arXiv subjects

Biman B. Nath

Publications and source records attributed to Biman B. Nath.

At least 19 recordsLinked to original sources

arXiv2026

Dust destruction in bubbles driven by multiple supernovae explosions

Dust lifetime derived from an isolated supernova (SN) evolution in the interstellar medium is known to be an order of magnitude shorter than the time needed to replenish dust mass by its production in various Galactic sources. We show, using 3-D numerical hydrodynamical simulations, that destruction of dust in the case of multiple SNe in a star cluster is markedly different from that in an isolated SN. We find that the mass of dust destroyed in the bubble does not grow for a considerable time, while SNe continue to explode. This regime is attained at saturation timescale, which is proportional to SNe rate in cluster. We show that the mass of dust destroyed in bubble per SN decreases for higher SN rate. Thus, the destruction efficiency -- defined as the ratio of the the total mass of dust destroyed by clustered SNe to that destroyed by the same number of isolated SNe -- in bubbles evolved in a homogeneous medium drops for massive clusters, e.g. around clusters with $M_\ast \gtrsim 4\times 10^4 M_\odot$ it is less $0.4$\%. For lower mass clusters, the efficiency is proportional to the average time delay between SNe. We found that each cluster with $M_\ast \lesssim 4\times 10^4 M_\odot$ destroys the same mass of dust as a single isolated SN. In a clumpy medium in bubbles formed around clusters with $M_\ast \sim 4\times 10^4 M_\odot$ and up to 4 times around $M_\ast \sim 8\times 10^5 M_\odot$. We argue that the interstellar dust swept up by multiple SNe almost completely survives in the shells of bubbles around such massive clusters. Therefore, the destruction of the interstellar dust is controlled by SNe in low-mass clusters. We point out that the interstellar dust lifetime for a given SN rate is at least a factor $\sim 10$ longer as compared to the estimates derived from an isolated SN. (abridged)

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arXiv2026

The Radio--X-ray Correlation of High-Redshift AGN: A Numerical Study of Inverse-Compton Scattering of the CMB Photons in Relativistic Jets

Relativistic jets from active galactic nuclei are expected to exhibit strong redshift evolution in their radiative output due to the increasing energy density of the cosmic microwave background (CMB). We investigate the role of inverse Compton (IC) scattering of CMB photons in regulating the radio and X-ray emission from large-scale jets using three-dimensional relativistic magnetohydrodynamic simulations coupled with a hybrid Eulerian-Lagrangian particle framework. By keeping the jet dynamics and ambient medium properties fixed across redshifts, we are able to isolate the impact of the cosmological evolution of the CMB on the jet radiation. From our simulations, we construct synthetic spectral energy distributions and intensity maps considering synchrotron and IC/CMB losses along with particle acceleration from shocks. We are able to reproduce the weak redshift dependence of radio luminosity and the strong enhancement of X-ray emission toward high redshift that is observed in radio-loud quasars. At high redshift, the X-ray luminosity follows the expected $(1+z)^4$ scaling, confirming IC/CMB as the dominant mechanism driving the X-ray enhancement. The resulting X-ray-to-radio flux ratio increases systematically with redshift and is consistent with observational constraints. Finally, we show that slower jets exhibit a stronger redshift evolution of the X-ray enhancement than faster jets, highlighting the critical role of jet propagation length scales and particle energy evolution. The simulations also naturally reproduce the steepening of the radio spectral index with redshift - the $α$-$z$ relation - thus providing a unified framework that allows to interpret the multiwavelength properties of high-redshift radio sources.

astro-ph.HE↗

arXiv2025

Misty, patchy, and turbulent: constraining the cool circumgalactic medium with mCC

The circumgalactic medium (CGM) is the largest baryon reservoir around galaxies, but its extent, mass, and temperature distribution remain uncertain. We propose that cool gas ($\sim 10^4$ K) in the CGM resides in clumpy structures referred to as cloud complexes (CCs) rather than uniformly filling the entire CGM volume. Each CC contains a mist of tiny cool cloudlets dispersed in a warm/hot medium ($\sim 10^5$ - $10^6$~K). Modeling CCs in the mist limit (unit area covering fraction within a CC) simplifies the calculation of observables like ion absorption columns, equivalent widths, compared to modeling individual cloudlets from first principles. Through Monte Carlo realizations of CCs, we explore how CC properties affect the observed variation in observables. We find that a power-law distribution of CCs ($dN_{\rm CC}/dR \propto R^{-1}$) with a total of $\sim 10^3$ CCs each with a radius of $\sim 10$ kpc and total cool gas mass of $\sim 10^{10} M_\odot$ reproduces MgII column density and equivalent width distribution trends with impact parameter for the COS-Halos sample (Werk+ 2013). We further show that the area-averaged MgII column density, combined with the area covering fraction, provides a robust proxy for estimating the cool CGM mass, independent of other model parameters. Modeling a larger number of (smaller size) cloudlets within a CC shows that line blending from individual cloudlets results in turbulent broadening on the CC scale. This work presents a practical framework for linking CGM models with observations of a multiphase CGM, illuminating the distribution of cool gas in galaxy halos.

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arXiv2025

Survival of dust in super-dusty galaxies at redshifts $z \approx 5-8$

The presence of dust in galaxies at redshifts $z>5$ is commonly connected with core collapse supernovae (SN). Galaxies with exceptionally large dust mass, of order of $1 - 3$\% of the galaxy stellar mass, have been detected during the last decade. The required SN dust yield is $\gtrsim 1~M_\odot$ per supernova, which is comparable to the theoretically predicted maximum. However, the reverse shock (RS) penetrating the SN ejecta significantly destroy the dust particles nucleating there through sputtering. The resulting net dust mass injected into the interstellar gas after processing by the RS turns out to be $\lesssim 0.1~M_\odot$ per SN. This makes the explanation of the existence of $z>5$ galaxies with dust masses as high as $M_d\gtrsim (0.01-0.03)M_\ast$ a challenging one. In this paper we present arguments in favor of an efficient inhibition of the sputtering behind the RS, caused by a strong radiation cooling from the dust in the supernova ejecta.

astro-ph.GA↗

arXiv2024

Can supernova from runaway stars mimic the signs of absorbing `super-virial' gas?

The recent detection of large column density absorption lines from highly ionized gas in a few directions through the circumgalactic medium (CGM) of the Milky Way (MW) has been puzzling. The inferred temperature from these absorption lines far exceeds the virial temperature of the MW, and the column densities are also too large to be easily explained. In this paper, we propose a novel idea to explain these observations and claim that they may not have originated from the CGM, but from a totally different type of source, namely, stellar ejecta from supernovae (SNe) above the Galactic disk that happen to lie in the line of sight to the background quasars. About $\sim 20\%$ of massive OB stars (progenitors of core-collapse supernovae) are known to be runaway stars that have high ejection velocities near the Galactic plane and can end up exploding as SNe above the Galactic disk. We show that the associated reverse shock in the supernova remnant in the early non-radiative phase can heat the ejecta to temperatures of $\gtrsim 10^7\,{\rm K}$ and can naturally explain the observed high column density of ions in the observed `super-virial' phase along with $α$-enriched super-solar abundance that is typical of core-collapse supernovae. However, SNe from runaway stars has a covering fraction of $\lesssim 0.7 \%$ and thus can only explain the observations along limited sightlines.

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arXiv2024

On the origin of the $10^7$ K hot emitting gas in the Circumgalactic medium of the Milky Way

The presence of the $\approx 10^6$ K gas in the circumgalactic medium of the Milky Way has been well established. However, the location and the origin of the newly discovered hot gas at `super-virial' temperatures of $\approx 10^7$ K have been puzzling. This hot gas has been detected in both absorption and emission; here we focus on the emitting gas only. We show that both the `virial' and the `super-virial' temperature gas as observed in \emph{emission} occupy disk-like extraplanar regions, in addition to the diffuse virial temperature gas filling the halo of the Milky Way. We perform idealized hydrodynamical simulations to show that the $\approx 10^7$ K emitting gas is likely to be produced by stellar feedback in and around the Galactic disk. We further show that the emitting gas at both super-virial and virial temperatures in the extraplanar regions is metal enriched and is not in hydrostatic equilibrium with the halo but is continuously evolving.

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arXiv2024

Beyond radial profiles: Using log-normal distributions to model the multiphase circumgalactic medium

Recent observations and simulations reveal that the circumgalactic medium (CGM) surrounding galaxies is multiphase, with the gas temperatures spanning a wide range at most radii, $\sim 10^4\ {\rm K}$ to the virial temperature ($\sim 10^6$ K for Milky Way). Traditional CGM models using simple density profiles are inadequate at reproducing observations that indicate a broad temperature range. Alternatively, a model based on probability distribution functions (PDFs) with parameters motivated by simulations can better match multi-wavelength observations. In this work, we use log-normal distributions, commonly seen in the simulations of the multiphase interstellar and circumgalactic media, to model the multiphase CGM. We generalize the isothermal background model by Faerman et al. 2017 to include more general CGM profiles. We extend the existing probabilistic models from 1D-PDFs in temperature to 2D-PDFs in density-temperature phase space and constrain its parameters using a Milky Way-like {\tt Illustris TNG50-1} halo. We generate various synthetic observables such as column densities of different ions, UV/X-ray spectra, and dispersion and emission measures. X-ray and radio (Fast Radio Burst) observations mainly constrain the hot gas properties. However, interpreting cold/warm phase diagnostics is not straightforward since these phases are patchy, with inherent variability in intercepting these clouds along arbitrary lines of sight. We provide a tabulated comparison of model predictions with observations and plan to expand this into a comprehensive compilation of models and data. Our modeling provides a simple analytic framework that is useful for describing important aspects of the multiphase CGM.

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arXiv2023

Dust-free starburst galaxies at redshifts $z>10$

One of the most distant galaxies GN-z11 was formed when the Universe was $\le$ 400 Myr old, and it displays a burst-like star formation rate $\sim 25~M_\odot$ yr$^{-1}$ with a metallicity $Z\sim 0.2\pm 0.1Z_\odot$. It resembles $z=2-3$ galaxies (at ``cosmic noon") except for the fact that the measured reddening $E(B-V)=0.01\pm 0.01$ indicates the presence of little or no dust. This marked absence of dust hints towards violent dynamical events that destroy or evacuate dust along with gas out of the galaxy on a relatively short time scale and make it transparent. We apply a 3D numerical model to infer possible physical characteristics of these events. We demonstrate that the energetics of the observed star formation rate is sufficient to tear apart the dusty veil on time scales of $20-25$ Myr. This can explain the apparent lack of evolution of UV luminosity function of galaxies between and $z\ge 10$ and $z\sim 7$, by compensating for the lower galaxy masses at higher redshift by the absence of dust. We show, however, that this is a temporary phenomenon and soon after the last of the supernovae explosions have taken place, the expanding shell shrinks and obscures the galaxy on time scales of $\approx 5-8$ Myr.

astro-ph.GA↗

arXiv2023

Disc-halo gas outflows driven by stellar clusters as seen in multiwavelength tracers

We consider the dynamics of and emission from growing superbubbles in a stratified interstellar gaseous disc driven by energy release from supernovae explosions in stellar clusters with {masses $M_{cl}= 10^5-1.6\times 10^6~M_\odot$}. Supernovae are spread randomly within a sphere of $r_c=60$ pc, and inject energy episodically with a specific rate $1/130~M_\odot^{-1}$ proportional to the star formation rate (SFR) in the cluster. Models are run for several values of SFR in the range $0.01$ to $0.1~M_\odot$ yr$^{-1}$, with the corresponding average surface energy input rate $\sim 0.04-0.4$ erg cm$^{-2}$ s$^{-1}$. We find that the discrete energy injection by isolated SNe are more efficient in blowing superbubbles: asymptotically they reach heights of up to 3 to 16 kpc for $M_{cl}=10^5-1.6\times 10^5~M_\odot$, correspondingly, and stay filled with a hot and dilute plasma for at least 30 Myr. During this time they emit X-ray, H$α$ and dust infrared emission. X-ray liminosities $L_X\propto {\rm SFR}^{3/5}$ that we derive here are consistent with observations in star-forming galaxies. Even though dust particles of small sizes $a\leq 0.03~μ$m are sputtered in the interior of bubbles, larger grains still contribute considerably ensuring the bubble luminosity $L_{\rm IR}/{\rm SFR}\sim 5\times 10^7 L_\odot M_\odot^{-1} ~{\rm yr}$. It is shown that the origin of the North Polar Spur in the Milky Way can be connected with activity of a cluster with the stellar mass of $\sim 10^5~M_\odot$ and the ${\rm SFR}\sim 0.1~M_\odot$ yr$^{-1}$ some 25--30 Myr ago. Extended luminous haloes observed in edge-on galaxies (NGC 891 as an example) can be maintained by disc spread stellar clusters of smaller masses $M_\ast \simlt 10^5~M_\odot$.

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arXiv2022

The non-thermal secondary CMB anisotropies from a cosmic distribution of radio galaxy lobes

Current and upcoming high angular resolution and multi-frequency experiments are well poised to explore the rich landscape of secondary CMB anisotropies. In this context, we compute for the first time, the power spectrum of CMB fluctuations from a cosmological distribution of evolving lobes of giant radio galaxies. We, also, explicitly take into account the non-thermal electron distribution, which has important implications for the inference of the CMB angular power spectrum. We calculate the mean global non-thermal y-distortion, \ynt. For observationally reasonable distribution of the jet luminosities in the range of $10^{45}-10^{47}$ ergs$^{-1}$, we find \ynt to be less than $10^{-5}$, and hence not violating the COBE limit as previously claimed. Using the unique spectral dependence of the non-thermal SZ, we show that a detection of \ynt can be within reach at the level of $\gtrsim 5σ$ from a future PIXIE-like experiment provided we understand the foregrounds precisely. The total non-thermal SZ power spectrum, $C^{NT}_\ell$, from the radio lobes peaks at $\ell \sim 3000$ with an amplitude $\sim 1\%$ of thermal SZ power spectrum from galaxy clusters. A a detection of the $C^{NT}_\ell$, with a PIXIE-like sensitivity experiment, can lead to $\sim 5σ$ constraint on the mass dependence of the jet luminosity with the constraint becoming, at least, ten times better for the proposed more ambitious CMB-HD survey. This will, further, lead to the tightest constrain on the central black hole mass -to- host halo mass scaling relations.

astro-ph.CO↗

arXiv2022

Heating of the intracluster medium by buoyant bubbles and sound waves

Active galactic nuclei (AGN) powered by the central Super-Massive Black Holes (SMBHs) play a major role in modifying the thermal properties of the intracluster medium (ICM). In this work, we implement two AGN heating models: (i) by buoyant cavities rising through stratified ICM (effervescent model) and, (ii) by viscous and conductive dissipation of sound waves (acoustic model). Our aim is to determine whether these heating models are consistent with ICM observables and if one is preferred over the other. We assume an initial entropy profile of ICM that is expected from the purely gravitational infall of the gas in the potential of the dark matter halo. We then incorporate heating, radiative cooling, and thermal conduction to study the evolution of ICM over the age of the clusters. Our results are: (i) Both the heating processes can produce comparable thermal profiles of the ICM with some tuning of relevant parameters. (ii) Thermal conduction is crucially important, even at the level of 10\% of the Spitzer values, in transferring the injected energy beyond the central regions, and without which the temperature/entropy profiles are unrealistically high. (iii) The required injected AGN power scales with cluster mass as $M_{\rm vir}^{1.5}$ for both models. (iv) The required AGN luminosity is comparable with the observed radio jet power, reinforcing the idea that AGNs are the dominant heating source in clusters. (v) Finally, we estimate that the fraction of the total AGN luminosity available as the AGN mechanical luminosity at $0.02r_{500}$ is less than 0.05\%.

astro-ph.CO↗

arXiv2022

Exploring diffuse radio emission in galaxy clusters and groups with the uGMRT and the SKA

Diffuse radio emission has been detected in a considerable number of galaxy clusters and groups, revealing the presence of pervasive cosmic magnetic fields, and of relativistic particles in the large-scale structure (LSS) of the Universe. Since cluster radio emission is faint and steep spectrum, its observations are largely limited by the instrument sensitivity and frequency of observation, leading to a dearth of information, more so for lower-mass systems. The unprecedented sensitivity of recently commissioned low-frequency radio telescope arrays, aided by the development of advanced calibration and imaging techniques, have helped in achieving unparalleled image quality. At the same time, the development of sophisticated numerical simulations and the availability of supercomputing facilities have paved the way for high-resolution numerical modeling of radio emission, and the structure of the cosmic magnetic fields in LSS, leading to predictions matching the capabilities of observational facilities. In view of these rapidly-evolving scenerio in modeling and observations, in this review, we summarise the role of the new telescope arrays and the development of advanced imaging techniques and discuss the detections of various kinds of cluster radio sources. In particular, we discuss observations of the cosmic web in the form of supercluster filaments, studies of emission in poor clusters and groups of galaxies, and of ultra-steep spectrum sources. We also review the current theoretical understanding of various diffuse cluster radio sources and the associated magnetic field and polarization. As the statistics of detections improve along with our theoretical understanding, we update the source classification schemes based on their intrinsic properties. We conclude by summarising the role of the upgraded GMRT and our expectations from the upcoming Square Kilometre Array (SKA) observatories.

astro-ph.CO↗

arXiv2022

Gamma-rays from the circumgalactic medium of M31

We discuss the production of $γ$-rays from cosmic rays (CR) in the circumgalactic medium (CGM) of Andromeda (M31) in light of the recent detection of $γ$-rays from an annular region of $\sim 5.5-120$ kpc away from the M31 disc. We consider the CRs accelerated as a result of the star-formation in the M31 disk, which are lifted to the CGM by advection due to outflow and CR diffusion. The advection time scale due to bulk flow of gas triggered by star formation activity in the M31 disc is comparable ($\sim$ Gyr) to the diffusion time scale with diffusion coefficient $\ge10^{29}$ cm$^2$ s$^{-1}$ for the propagation of CR protons with energy $\sim 412$ GeV that are responsible for the highest energy photons observed. We show that a leptonic origin of the $γ$-rays from cosmic ray (CR) electrons has difficulties, as the inverse Compton time scale ($\sim$Myr) is much lower than advection time scale ($\sim$Gyr) to reach $120$ kpc. Invoking CR electrons accelerated by accretion shocks in the CGM at $\sim100-120$ kpc does not help since it would lead to diffuse X-ray features that are not observed. We, therefore, study the production of $γ$-rays via hadronic interaction between CR protons and CGM gas with the help of numerical two-fluid (thermal + CR) hydrodynamical simulation. We find that a combination of these mechanisms, that are related to the star formation processes in M31 in the last $\sim $ Gyr, along with diffusion and hadronic interaction, can explain the observed flux from the CGM of M31.

astro-ph.GA↗

arXiv2022

Constraints on cosmic rays in the Milky Way circumgalactic medium from OVIII observations

We constrain the cosmic ray (CR) population in the circumgalactic medium (CGM) of Milky Way by comparing the observations of absorption lines of OVIII ion with predictions from analytical models of CGM : precipitation (PP) and isothermal (IT) model. For a CGM in hydrostatic equilibrium, the introduction of CR suppresses thermal pressure, and affects the OVIII ion abundance. We explore the allowances given to the ratio of CR pressure to thermal pressure ($\rm{P}_{\rm{CR}}/\rm{P}_{\rm{th}}=η$), with varying boundary conditions, CGM mass content, photoionization by extragalactic ultraviolet background and temperature fluctuations. We find that the allowed maximum values of $η$ are : $η\lesssim10$ in the PP model and $η\lesssim6$ in the IT model. We also explore the spatial variation of $η$ : rising ($η=Ax$) or declining ($η=A/x$) with radius, where A is the normalization of the profiles. In particular, the models with declining ratio of CR to thermal pressure fare better than those with rising ratio with suitable temperature fluctuation (larger $σ_{\rm ln T}$ for PP and lower for IT). The declining profiles allow $A\lesssim8$ and $A\lesssim10$ in the case of IT and PP models, respectively, thereby accommodating a large value of $η\,(\simeq 200)$ in the central region, but not in the outer regions. These limits, combined with the limits derived from $γ$-ray and radio background, can be useful for building models of Milky Way CGM including CR population. However, the larger amount of CR can be packed in cold phase which may be one way to circumvent these constraints.

astro-ph.GA↗

arXiv2022

Star-dust geometry main determinant of dust attenuation in galaxies

Analysing a large representative sample of local galaxies (8707), we find that the variation in the shape of their dust attenuation curves is driven primarily by their structure, i.e., distribution of stars (and dust) within them. The attenuation curve for spheroid dominated galaxies, as compared to the disc dominated ones, is nearly twice as steep. Both structural types cover distinct ranges of attenuation slope values. Similar findings are reflected in the case of star-forming and passive galaxies. Spheroids and passive galaxies witness minimal attenuation in the optical compared to UV wavelengths underlining the lack of dusty birth-clouds that define complex star-dust geometry. The distinction in the attenuation properties of spheroids and discs is maintained in each stellar mass range emphasising that structure is the primal cause of variation. However, within a structural group, the attenuation curve becomes shallower with both the increase in total stellar mass and optical depth of the galaxy. Overall, with the extinction curve fixed to be the same for all galaxies, the star-dust geometry emerges to be the prime determinant of the variation in their attenuation properties.

astro-ph.GA↗

arXiv2022

Dust in Clusters of Galaxies

The presence of dust in the intracluster medium (ICM) has been a long-standing problem that is still awaiting elucidation. Direct observational diagnostics are rather challenging (though not impossible) either because of a sparse distribution of dust in the intracluster space that makes extinction measurements difficult or because of a low surface brightness of infrared emission from dust. Complex indirect approaches are currently available that can overcome uncertainties and provide a reasonable understanding of the basic regulations of the physical state of dust in the ICM. Contrary to the common opinion that the hot ICM does not allow dust to survive and manifest, many sparse observational data either directly point out that dust exists in the intracluster space or its presence is consistent with the data. Highly divergent data in direct evidence and highly uncertain indirect indicators are often connected either with dust fragility in a hot environment, the possible compactness of spatial (clumpy) dust distribution in the ICM, or dynamical features of dust transport. The source of dust is obviously connected with galaxies, and it turns out that in most cases, dust is carried from galaxies into the ICM while being thermally and dynamically shielded against the hostile influence of high-energy ions. In this review, we briefly discuss related issues from observational and theoretical points of view, including the transport of dust into the ICM, and the associated destructive and protective mechanisms and their characteristic time scales.

astro-ph.GA↗

arXiv2022

Non-thermal Sunyaev-Zeldovich signal from radio galaxy lobes

Energetic electrons in the lobes of radio galaxies make them potential sources for not only radio and X-rays but also Sunyaev-Zeldovich (SZ) distortions in the cosmic microwave background (CMB) radiation. Previous works have discussed the energetics of radio galaxy lobes, but assuming thermal SZ effect, coming from the non-thermal electron population. We use an improved evolutionary model for radio galaxy lobes to estimate the observed parameters such as the radio luminosity and intensity of SZ-distortions at the redshifts of observation. We, further, quantify the effects of various relevant physical parameters of the radio galaxies, such as the jet power, the time scale over which the jet is active, the evolutionary time scale for the lobe, etc on the observed parameters. For current SZ observations towards galaxy clusters, we find that the non-thermal SZ distortions from radio lobes embedded in galaxy clusters can be non-negligible compared to the amount of thermal SZ distortion from the intra-cluster medium and, hence, can not be neglected. We show that small and young (and preferably residing in a cluster environment) radio galaxies offer better prospects for the detection of the non-thermal SZ signal from these sources. We further discuss the limits on different physical parameters for some sources for which SZ effect has been either detected or upper limits are available. The evolutionary models enable us to obtain limits, previously unavailable, on the low energy cut-off of electron spectrum ($p_{min} \sim 1\hbox{--}2$) in order to explain the recent non-thermal SZ detection. Finally, we discuss how future CMB experiments, which would cover higher frequency bands ($>$400 GHz), may provide clear signatures for non-thermal SZ effect.

astro-ph.HE↗

arXiv2022

Radio Halo of NGC 4631: Comparing Observations and Simulations

We present low frequency observations at $315$ and $745$ MHz from the upgraded Giant Metrewave Radio Telescope (uGMRT) of the edge-on, nearby galaxy NGC $4631$. We compare the observed surface brightness profiles along the minor axis of the galaxy with those obtained from hydrodynamical simulations of galactic outflows. These are 3D simulations that replicate star-formation in a Milky-Way mass galaxy and follow magnetized outflows emerging from the disk. We detect a plateau-like feature in the observed emission at a height of $2-3$ kpc from the mid-plane of the galaxy, in qualitative agreement with that expected from simulations. This feature is believed to be due to the compression of magnetic fields behind the outer shocks of galactic outflows. We model the observed surface brightness profiles by assuming an exponential as well as a Gaussian fitting model. Using $χ^2$ statistics, we find that the exponential model fits the profiles better and we use it to determine the scale heights. We estimate the scale height for the synchrotron radio emission to be $\sim 1$ kpc. The timescales for advection due to outflows and diffusion of cosmic ray electrons are $\gtrsim 5$ and $\sim 160$ Myr, respectively. Because advection acts on a timescale much shorter than diffusion, we conclude that in NGC $4631$ advection, rather than diffusion, plays the dominant role in the formation of radio halo. The spectral index image with regions of flatter radio spectral index in the halo appears to indicate possible effects of gas outflow from the plane of the galaxy.

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