields observed at the surface of 5−10% of early-type stars ("fossil fields") are thought to be inherited during the star formation process (e.g., Donati & Landstreet 2009).It is also possible that some of these fields are created via interactions with a companion star (Ferrario et al. 2009).Finally, dynamo action might be at play in the interiors of early-type stars as well, in their convective cores (Augustson et al. 2016), nearsurface convective zones (Cantiello & Braithwaite 2011), and radiative regions (Spruit 2002;Wheeler et al. 2015;Fuller et al. 2019).
As we outlined in the introduction, state-of-the-art cosmological models and some galactic dynamo models approach stellar magnetic seeding by introducing a magnetic field corresponding to roughly 1% the SN energy in feedback regions.The motivation behind this choice is based on estimates from galactic SN remnants.For instance, Beck et al. (2013) estimate an average SN remnant with a radius of 5 pc and magnetic field strength 10 −4 G, citing the review of magnetism around local SN remnants by Reynolds et al. (2012).This estimate corresponds to about 6 × 10 48 erg in total magnetic energy.However, assuming that such strong magnetic fields at the SN stage come directly from the progenitor implies paradoxically high magnetization for the star.If the interstellar medium at the typical radius of a SN remnant R SN contains E mag,inj = 0.01 E SN , with E SN = 10 51 erg, then the magnetic energy density of the progenitor would be E mag,inj divided by the volume of the star: assuming no magnetic energy was lost or gained during the explosion.For an O-type star with a radius R * = 100 R and E mag,inj = 10 49 erg, u mag, * 10 10 erg cm −3 .Since u mag = B 2 /8π, the typical magnetic field of the progenitor would be roughly B * = 10 5 −10 6 G.This number is at least one order of magnitude larger than the maximum magnetic field observed at the surface of early-type stars (e.g., Petit et al. 2019).This inconsistency is exacerbated assuming a smaller radius for the progenitor star, or dissipative losses during the evolution of the SN.
Clearly, the typical values for SN-injected magnetic fields used so far in cosmological simulations are unreasonably high; although, they match the observed magnetic fields around local SN remnants, which are on the order of a few microgauss to a milligauss.However, there is currently no theory that involves magnetization of the SN ejecta to explain these values.Instead, current theories involve the amplification of the magnetic field surrounding the remnant, through either compression and turbulence (Inoue et al. 2009), or a cosmic-ray-driven dynamo (e.g., Xu & Lazarian 2017).In the following section, we attempt to make a physically meaningful estimate for magnetic seeding from SNe.

The maximum SN-injected magnetic energy from a stellar population
We construct a simple model for the magnetization of a primordial cloud by SN ejecta.In this model, we assume that the stars return all of their magnetic field to the ISM.In doing so, we implicitly require that the entire mass of the star is ejected in the ISM, neglecting the mass and magnetization of the resulting compact object.Obviously, this assumption is not true for most massive stars, but it helps our purpose of calculating an upper limit to the magnetic field that can be injected by the population.We also neglect dissipative losses or other forms of magnetic energy gain, so that the magnetic energy is conserved.
The model essentially consists of the following relation for the total magnetic energy generated by a cluster of mass M cl : where ξ(m) is the assumed initial mass function (IMF) of the primordial stellar population, B * the magnetic field of a star of mass m1 , R * (m) its radius, M SN the minimum mass for SNe, and M max (M cl ) the maximum stellar mass of a cluster of mass M cl .
We subsequently estimate E mag,cl for different masses of the stellar cluster.All the quantities that enter Eq. ( 2) are only partially known, with ξ(m) and B * being the least understood so far.However, we can make order-of-magnitude estimates based on what is currently known.

IMF of the star cluster
The IMF of stars in primordial, low-metallicity environments is highly uncertain.Numerical work (e.g., Clark et al. 2008;Dopcke et al. 2013;Chon et al. 2021) has shown that for metallicities below Z/Z 10 −5 , fragmentation becomes inefficient and the stellar IMF favors high-mass stars, with an almost flat distribution at high masses.Above this metallicity limit, simulations recover a Salpeter-like power-law slope in high masses (Salpeter 1955).Therefore, here we use two IMF models: (i) a Salpeter IMF, ξ S (m) = ξ 0 m −2.35 (which for m > 0.5 M coincides with the Kroupa 2001 and Chabrier 2003 models); and (ii) a top-heavy IMF, with a high-mass slope of −1.5: ξ TH (m) = ξ 0 m −1.5 .As limits in the integration of the IMF, we set M SN = 8 M and M max (M cl ) = 1.2 • M 0.45 cl (Larson 2003).Finally, for R * (m), we use the simple relation for nonconvective stars, R * (m) ∝ m 0.57 .The dispersion of each distribution with mean µ i is set to σ i = 0.1µ i .

Stellar magnetic field distribution
For the distribution of B * we use two different models, starting from assumptions about the surface magnetic field of the stars.The first model is driven from local observations of earlytype stars, which show a very low incidence of detectable magnetization.Specifically, only about 5 to 10% of the observed stars host magnetic fields of 10 3 −10 4 G, while the rest fall below about 100 G (Hubrig et al. 2011;Alecian et al. 2014;Morel et al. 2014;Wade et al. 2014;Petit et al. 2019).In fact, there are indications that OB stars present a magnetic dichotomy similar to that observed for Ap stars.Interestingly, the incidence of high magnetization does not seem to correlate with the stellar parameters (Wade & MiMeS Collaboration 2015).Therefore, our first model approximates the distribution of stellar surface magnetic fields, B * ,sur , as a bimodal for all the stars contained in the integral of Eq. ( 2).The parameters of the distribution are the two means, µ 1 and µ 2 , which we vary from 10 G < µ 1 < 100 G and 500 G < µ 2 < 5 × 10 3 G, respectively, and f 2 , the incidence of highly magnetized stars, which takes values from 0.05 < f 2 < 0.2.
Recently, Farrell et al. (2022) showed that the surface magnetic field distribution of AB stars is well fitted by a Gaussian with a mean at about 1 kG.Although their study involved less massive stars, we explore a second model where all OB stars are strongly magnetized, in order to push our calculations toward an upper limit.In this case, B * ,sur is a single Gaussian with a peak that we vary between 10 3 < µ < 10 4 G.The two panels of Fig. 1 illustrate the resulting range of B * distributions for the two models, for a stellar population of 10 5 M .The different line colors correspond to different sets of parameters.
Going from the surface to the global field strength requires an assumption as to the internal magnetic field distribution.Here, we assume a dipole distribution of the magnetic field strength as a function of radius r: Since the dipole field diverges at the center, in Eq. ( 3) the integration starts from a core radius, R core .O-and B-type stars rotate rapidly and possess convective cores, which means that a dynamo could act in their interior (Augustson et al. 2016).
Although a core dynamo cannot be an explanation for the strong surface fields (since there is no time for such a field to reach the surface -Charbonneau & MacGregor 2001), a fossil field from a convective core could still be partly expelled with the SN.Equipartition fields in the convective cores of OB stars are expected to be in the range 10 5 −10 6 G (Augustson et al. 2016).
With this in mind, we estimate R core as the largest of two radii: (i) the radius at which the magnetic field strength from Eq. ( 3) reaches the maximum stable field according to dynamo models (e.g., Augustson et al. 2016), B * (R core ) = 10 6 G; or (ii) the mean core radius for massive stars, R core = 0.2 R *2 .Inside the core, the magnetic field is considered constant, either at the maximum value of 10 6 G or to the value reached at R core from Eq. (3).

Results
Figure 2 shows the result of this calculation in terms of the fraction of injected magnetic energy E mag,inj over total SN energy E SN,tot .This quantity is equivalent to the factor employed in numerical simulations of magnetic field injection.Even at the highest limit of this range, assuming a top-heavy IMF and all the stars strongly magnetized (red line of the bottom panel), does not exceed 10 −7 .This strong upper limit value is still 10 5 times lower than the number typically assumed in cosmological models.
Figure 3 shows the same results in terms of the magnetic field strength.Here, we have carried out two calculations, corresponding to two approaches of distributing the SN ejecta.One assumes that all the stars deposit their magnetic energy in a single superbubble with a radius of 300 pc (so that their magnetic energies are added), and this is shown as solid lines.The other, shown in dashed lines, assumes that each star explodes in isolation, forming a SN remnant with an average radius of 30 pc (so that their magnetic energies are averaged).In reality, both scenarios are possible, depending on the clustering of the stars.Here, we see that an extremely massive cluster, with a top-heavy IMF and all of its early-type, strongly magnetized stars, depositing their magnetic field in a single superbubble of just a 300 pc radius, would yield a magnetic field of about 10 −7 G.More typical values instead are between 10 −10 −10 −8 G, which is comparable to other primordial seeding mechanisms.

Discussion
We have shown that the magnetic fields used in cosmological simulations (e.g., Beck et al. 2013;Vazza et al. 2017;Katz et al. 2019;Garaldi et al. 2021;Martin-Alvarez et al. 2021) under the SN-injection scenario are exceedingly high.However, in cosmicray-driven galactic dynamo simulations, Hanasz et al. (2009) and Kulpa-Dybeł et al. (2011, 2015) only introduced magnetization in 10% of the SN events.This approach could be considered equivalent to our bimodal distribution of stellar magnetization, with the employed values (10 −5 µG) lower than our average estimates.Butsky et al. (2017) used magnetic seeds with strengths close to those we predict in this Letter: they injected about 6 × 10 −7 of the total SN energy as magnetic energy.All the above works found these small values to be sufficient for driving a galactic dynamo.
The upper limits we calculate here are sensitive to the assumed magnetic field distribution in the stellar interior and to the maximum-allowed surface magnetic field strengths.Assuming a constant magnetic field, B * = B sur , in the entire volume of the star instead of a truncated dipole lowers our estimates for the magnetic energy injection by roughly an order of magnitude.Allowing for extreme surface magnetic fields on the order of a few times 10 5 G increases the maximum magnetic yield by roughly a factor of 200.We note that our model does not take any dissipative process into account.Moreover, we do not consider the magnetization of the compact objects left behind by the SNe: they could retain a significant amount of the magnetic energy assumed to be injected into the ISM.
Finally, here we have examined the effect of the short-lived, explosive, early-type stars on magnetic field seeding.However, stellar magnetic field injection is also possible through a dynamo-active accretion disk around a protostar.These dynamogenerated fields are strong enough to launch a wind, which in turn could magnetize the ISM (e.g., Brandenburg 2000;von Rekowski et al. 2003;Dyda et al. 2018).Such disks are long-lived for low-mass stars, which can keep seeding the ISM with low levels of magnetization on small scales.This scenario is worth investigating with dedicated numerical models.

Conclusions
We conclude that stellar magnetic fields purely advected through SN explosions can tap only 10 −10 −10 −7 of the SN explosion energy.The highest limit of these estimates is five orders of magnitude lower than what is currently used in cosmological simulations, even without accounting for diffusion or a residual magnetization of the compact object.Therefore, SNinjected magnetic fields can only provide the seed for additional L6, page 4 of 5 processes, such as a turbulent dynamo, which can amplify them to the near-equipartition values measured in local galaxies.Since early galaxies do host strong turbulence, this scenario can be probed with high-resolution simulations of these systems.The obvious next step for this investigation involves resolved numerical simulations of SN-injected fields that can account for magnetic diffusion and model possible dynamo action.

Fig. 1 .
Fig. 1.Range of stellar surface stellar magnetic field distributions used in the two models.Top: bimodal distribution model.Bottom: single Gaussian model.Both models are shown here for a cluster of 10 5 M and a Salpeter IMF.Each color corresponds to a set of parameters for each distribution: (µ 1 , µ 2 , f 2 ) for the bimodal, and µ for the Gaussian.The dispersion of each distribution with mean µ i is set to σ i = 0.1µ i .

Fig. 2 .
Fig. 2. Magnetic energy injected by a stellar population of mass M cluster over SN energy of the same population.The latter is calculated as 10 51 erg times the total number of massive stars.The shaded area shows the range of values assuming different magnetic field distributions for the stars.The top panel assumes bimodal magnetic field distributions (top panel of Fig. 1) and the bottom one assumed Gaussian ones (bottom panel of Fig. 1).The two lines in each panel show results for different IMFs.

Fig. 3 .
Fig. 3. Total magnetic field in a single superbubble, B SB , or mean magnetic field in SN remnants, B SN , injected by a stellar population of mass M cluster .The top panel assumes bimodal magnetic field distributions (top panel of Fig. 1) and the bottom one assumes Gaussian ones (bottom panel of Fig. 1).Results are shown for different IMFs, with the shaded areas indicating the range over different stellar magnetic field distributions.