# Empirical relations in thermal Comptonization

Richárd Bagi, Filip Alamaa, Felix Ryde

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Richárd Bagi, Filip Alamaa, Felix Ryde. Empirical relations in thermal Comptonization. Monthly Notices of the Royal Astronomical Society, 2025, 536, pp.603-608. 10.1093/mnras/stae2664. insu-05101715

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# Empirical relations in thermal Comptonization

Richárd Bagi<sup>1,1*</sup> Filip Alamaa<sup>1,1,2</sup> and Felix Ryde<sup>1,3</sup>

Accepted 2024 November 27. Received 2024 November 27; in original form 2024 September 19

## ABSTRACT

In many astrophysical systems, photons interact with matter through thermal Comptonization. In these cases, under certain simplifying assumptions, the evolution of the photon spectrum is described by an energy diffusion equation such as the Kompaneets equation, having dependencies on the seed photon temperature, $\theta_i$, the electron temperature, $\theta_e$, and the Compton y-parameter. The resulting steady-state spectrum is characterized by the average photon energy and the Compton temperature, which both lack analytical dependencies on the initial parameters. Here, we present empirical relations of these two quantities as functions of $\theta_i$, $\theta_e$, and y, obtained by evaluating the steady-state solution of the Kompaneets equation accounting for energy diffusion and electron recoil. The relations have average fractional errors ~1 per cent across a wide range of the initial parameters, which make them useful in numerical applications.

**Key words:** radiation mechanisms: general – methods: miscellaneous.

## 1 INTRODUCTION

In many astrophysical settings, low-energy photons are injected into a cloud of plasma with hot, non-relativistic, thermal electrons. Before escaping, the photons increase their energy due to Compton scatterings with the electrons, which broadens the initial thermal photon distribution towards higher energies. The evolution of the photon distribution in this case is given by the Kompaneets equation (Kompaneets 1957) and the astrophysical applications include cosmology (Sunyaev & Zeldovich 1970), studies of the intergalactic medium (Sunyaev & Zeldovich 1972), accretion discs (Sunyaev & Titarchuk 1980; Kaufman & Blaes 2016; Narayan et al. 2016), X-ray bursts (Titarchuk 1994), and gamma-ray bursts (Vurm & Beloborodov 2016; Samuelsson, Lundman & Ryde 2022; Samuelsson & Ryde 2023).

The detailed shape of the observed photon spectrum escaping the cloud has historically invoked a lot of analytical (e.g. Shapiro, Lightman & Eardley 1976; Sunyaev & Titarchuk 1980; Becker 2003) and numerical (e.g. Katz 1976; Pozdnyakov, Sobol & Sunyaev 1976, 1983) efforts. Although impressive works, the expressions found are often cumbersome and limited to specific parameter ranges. This can make them difficult to include in numerical applications.

In this paper, we present empirical expressions for the mean photon energy, $\bar{\epsilon}$, and the Compton temperature, $\theta_c$, of the escaping radiation spectrum as a function of the seed photon temperature, $\theta_i$, the plasma electron temperature $\theta_e$, and the Compton y-parameter in the cloud. Using the relations, one can obtain important information regarding the emitted spectrum from the initial parameters alone. For instance, an empirical relation for $\bar{\epsilon}$ is useful in the Kompaneets radiation-mediated shock (RMS) approximation (KRA; Samuelsson

et al. 2022), where photon rich RMSs are modelled via thermal Comptonization. The expressions are simple, accurate and easy to implement numerically.

The paper is structured as follows. In Section 2, we outline the method used and in Section 3, we give the empirical relations. We discuss and conclude in Sections 4 and 5.

## 2 SCENARIO, ASSUMPTIONS, AND METHOD

### 2.1 Kompaneets equation

In this study, we consider the following scenario. Relatively cold, thermal photons at temperature $\theta_i$ are continuously injected into a region with a hot plasma, at a constant temperature $\theta_e$. Photons continuously escape the region with an escape probability that is energy independent. The hot electrons scatter the cold photons to higher energies, which broadens the photon energy distribution.

Under the assumptions that the plasma is non-relativistic, the spectral evolution within the Comptonization region (the cloud) is given by the Kompaneets equation. Neglecting induced scatterings, it reads

$$t_{sc} \frac{\partial n}{\partial t} = \frac{1}{\epsilon^2} \frac{\partial}{\partial \epsilon} \left[ \epsilon^4 \left( \theta_e \frac{\partial n}{\partial \epsilon} + n \right) \right] + s, \quad (1)$$

where $t_{sc}$ is the Thompson scattering time, $n$ is the photon phase space density, $\epsilon \equiv h\nu/m_ec^2$ is the photon energy, $\theta_e \equiv kT_e/m_ec^2$ is the plasma temperature, and $s$ is a source term. Throughout the paper, temperatures and photon energies are given in units of electron rest-mass energy. Neglecting induced scatterings is appropriate as long as the brightness temperature of the plasma does not greatly exceed the electron temperature, which is common in most astrophysical cases (Goldreich, McCary & Rees 1968).

<footnote n="1"><sup>1</sup>Department of Physics, KTH Royal Institute of Technology, and The Oskar Klein Centre, SE-10691 Stockholm, Sweden</footnote>

<footnote n="2"><sup>2</sup>Sorbonne Université, CNRS, UMR 7095, Institut d'Astrophysique de Paris (IAP), 98 bis boulevard Arago, F-75014 Paris, France</footnote>

<footnote>* E-mail: bagi@kth.se</footnote>
The average photon energy gain across the region is determined by the Compton y-parameter, which is given by the product of the energy gain per scattering and the number of scatterings as

$$y = 4\theta_e \bar{N}_{sc}. \qquad (2)$$

Here, $4\theta_e$ is the average fractional energy gain in a scattering for a low-energy photon surrounded by electrons with a Maxwellian distribution at temperature $\theta_e$ and $\bar{N}_{sc}$ is the average number of scatterings.

The source term accounts for injection into, and escape from, the Comptonization region. Under the assumption that the number of photons in the Comptonization region is constant, as expected in steady-state, we employ a source term given by

$$s(\epsilon) = s_{in} - s_{out} = \left(\frac{4\theta_e}{y}\right) \left(\frac{\int N_{\epsilon} d\epsilon}{\int N_{\epsilon,in} d\epsilon}\right) n_{in} - \frac{4\theta_e}{y} n, \qquad (3)$$

where $N_\epsilon = \epsilon^2 n$ is the specific photon number density. The outgoing source term $s_{out} = (4\theta_e/y)n$ is chosen such that each photon has a probability $\bar{N}_s^{-1}$ of escaping the Comptonization region per scattering time. The coefficient in $s_{in}$ assures that the number of photons in the Comptonization region is conserved. Finally, $n_{in}$ is the source photon phase space density, which is assumed to be in a Wien distribution at temperature $\theta_i$ (Wien 1896).

## 2.2 Average photon energy and Compton temperature

The average photon energy in a given spectrum is given by

$$\bar{\epsilon} = \frac{\int \epsilon N_{\epsilon} d\epsilon}{\int N_{\epsilon} d\epsilon}, \qquad (4)$$

where the integrals are taken over the whole energy space. In the special case of a Wien spectrum with temperature $\theta$, this mean photon energy becomes

$$\bar{\epsilon} = 3\theta. \qquad (5)$$

The Compton temperature, defined as the plasma temperature where the net energy exchange due to scatterings between electrons and photons is zero, has a similar expression as $\bar{\epsilon}$. Apart from an extra factor 1/4, there is an additional weight factor $\epsilon$ in the integrals (Beloborodov & Illarionov 1995) and is given by

$$\theta_c = \frac{1}{4} \frac{\int \epsilon^2 N_{\epsilon} d\epsilon}{\int \epsilon N_{\epsilon} d\epsilon}. \qquad (6)$$

## 2.3 Steady-state spectrum

Solving equation (1) with the source term given in equation (3) results in a steady-state spectrum that is uniquely determined by $\theta_i$, $\theta_e$, and $y$. If one is only interested in the spectral shape and not the absolute value of the photon energies, the shape of the steady-state spectrum is determined by only two parameters, $R$ and $y$ (Bagi 2023), where $R$ is defined as

$$R = \frac{\theta_e}{\theta_i}. \qquad (7)$$

However, a third parameter is needed to decide on the energy scale, so we handle $\theta_e$ as a free parameter as well.

For values of $y \lesssim 1$, the spectrum roughly consists of a power law between two characteristic energies with smooth cut-offs at the boundaries. The low characteristic energy is at $3\theta_i$ and consists of photons escaping the region with negligible upscattering. The high characteristic energy is a cut-off at $3\theta_e$, since photons with energies

<figure><img src="image_2.png" /><figcaption>Figure 1. Example of a steady-state spectrum as a result of thermal Comptonization. The mean photon energy $\bar{\epsilon}$, four times the Compton temperature $4\theta_c$, as well as the mean photon energies $3\theta_i$ and $3\theta_e$, corresponding to the initial and the newly built Wien peak, are marked. The parameter values are $R = 132$ and $y = 3.16$. The procedure how the spectra are produced is detailed in Section 2.4.</figcaption></figure>

greater than that are affected by electron recoil. When $y \gg 1$, most photons significantly increase their energy and the spectrum is dominated by a Wien peak at around $3\theta_e$ due to a pile-up of high-energy photons. At $y \gtrsim 1$, spectra are more complex, consisting of a combination of a power law at lower frequencies and the Wien peak at $3\theta_e$. An example of such a spectrum is given in Fig. 1. Finally, when $y = 1$, the power-law part of the spectrum is flat in a $v_F v \propto \epsilon^4 n$ representation, implying an equal amount of radiation energy per decade energy.

The number of scatterings that are needed to reach steady state, $N_{sc}^{ss}$, can be estimated by evaluating the number of scatterings required by a cold photon to increase its energy to $\epsilon_{max} = 3\theta_e$. This gives

$$N_{sc}^{ss} \approx \frac{\ln(R)}{4\theta_e}. \qquad (8)$$

## 2.4 Numerical method

The Kompaneets equation (1), with a source term as equation (3) is solved here using the solver developed in Chang & Cooper (1970). Apart from $\theta_e$, $\theta_i$, and $y$, one also needs to specify the total simulation time, $t$. In each simulation, $t$ is chosen much larger than $t_{sc} N_{sc}^{ss}$ such that the simulation has reached steady state.

To obtain the empirical relations, a large number of simulations were run for a range of different initial conditions. Given that the two quantities of interest, $\bar{\epsilon}$ and $\theta_c$, depend linearly on $\theta_e$ (Bagi 2023), we do not need to probe different values of $\theta_e$. Instead, the simulation was run for 100 different $y$ values between $10^{-2}$ and $10^2$ and 97 different $R$ values between 1 and $10^5$ for a total of 9 700 simulation runs. For each simulation run, $\bar{\epsilon}$ and $\theta_c$ were evaluated at the end using equations (4) and (6).

The obtained data set, i.e. the results of the simulations, is shown in Fig. 2. Utilizing this new data set, we aimed to find continuous functions describing $\bar{\epsilon}$ and $\theta_c$ and their dependencies on the three physical variables $\theta_e$, $R$, and $y$.
<figure><img src="image_3.png" /><figcaption>Figure 2. Contour plots of the mean photon energy $\bar{\epsilon}$ (left) and the Compton temperature $\theta_C$ (right) from the Kompaneets simulations. For simplicity, we plot $\bar{\epsilon}/3\theta_e$ and $\theta_C/\theta_e$. Here, log denotes logarithm in base 10.</figcaption></figure>

# 3 EMPIRICAL RELATIONS AND THEIR ACCURACY

## 3.1 Empirical relation for the mean photon energy

By performing an empirical analysis on the data set, we found that the mean photon energy could be described as a broken power law given by

$$ \bar{\epsilon}(\theta_e, y, R) = 3\theta_e [f(y)R^{k_1(y)} + (1 - f(y))R^{k_2(y)}]^{-1}, \quad (9) $$

where the included functions $f(y)$, $k_1(y)$, and $k_2(y)$ are constructed such that in the cases of both weakly modified seed spectrum ($y \ll 1$) and saturated Comptonization ($y \gg 1$), the mean photon energy would converge to that of the corresponding Wien spectrum. Furthermore, the expression is structured so that the first term $3\theta_e [f(y)R^{k_1(y)}]^{-1}$ describes $\bar{\epsilon}$ at low values of $R$, while the second term $3\theta_e [(1 - f(y))R^{k_2(y)}]^{-1}$ applies in the region of high values of $R$. Then the exact parameters were found through fitting the variables on the whole data set. The results became

$$ f(y) = \frac{1}{0.916y^{-1.03} + 0.527y^{1.95}}, \quad (10) $$

$$ k_1(y) = \frac{1}{0.936y^{1.88} + 1.10y^{-0.896}} - \frac{1}{0.146y^{0.896} + 14.5y^{-1.10}} \quad (11) $$

and

$$ k_2(y) = \tanh (1.20y^{-0.976} + 0.270y^{-3.17}). \quad (12) $$

Keeping the definition of $R$ as given in equation (7) in mind, we can factor out $1/R$ to obtain the temperature of the initial Wien spectrum as a factor in front of the broken power-law term:

$$ \bar{\epsilon}(\theta_i, y, R) = 3\theta_i [f(y)R^{k_1(y)-1} + (1 - f(y))R^{k_2(y)-1}]^{-1}. \quad (13) $$

In this expression, the last factor provides a direct measure of the increase in the mean photon energy relative to its initial value according to equation (5).

## 3.2 Empirical relation for the Compton temperature

We performed a similar analysis on the Compton temperature as well. Once again, a broken power-law description with similar limit requirements was constructed. Thus the Compton temperature can be described as

$$ \theta_C(\theta_e, y, R) = \theta_e [f_C(y)R^{k_{C,1}(y)} + (1 - f_C(y))R^{k_{C,2}(y)}], \quad (14) $$

where

$$ f_C(y) = 1 - 10^{1/(17.2y^{4.45} + 0.208y^{-1.51})} - 1/(1.03y^{1.51} + 0.667y^{-1.28}), \quad (15) $$

$$ k_{C,1}(y) = -\tanh\left(\frac{1}{5.03y^{0.708}}\right) - 1 + \frac{1}{0.915y^{0.862} + 0.691y^{-1.83}}, \quad (16) $$

and

$$ k_{C,2}(y) = -\tanh\left(\frac{1}{6.55y^{2.45}}\right) + \left(\frac{1}{1.37y^{0.413} + 0.202y^{-3.27}}\right)^{5.96}. \quad (17) $$

Here, we have the same structure as in the case of $\bar{\epsilon}: k_{C,1}(y)$ is the power-law index dominating in the low $R$ region and $k_{C,2}(y)$ is governing in the high $R$ region. Using the same factorization as above, we can then relate this function to $\theta_i$, the initial value of $\theta_C$ before the broadening process as

$$ \theta_C(\theta_i, y, R) = \theta_i [f_C(y)R^{k_{C,1}(y)+1} + (1 - f_C(y))R^{k_{C,2}(y)+1}], \quad (18) $$

where the last factor indicates the increase in the Compton temperature relative to its initial value.

## 3.3 Accuracy of the empirical relations

One way to study the accuracy of the empirical relations in equations (9) and (14) is to consider the fractional error, which is the ratio between the modelled values and the values in the data set given by

$$ e = \frac{f_{\text{model}}}{f_{\text{data}}}, \quad (19) $$
of the two studied functions. The closer the value is to unity, the more accurate the description is. The fractional errors of the two functions $\bar{\epsilon}(\theta_e, y, R)$ and $\theta_C(\theta_e, y, R)$ are shown in Fig. 3. The left panel shows that the model for $\bar{\epsilon}$ achieves errors up to 6 per cent, and is otherwise in good accordance with the simulated data. In contrast, the model for $\theta_C$ exhibits somewhat larger errors, as seen in the right panel. It shows deviations reaching up to 16 per cent, especially in the parameter ranges $y \in (0.25, 0.56)$ and $R \in (2, 70)$, independently of $\theta_e$. In the complementary region though, the model is in good agreement with the simulated values.

As noted by Rybicki & Lightman (1979), the central part of a Comptonization spectrum ($3\theta_i \ll \epsilon \ll 3\theta_e$) follows a simple power-law distribution for $y \gg 1$ and for $y \ll 1$. As mentioned above, for $y \sim 1$, spectra can be quite complex, where one needs to take linear combinations of power-law distributions to find the steady-state solution of the spectrum. This complexity contributes to greater uncertainties in the empirical relations in this region. Consistent with this expectation, fractional errors are indeed most significant near or just below $y = 1$ for both $\bar{\epsilon}$ and $\theta_C$.

It may be instructive to estimate the average fractional error for each function across the probed parameter space. One estimate of this can be obtained by taking the average of all individual fractional errors for the 9700 simulations in the data set. This results in an average value for $|e - 1|$ of 0.86 per cent for $\bar{\epsilon}$, and 2.7 per cent for $\theta_C$.

Within high-energy astrophysics, it is common to measure the mean photon energy and the electron temperature with an accuracy of about 10–20 per cent (O’Brien et al. 2001; Roberts et al. 2005; Skinner et al. 2008; Poutanen & Veledina 2013; Wallbank et al. 2022). With this in mind, our results should be accurate enough for many astrophysical contexts.

# 4 DISCUSSION

## 4.1 Analytical relation between $\theta_C$ and $\bar{\epsilon}$

It is possible to derive an analytical relation between the Compton temperature $\theta_C$ and $\bar{\epsilon}$ (see e.g. Samuelsson & Ryde 2023). The relation is given as

$$ \theta_C = \theta_e \left[ 1 - \frac{1}{y} \left( 1 - \frac{3\theta_i}{\bar{\epsilon}} \right) \right]. \qquad (20) $$

This means that having determined the mean photon energy $\bar{\epsilon}$, one should, in principle, be able to find the Compton temperature unambiguously. Unfortunately, it is not very useful in practice. The stability of the formula can be studied using the derivative given as

$$ \frac{\partial \theta_C}{\partial \bar{\epsilon}} = - \frac{3\theta_i \theta_e}{y \bar{\epsilon}^2} \frac{1}{\bar{\epsilon}^2} = - \frac{R}{3y} \left( \frac{3\theta_i}{\bar{\epsilon}} \right)^2. \qquad (21) $$

For small values of $y$, the mean photon energy is close to the mean energy of the source photons, $\bar{\epsilon} \gtrsim 3\theta_i$, which implies that $\partial\theta_C/\partial\bar{\epsilon} \propto R/y \gg 1$. Thus, for $y \ll 1$, the analytical expression becomes numerically very sensitive and a small uncertainty in $\bar{\epsilon}$ leads to large jumps for $\theta_C$. In accordance with this expectation, we find that the relation in equation (20) is only accurate when the energy gain is sufficiently high, $y > 1$, or as long as the broadening is small, $R \lesssim 10$. Since this relation only covers half of the parameter space accurately, we have chosen to develop the empirical formula given in equation (14), instead.

## 4.2 Estimating the power-law index

At photon energies $3\theta_i \ll \epsilon \ll 3\theta_e$, the Comptonized spectrum can be approximated with a power law, $n \propto \epsilon^\alpha$. According to Shapiro et al. (1976) and Rybicki & Lightman (1979), the power-law index in this region can be estimated by

$$ \alpha = -\frac{3}{2} \pm \sqrt{\frac{9}{4} + \frac{4}{y}}, \qquad (22) $$

where the positive root is appropriate when $y \gg 1$, the negative root is appropriate when $y \ll 1$, and one must take a non-trivial linear combination of these solutions in the region where $y \sim 1$.

When the spectrum is sufficiently broad, $R \gg 1$, the power-law segment may carry the majority of the photons and the energy. Another way of estimating the power-law index in these cases is to approximate the spectrum with a single power law between $3\theta_i$ and $3\theta_e$. With this approximation, the mean energy is given as (Bagi 2023)

$$ \bar{\epsilon} = \frac{(\alpha + 3) (3\theta_e)^{\alpha+4} - (3\theta_i)^{\alpha+4}}{(\alpha + 4) (3\theta_e)^{\alpha+3} - (3\theta_i)^{\alpha+3}}, \qquad (23) $$

which, using equation (7), can be rewritten as

$$ \bar{\epsilon} = 3\theta_e \frac{(\alpha + 3) (1 - R^{-\alpha-4})}{(\alpha + 4) (1 - R^{-\alpha-3})}. \qquad (24) $$

Using the empirical formula for $\bar{\epsilon}$ given in equation (9), one can solve for the power-law index $\alpha$ numerically. This will give an estimate of the average power-law index even in the cases when $y \sim 1$.

## 4.3 Applications of the empirical relations

The empirical relations given in equations (9) and (14) can be used whenever the spectrum of the radiation field resembles a Comptonized spectrum and when an efficient and rapid method to determine the energetics of the spectrum is needed.

One application of these relations is in the study of gamma-ray bursts, as mentioned in the introduction. Prompt gamma-ray burst spectra can, in many cases, be interpreted as being formed by RMSs. They thereby resemble Comptonized spectra (Ito et al. 2018; Samuelsson & Ryde 2023). This is the motivation for the KRA, which can be used to make time-efficient fits of RMS spectra (Samuelsson et al. 2022). In KRA, the average photon energy of the downstream is needed in the fitting routine. Equation (9) is perfectly suited for this application as it gives an efficient and rapid estimate. This method has recently been implemented in the data analysis in Wistemar, Alamaa & Ryde (in preparation).

Another possible example of the use of the empirical relations is in the analysis of X-ray binaries (Maraschi & Molendi 1990; Gierliński et al. 1999). For instance, when an external plasma cloud or wind interacts with the radiation field of the compact object, the equilibrium between photon-induced cooling and heating results in the plasma attaining the Compton temperature of the radiation field (Beloborodov 1999; Zdziarski, Misra & Gierliński 2010). The relation described in equation (14) offers an efficient method for characterizing the plasma temperature under these conditions.

## 4.4 Symbolic regression using machine learning

A machine learning technique performing symbolic regression has recently been developed (Burlacu, Kronberger & Kommenda 2020) and used, i.e. within dark matter cosmology (Bartlett et al. 2024a, b). This machine learning technique is capable of reading in data sets
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Fractional errors of the empirical formulae for both $\bar{\epsilon}$ (left) and $\theta_C$ (right). The fractional errors are obtained using equation (19). Here, log denotes logarithm in base 10.</figcaption></figure>

with initial input parameters, and returning an analytical function of the initial parameters describing the data set with a high accuracy. The returned function does, however, not take any physical considerations into account and is therefore harder to interpret. Performing such a regression analysis on our data provides functions, that have fractional errors of the same magnitude as ours, and that have the largest deviations in the same regions.

## 5 SUMMARY AND CONCLUSIONS

In this paper, we have derived empirical relationships in thermal Comptonization, a process where photons gain energy through inverse Compton scattering with thermal, non-relativistic electrons. Such a process is described by the Kompaneets equation. The study has focused on the steady-state mean photon energy, $\bar{\epsilon}$, and the Compton temperature, $\theta_C$, accounting for energy diffusion and electron recoil. These quantities depend on the seed photon temperature, $\theta_i$, the electron temperature, $\theta_e$, and the Compton y-parameter. A typical steady-state spectrum has been illustrated in Fig. 1. The aim has been to find continuous functions that are easy to implement numerically. By numerically solving the Kompaneets equation (1) for a wide range of the parameters, we have obtained the distributions of $\bar{\epsilon}$ and $\theta_C$ shown in Fig. 2. Using physical considerations and an empirical study, we have developed accurate and computationally efficient empirical formulae predicting $\bar{\epsilon}$ and $\theta_C$ from the initial parameters, see equations (9) and (14) respectively. The formulae describe the studied quantities with fractional errors ranging up to 6 per cent and 16 per cent for the two models respectively, with average fractional errors ~1 per cent in the studied parameter space. The error distributions are shown in Fig. 3. These results can be applied in various astrophysical contexts, including gamma-ray bursts, X-ray bursts, and accretion discs around compact objects.

## ACKNOWLEDGEMENTS

We acknowledge support from the Swedish National Space Agency (196/16) and the Swedish Research Council (Vetenskapsrådet, 2018–

03513 and 2022–00347). We thank Dr Deaglan Bartlett for fruitful discussions and the anonymous referee for useful comments.

## DATA AVAILABILITY

There are no new data associated with this article.

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