2.1. Compton scattering of unpolarised photons

For an unpolarised incident photons, the $\sigma_\perp$ and $\sigma_\parallel$ can be determined by averaging the $\cos^2\Theta$ -term of equation (1) over $\theta_e$ using equation (2) for $\theta'_{\!\!e}$ = 0 and $\pi/2$ , respectively, and it is expressed as (see, e.g. McMaster Reference McMaster1961):

\begin{align*} d\sigma_\perp^{unpol} = \frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2\left[\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}\right], \end{align*}

\begin{align*}d\sigma_\parallel^{unpol} = \frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2\left[\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}-2\sin^2\theta\right], \end{align*}

here, $\langle \cos^2\theta_e\rangle$ = $\langle \sin^2\theta_e\rangle$ =0.5; and $\langle \cos\theta_e\rangle$ = $\langle \sin\theta_e\rangle$ =0, as for the unpolarised photons the $\theta_e$ is distributed isotropically.

The angle of polarisation: Similarly, we estimate the $\sigma_{\perp+45}^{unpol}$ and $\sigma_{\parallel+45}^{unpol}$ with having $\theta'_{\!\!e}$ = $\pi/4$ and $3\pi/4$ , respectively, its values are $\sigma_{\perp+45}^{unpol} = \sigma_{\parallel+45}^{unpol} = \frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2\left[\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}-\sin^2\theta\right]$ . Thus, the Stokes parameter U is zero, which gives $\chi = 0$ . Therefore, after single scattering, the scattered photons are polarised along the $(k \times k^{\prime})$ direction or in another words, the plane of polarisation of scattered photons is along the perpendicular to the scattering plane.

The degree of polarisation: The degree of linear polarisation of the scattered photons after single scattering of unpolarised photons is written (using equation (5)) as (see, e.g. McMaster Reference McMaster1961; Matt et al. Reference Matt, Feroci, Rapisarda and Costa1996; Lei et al. Reference Lei, Dean and Hills1997):

(7) \begin{equation} P = \frac{\sin^2\theta}{\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}-\sin^2\theta}.\end{equation}

Here, P = 0, for $\theta$ = 0, and $P = \frac{1}{\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}-1}$ for $\theta$ = 90 $^{\circ}$ . In Thomson limit (precisely defined as $\frac{h\nu}{\gamma \ m_e c^2} \ll 1$ , here $\gamma = 1/\sqrt{(1 - \frac{\text v^2}{c^2})}$ is the electrons Lorentz factor, $\text v$ is the speed of electron), one has $\frac{k}{k^{\prime}} \sim 1$ (see equation (3)), thus P = 1 for $\theta = 90^{\circ}$ . Hence, in Thomson regime the single scattered unpolarised (incident) photons at $\theta =90^{\circ}$ are completely polarised in a perpendicular plane to the scattering plane.

Modulation curve: The modulation curve is a distribution of the $(\theta'_{\!\!e})$ -angle. The differential cross section for unpolarised incident photons can be expressed by using equations (1) and (2) as:

\begin{align*}d\sigma = \frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2\left[\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}- 2\sin^2\theta \sin^2\theta'_{\!\!e}\right],\end{align*}

here, we consider once again $\langle \cos^2\theta_e\rangle$ = $\langle \sin^2\theta_e\rangle$ =0.5, and $\langle \cos\theta_e\rangle$ = $\langle \sin\theta_e\rangle$ =0. The above expression after rearranging the term can be written as (using, $\cos 2\theta'_{\!\!e} = 1 -2\sin^2\theta'_{\!\!e})$

(8) \begin{equation} d\sigma =\frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2\left[\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}-\sin^2\theta \right] (1 + P \cos 2\theta'_{\!\!e})\end{equation}

By comparison to the above expressed modulation curve from the commonly used expression for modulation curve in literature (e.g. equation (4.10) of Lei et al. Reference Lei, Dean and Hills1997, or, equation (2) of Chattopadhyay et al. Reference Chattopadhyay, Vadawale, Rao, Sreekumar and Bhattacharya2014, note there, authors have measured the corresponding $\theta'_{\!\!e}$ with respect to the scattering plane), we again find that the angle of polarisation for scattered photons after single scattering of unpolarised incident photons is zero, that is, the linear polarisation is along the perpendicular direction to the scattering plane.

For polarisation-insensitive detector: The cross section for a polarisation-insensitive detector can be written (by using equations (1) and (2), and now with having additional $\langle \cos^2\theta'_{\!\!e}\rangle$ = $\langle \sin^2\theta'_{\!\!e}\rangle$ =0.5) as:

(9) \begin{equation} d\sigma_{ins-detct} = \frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2\left[\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}-\sin^2\theta\right]\end{equation}

Also, here $d\sigma_{ins-detct} = (d\sigma_\perp^{unpol} + d\sigma_\parallel^{unpol})/2 = (d\sigma_{\perp+45}^{unpol} + d\sigma_{\parallel+45}^{unpol})/2 $ .

2.2. Compton scattering of polarised photons

For a completely polarised incident photons with polarisation angle $\phi$ (i.e. $\theta_e = \phi$ ) the cross section can be obtained by averaging the equation (1) over $\theta'_{\!\!e}$ (see, e.g. Lei et al. Reference Lei, Dean and Hills1997, and reference therein), and it is written as:

(10) \begin{equation} \frac{d\sigma}{d\Omega}^{pol} = \frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2\left[\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}-2\sin^2\phi \sin^2\theta \right],\end{equation}

here we consider $\langle \cos^2\theta'_{\!\!e}\rangle$ = $\langle \sin^2\theta'_{\!\!e}\rangle$ = 0.5, and $\langle \cos\theta_e\rangle$ = $\langle \sin\theta_e\rangle$ = 0. However, it is expected that the distribution of $\theta'_{\!\!e}$ is no longer isotropic but depends on the cross section, equation (1) (see, e.g. Matt et al. Reference Matt, Feroci, Rapisarda and Costa1996). Similar to the unpolarised incident photons case, we compute the $d\sigma_\perp$ and $d\sigma_\parallel$ for polarised incident photons with having $\theta'_{\!\!e}$ = 0 and $\pi/2$ , respectively, which is written as (see, e.g. McMaster Reference McMaster1961):

\begin{align*}\begin{aligned}d\sigma_\perp^{pol} =\frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2\left[\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}-2+4 \cos^2\phi \right]\\[5pt] d\sigma_\parallel^{pol}=\frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2\left[\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}-2+ 4\sin^2\phi \cos^2\theta \right]\end{aligned}\end{align*}

The angle of polarisation: The $d\sigma_{\perp+45}$ and $d\sigma_{\parallel+45}$ for polarised incident photons are $d\sigma_{\perp+45}^{pol} = \frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2\left[\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}-2+2(\cos^2\phi +\sin^2\phi\cos^2\theta +\sin2\phi \cos\theta) \right]$ , and $d\sigma_{\parallel+45}^{pol} = \frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2[\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}-2+2(\cos^2\phi +\sin^2\phi\cos^2\theta -$ $\sin2\phi \cos\theta) ]$ . Here, $\theta'_{\!\!e}$ = $\pi/4$ and $3\pi/4$ for $d\sigma_{\perp+45}^{pol}$ and $d\sigma_{\parallel+45}^{pol}$ , respectively. The Stokes parameters U $\&$ Q are expressed as:

\begin{align*} U = \frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2\left[4\sin2\phi \cos\theta) \right] =d\sigma_{\perp+45}^{pol} - d\sigma_{\parallel+45}^{pol} \end{align*}

\begin{align*} Q = \frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2\left[4\cos^2\phi-4\sin^2\phi\cos^2\theta\right] =d\sigma_{\perp}^{pol} - d\sigma_{\parallel}^{pol}\end{align*}

The angle of polarisation can be obtained by using expression (4). In practice, the average angle of polarisation $\langle \chi\rangle$ is interested, it is written as (e.g. Li et al. Reference Li, Narayan and McClintock2009):

(11) \begin{equation} \tan2\langle \chi\rangle = \frac{\langle U \rangle}{\langle Q \rangle},\end{equation}

here, $\langle U \rangle$ and $\langle Q \rangle$ are averaged of U and Q over angle, respectively. In Thomson limit, we find that the magnitude of $\langle U \rangle$ is almost one order less than the magnitude of $\langle Q \rangle$ , that is, $|\langle U \rangle| << |\langle Q \rangle|$ . Thus, $\langle \chi \rangle$ $\sim$ 0 or $\pi/2$ for a positive or negative value of $\langle Q \rangle$ , respectively.

The degree of polarisation: On average $|\langle U \rangle| < < |\langle Q \rangle|$ , but we notice also $U > Q$ for a range of $\theta$ , for example, see Fig. A1. Hence we define the PD with considering two extreme cases, case A: $|Q| >> |U|$ and case B: $|Q| \sim |U|$ . For case A, the degree of polarisation for single scattered photons of polarised incident photons is expressed by using equation (5) as:

(12) \begin{equation}P_A = \frac{Q}{I} = \frac{2 - 2\sin^2\phi(1+\cos^2\theta)}{\frac{k^{\prime}}{k}+\frac{k}{k^{\prime}}-2\sin^2\phi\sin^2\theta}\end{equation}

For case B, it is expressed by using equation (4) as (see, e.g. Matt et al. Reference Matt, Feroci, Rapisarda and Costa1996; Lei et al. Reference Lei, Dean and Hills1997):

(13) \begin{equation}P_B = \frac{\sqrt{Q^2+U^2}}{I} = \frac{2 - 2\sin^2\phi\sin^2\theta}{\frac{k^{\prime}}{k}+\frac{k}{k^{\prime}}-2\sin^2\phi\sin^2\theta}\end{equation}

In Thomson regime, $P_B$ = 1.

Some interesting facts: (i) $\theta$ = 0: For $\theta = 0$ , $Q \propto \cos\!(2\phi)$ and $U \propto \sin(2\phi)$ . Since for $\theta=0$ one has $k=k^{\prime}$ which gives $P_B$ = 1 for all $\phi$ . However, U = 0 for $\phi = 0, \pi/2, \pi$ , and in this case PD would be determined by $P_A$ . Here, $P_A$ = 1 for $\phi = 0, \pi$ , and $P_A$ = −1 for $\phi = \pi/2$ . Conclusively, $|P|$ = 1 for $\theta = 0$ . (ii) $\phi$ = 0 , $\pi/2$ : For $\phi$ = 0 or $\pi/2$ , $U = 0$ . The PD is $P_A = \frac{2}{\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}}$ and $\frac{-2\cos^2\theta}{\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}-2\sin^2\theta}$ for $\phi$ = 0 and $\pi/2$ , respectively (for $\phi =0$ , see McMaster Reference McMaster1961). In Thomson limit, $P_A $ = 1 and −1; $\chi$ = 0 and $\pi/2$ for $\phi = 0$ and $\pi/2$ , respectively (see also Fig. 5), also by definition of Q, here $\theta'_{\!\!e}$ = 0 and $\pi/2$ , respectively. And in words, the polarisation properties of scattered photons are same to the incident photons. (iii) $\phi$ = $\pi$ /4: For $\phi = \pi/4$ , it is expected that the incident polarised photons behave like unpolarised photons. It means that the degree of polarisation of scattered photons will be described by expression (7). We note that for $\phi = \pi/4$ the only $P_A$ reduces to the expression (7).

Modulation curve: The expression for modulation curve is written by using equation (1) as:

(14) \begin{align} \frac{d\sigma}{d\Omega}^{pol} &= \frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2\left[\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}-2 + 4(\cos^2\phi \cos^2\theta'_{\!\!e} \right. \nonumber \\[5pt] & \quad \left. + \sin^2\phi \sin^2\theta'_{\!\!e} \cos^2\theta - \frac{1}{2}\sin2\phi\sin2\theta'_{\!\!e} \cos\theta) \right].\end{align}

After rearranging the term, the above equation can be written as:

\begin{align*}\frac{d\sigma}{d\Omega}^{pol} = \frac{1}{4}r_o^2\left(\frac{k^{\prime}}{k}\right)^2& \left[\frac{k}{k^{\prime}}+\frac{k^{\prime}}{k}-2\sin^2\phi \sin^2\theta \right] \left[1 + P_A \left(\cos2\theta'_{\!\!e} \right. \right. \nonumber \\[5pt] & \left. \left. - \frac{\sin 2\phi \cos\theta}{\cos^2\phi-2\sin^2\phi\cos^2\theta} \sin 2\theta'_{\!\!e} \right)\right]\end{align*}

2.3. A special case for polarisation measurement at $\theta$ = 0

For $\theta$ = 0, the cross section (1) can be written simply as:

(15) \begin{equation} \frac{d\sigma}{d\Omega} = r_o^2 \cos^2(\theta_e+\theta'_{\!\!e}),\end{equation}

here we use $\frac{k^{\prime}}{k} = 1$ , for $\theta$ = 0, which is true for a electron is in either rest or motion. In this case, $k^{\prime}_{e}$ will also lie on the perpendicular plane to the k, and thus for a plane the solid angle becomes $d\Omega$ = $d\theta'_{\!\!e}$ . Hence, $d\sigma$ $\propto$ $\cos^2(\theta_e+\theta'_{\!\!e}) d\theta'_{\!\!e}$ , which has properties that the distribution of $\theta'_{\!\!e}$ replicates the polarised distribution of $\theta_e$ . It can be understood as (i) for unpolarised incident photons, $\theta_e$ is isotropically distributed; thus, the averaged cross section over $\theta_e$ for $\theta'_{\!\!e}$ is simply a constant, or

\begin{align*}\frac{d\sigma}{d\theta'_{\!\!e}} = constant. \end{align*}

(ii) for polarised incident photons, $\theta_e$ = constant = $\phi$ , and the cross section becomes

\begin{align*}\frac{d\sigma}{d\theta'_{\!\!e}} = r_o^2 \cos^2(\theta'_{\!\!e}+\phi) = \frac{r_o^2}{2} \left(1 + \cos\!(2(\theta'_{\!\!e}+\phi))\right),\end{align*}

which is a modulation curve for the scattered photons with degree of polarisation P = 1 and the angle of polarisation $\chi = \phi$ .

In general for the partially polarised incident photons, in which P fraction is the polarised photons with polarisation angle $\phi$ and $(1-P)$ fraction is the unpolarised photons, the modulation curve can be written as:

(16) \begin{equation} \frac{d\sigma}{d\theta'_{\!\!e}} = A + B \cos\!(2(\theta'_{\!\!e}+\phi))\end{equation}

here, A and B are a normalisation factor, and clearly, $P = B/A$ and $\chi = \phi$ . The above expression is similar to the equation (4.10) of Lei et al. (Reference Lei, Dean and Hills1997).

2.4. Lorentz invariance of the Stokes parameters

We know that the field of the radiation is transverse in any reference frame, and the Lorentz boost subjects to an aberration effect of radiation. Since the electric vector always lies on the perpendicular plane to radiation propagation direction, and these electric vectors will be transformed from one frame to another Lorentz-boosted frame with the same rule. Thus, if the radiation is completely polarised in one reference frame, then it will be completely polarised in any Lorentz-boosted frame. In other words, the degree of polarisation of photons in any Lorentz-boosted frame is same to the magnitude of PD in the electron rest frame. Later, we will argue that in Compton scattering the angle of polarisation for photons remains the same in any Lorentz-boosted frame. Hence, the Stokes parameters are invariant under Lorentz transformations (see, e.g. Landau & Lifshitz Reference Landau and Lifshitz1987; Krawczynski Reference Krawczynski2012, references therein).

4.1. General modulation curve to estimate the PD & PA

In Section 2.3, we showed that one can know the polarisation properties of incident photons $P \ \& \ \chi$ by mapping the distribution of $\theta'_{\!\!e}$ of scattered photons at $\theta$ $\sim$ 0 after single scattering. In general, one can estimate the polarisation properties of scattered photons with any average scattering no., say $\langle N_{sc} \rangle$ by mapping the distribution of $\theta'_{\!\!e}$ of scattered photons of $\langle N_{sc} + 1 \rangle$ scattering no. at $\theta$ $\sim$ 0. Mathematically, we are essentially using here a probability $\propto \cos^2(\theta_e|_{\langle N_{sc}+1\rangle} + \theta'_{\!\!e}|_{\langle N_{sc}+1\rangle})$ for constructing the distribution of $\theta'_{\!\!e}|_{\langle N_{sc}+1\rangle}$ for a known $\theta_e|_{\langle N_{sc}+1\rangle}$ (see equation (15)). From equation (6), we can write $\theta_e|_{\langle N_{sc}+1 \rangle} = (\theta'_{\!\!e} \pm \chi)|_{\langle N_{sc}\rangle}$ , here the subscript with vertical bar is used for the quantity related to that scattering number, $\chi$ is computed using equation (11). Hence, now without going for the calculations of $\langle N_{sc} + 1\rangle$ th scattering we can estimate the polarisation properties of $\langle N_{sc}\rangle$ th scattered photons by constructing the modulation curve of $\eta$ -angle for a known value of $(\theta'_{\!\!e} \pm \chi)|_{\langle N_{sc}\rangle}$ using the probability $p(\eta)$ :

(17) \begin{equation} p(\eta) \propto \cos^2(\eta+(\theta'_{\!\!e} \pm \chi)|_{\langle N_{sc}\rangle}) d\eta.\end{equation}

We have used this $p(\eta)$ in the MC calculations to estimate the polarisation properties of the scattered photons.

4.2. MC results for unpolarised incident photons

In Fig. 2 we show the PD as a function of scattering angle for the unpolarised monochromatic incident photons for five different energies 3, 300, 900, 3 000, 9 000 keV. The solid gray curves are for analytic PD expressed by equation (7), the MC results are consistent with analytic ones. Clearly in Thomson regime, at $\theta =$ 90 $^{\circ}$ the single scattered unpolarised (incident) photons are completely polarised. In Fig. 3, the modulation curves for a given $\theta$ have been shown for two different photon energies 3 (in left panel) and 300 (in middle panel) keV. In both cases, the MC results match with analytic one, equation (8) for a given $\theta$ which is shown by gray curve. By comparing with equation (16) for all curves of left and middle panels we have $\phi$ = 0 $^{\circ}$ , which signifies that the scattered photons are polarised in a perpendicular direction to the scattering plane. In the right panel we show the averaged modulation curve over $\theta$ for two photon energies 3 and 300 keV. The gray curves for 3 and 300 keV are for equation $4.5\times 10^6(1+0.33\cos 2\theta'_{\!\!e})$ and $4.45\times 10^6(1+0.28\cos 2\theta'_{\!\!e})$ , respectively. By comparing with equation (16) the estimated PD of single scattered 3 and 300 keV unpolarised photons are $\sim$ 0.33 and 0.28. Since we know the distribution of $\theta$ and know the PD as a function of $\theta$ , we have computed the averaged PD weighted over the $\theta$ . And we find the PD for 3 and 300 keV photons are $\sim$ 0.28 and 0.24, respectively; thus, the both methods almost agree with each other. In addition, for 3 keV photons (which is in Thomson regime) we compute the averaged PD analytically as $\langle P \rangle = \frac{\int_0^\pi P (1+ \cos^2\theta) d\theta}{\int_0^\pi (1+\cos^2\theta) d\theta}$ = 1/3 with considering $d\sigma \propto (1+\cos^2\theta)$ in Thomson regime. Here, we reemphasise that the high value of PD of single scattered unpolarised (incident) photons is due to the fixed scattering plane (see also case I with $\theta_i \equiv [0, \pi]$ of Section 6) and if one accounts the effect of orientation of the scattering plane, then the PD magnitude will get reduced, see the Section 6 for details.

Figure 2. The degree of polarisation after single scattering of unpolarised (incident) photons as a function of scattering angle for five different photon energies. Here, the curves 1, 2, 3, 4, and 5 are for incident photon energies 3, 300, 900, 3 000, and 9 000 keV, respectively. The gray solid curves are analytic one expressed by equation (7).

Figure 3. The modulation curves after single scattering of unpolarised (incident) monochromatic photons. The left and middle panels are for the fixed $\theta$ and for the incident photon energy 3 and 300 keV, respectively, while the right panel is for the averaged $\theta$ . In left and middle panels the blue, orange, green, red, and cyan curves are for $\theta$ = 0, 30, 45, 60, and 90 degree, respectively. The gray solid curves are the analytical one expressed by equation (8) for a given $\theta$ . In right panel, the red and green curves are for photon energies 3 and 300 keV, respectively. The gray curves for 3 and 300 keV are $4.5\times10^6(1+0.33\cos 2\theta'_{\!\!e})$ and $4.45\times 10^6(1+0.28\cos 2\theta'_{\!\!e}),$ respectively, which reflects the PD = 0.33 and 0.28 by comparing the equation (16) for 3 and 300 keV, respectively.

4.3. MC results for polarised incident photons

In Section 2.2, we argue that the cross section for a completely polarised incident photon of polarisation angle $\phi$ is obtained with an assumption of isotropic distribution of $\theta'_{\!\!e}$ . We examine this assumption in MC calculations. We compute the averaged cross section over $\theta'_{\!\!e}$ as a function of $\theta$ (using the equation (1)) for photon energies 3 and 300 keV, the results are shown in Fig. 4. We find that the computed cross sections are significantly deviated from the equation (10) except for $\phi$ $\sim$ 0 and 90 $^{\circ}$ , where it shows a slight deviation. Actually in case of $\phi$ $\sim$ 0 and 90 $^{\circ}$ , the equation (1) nearly reduces to the equation (10), as here also $\theta'_{\!\!e}$ = 0 and 90 $^{\circ}$ (see the facts (ii) of Section 2.2) for $\phi$ $\sim$ 0 and 90 $^{\circ}$ , respectively. Hence, the distribution of $\theta'_{\!\!e}$ -angle is no longer isotropic (e.g. Matt et al. Reference Matt, Feroci, Rapisarda and Costa1996), as stated in the previous section.

Figure 4. The Klein–Nishina cross section as a function of $\theta$ for a completely polarised incident photon of polarisation angle $\phi = \theta_e$ . The dashed curves are calculated one by MC method using the equation (1) while the gray solid curves are analytic one expressed by equation (10) and the observed mismatched is due to the approximation involved in equation (10), see the text for details. The left and right panels are for incident photon energy 3 and 300 keV, respectively, and the dashed green, orange, blue, red, and black curves are for $\theta_e$ = 0, 30, 45, 60, and 90 degree, respectively.

In Fig. 5, we show the PD as a function of $\theta$ for single scattered completely polarised photons of energy 3 keV. We find that the MC results agree with the corresponding theoretical PD (shown by gray solid curve) expressed by equation (12). Here we remind that the positive value of PD signifies that the scattered photons polarisation is perpendicular to the scattering plane while the polarisation is along the scattering plane for negative value. In Fig. 6, we show the modulation curves of single scattered polarised photons for a fixed $\theta$ . We have computed the results for four different polarisation angles of incident photons of energy 3 keV. For $\phi$ = 0, the equation (14) predicts that the modulation curve is independent of $\theta$ in Thomson regime, we find same here see the top left panel. In general, these modulation curves will be described by the equation (14).

Figure 5. The degree of polarisation after single scattering of completely polarised (incident) photons of energy 3 keV as a function of scattering angle for five different polarisation angle. Here, the dashed green, orange, blue, red, and black curves are for $\theta_e$ = 0, 30, 45, 60, and 90 degree, respectively, and the solid gray curves are an analytical one expressed by equation (12).

Figure 6. The modulation curves after single scattering of completely polarised (incident) photons of energy 3 keV for a given $\theta$ . The solid blue, orange, green, red, and cyan curves are for $\theta$ = 0, 30, 45, 60, and 90 degree, respectively. The first (top), 2nd, 3rd, and 4th (last) panels are for $\theta_e$ = 0, 30, 45, and 90 degree, respectively. The curves are described analytically by equation (14).

6.1. Averaged degree of polarisation

The partially polarised incident photons of degree of polarisation $p_I$ and the angle of polarisation $\phi_I$ can be an average behaviour of total n type of incident photons which have different degree and angle of polarisation, say $p_j$ and $\phi_j$ for $j^{th}$ type incident photons, respectively. In context of Compton scattering, it can be expressed for $\xi$ -angle variable as (using equation (16)):

(18) \begin{equation} p_I \cos\!(2(\xi+\phi_I)) + 1 = \sum_{J=1}^n \frac{1}{n}[p_j \cos\!(2(\xi+\phi_j)) + 1 ],\end{equation}

For example, one has unpolarised photons which is a combination of two types of photon with $p_1 =p_0, \phi_1 =\phi_0$ and $p_2 = p_0$ , $\phi_2=\phi_o+\pi/2$ . It would be true for any value of $p_0$ (e.g. $p_0 = 1$ ). We use this characteristic to estimate the PD and PA for emergent photons either after single scattering or multi scattering. For this we compute the $\chi_j$ (PA of the $j\mathrm{ th}$ type scattered photons) using equation (11), as $\tan 2\chi_j = \frac{U}{Q}$ , and compute the averaged $p_j$ (over $\phi$ , PA of the incident photons) as a function of $\theta$ as, $p_j$ = $\frac{\langle Q\rangle}{\langle I\rangle}$ .

6.2. Case I: Fixed k direction (along z-axis) for a given scattering plane

We first consider a simplistic situation in which the unpolarised photons incident along the z-direction, and we fix the scattering plane to $\phi_s$ -plane, that is, the ( $k\times k^{\prime}$ ) direction is ( $\pi/2, \phi_s+\pi/2$ ). Since, the scattering plane is the same as the interested meridian $\phi_s$ -plane; thus, the scattering angle is $\theta = \theta_s$ . And so P will follow equation (7) with $\theta = \theta_s$ . The $\chi$ is always 90 degree (with the choice of ( $k\times k^{\prime}$ )-next), it means that the $k^{\prime}_{e}$ lies parallel to the (x, y)-plane. The result (for $\theta_i$ =0) is shown in the 2nd column of Table 1. In addition, similarly, for a fully polarised incident photons the PD will be determined by equation (12).

Table 1. The PD (P) and PA ( $\chi$ ) of the emergent photons as a function of $\theta_s$ (or i) for cases I and II. The results for case II are presented for six different scattered meridian planes $\phi_i + \phi_{is}$ with $\phi_{is}$ = 15, 30, 45, 60, 75, and 90 degree. While the results for case I are true for any scattered meridian plane, as here $\phi_s$ -plane = $\phi_i$ -plane, and computed for two different ranges of $\theta_i$ = 0 and [0, $\pi$ ].

1: here $\phi_{si}$ is measured in anti-clockwise direction, if it is measured in clockwise direction then the angle of polarisation becomes $\pi-\chi$ .

Instead of taking incident photons along the $z-$ axis (or $\theta_i=0$ ), we also consider any direction of k on incident meridian $\phi_i-$ plane, that is, the range of $\theta_i$ is $\theta_i \equiv [0,\pi]$ . And we assume again that the scattering plane is the same as the interested meridian $\phi_s$ -plane. In this case, we expect a constant P $\sim$ 0.33, as shown by the red curve in the right panel of Fig. 3. We find almost the same and show the results in the 3rd column of Table 1.

6.3. Case II: ( $k \times k^{\prime}$ ) lies on the meridian plane of k

In the previous case, the scattering plane and interested meridian plane are the same. Next, we consider that incident photons and the ( $k \times k^{\prime}$ ) both lie on the same $\phi_i$ -meridian plane, so ( $k \times k^{\prime}$ )’s direction for a given k is ( $\theta_i+\pi/2, \phi_i$ ). Now the photons will scattered any meridian ( $\phi_i+\phi_{is}$ )-plane, in which $\phi_{is}$ is the angle between incident and scattered meridian plane and the range of $\phi_{is}$ is $\equiv$ [ $-\pi, \pi$ ]. For $\theta_i$ = 0 and $\pi/2$ , the scattering plane of k is $(\phi_i \pm \pi/2)$ -plane and $(\theta = \pi/2)$ -plane, respectively. In general, the scattering plane of k will must pass the line ( $\pi/2,\phi_i+\pi/2$ ) (i.e. rotated y-axis with angle $\phi_i$ about z-axis, say y ^′-axis), or in other words it is a rotation of $(\phi_i \pm \pi/2)$ -plane about y ^′-axis with same angle $\theta_i$ . Thus, except for scattered meridian $(\phi_i \pm\pi/2)$ -plane, the k ^′ arises uniquely on a different scattering plane with different $\theta$ . For example, for the meridian $(\phi_i+\pi/6)$ -plane the $\theta$ varies from $\pi/6$ to zero, and $\chi$ varies from zero to $\pi/6$ , when $\theta_s$ varies from $\pi/2$ to zero. The degree of polarisation will simply determine by using equation (7), and in general, for a given scattered meridian $(\phi_i+\phi_{is})$ -plane the P varies from $P(\theta=\phi_{is})$ of equation (7) to zero for the variation of $\theta_s$ from $\pi/2$ to zero, respectively. The results for P as a function of $\theta_s$ for 6 different meridian planes with $\phi_{is}$ = 15, 30, 45, 60, 75, and 90 degree are shown in Table 1.

6.4. Case III: Any fixed incident photon direction and a random scattering plane

Next, we fix the k direction ( $\theta_i,\phi_i$ ) and take all possible directions of ( $k \times k^{\prime}$ ) on a perpendicular plane to k. Like case II, the incident photon is scattered in all directions. For a given meridian plane, each k ^′ is arisen from a different direction of ( $k \times k^{\prime}$ ), and clearly the different $\theta$ . The PD can be determined using equation (7) for known $\theta$ . Since, the scattering plane is unique for a given k ^′, like PD, PA will be also unique. We have computed the P and $\chi$ for the meridian plane ranges from $\phi_i$ to $\phi_i +\pi$ , and for each meridian plane $\theta_i$ ranges from 0 to $\pi/2$ . It should be noted that case III is identical to case I for $\theta_i$ = 0.

6.5. Case IV: k lies on the surface of cone at centre with opening angle $\theta_i$ and a random scattering plane

We extend the case III with considering that $\theta_i$ of k is still fix but now $\phi_i$ can take any value in [0,2 $\pi$ ], so essentially the vector k is rotating on the surface of the cone of opening angle $\theta_i$ which is situated at the centre. Unlike Case II or III, here on a given meridian plane the k ^′ is arisen by any k with appropriate scattering angle; thus, PD and PA of k ^′ would be estimated by averaging method. The averaged P and $\chi$ of the k ^′ are computed using equation (18) for a given $\theta_s$ , where now, n is total no. of scattered photons with $\theta_s$ , the $p_j$ is determined for each scattered photon for known $\theta$ by using equation (7). Since, for unpolarised incident photons the $k^{\prime}_{e}$ lies along the $(k\times k^{\prime})$ which gives $\phi_j$ = 0, but we have fixed the reference $(k\times k^{\prime})\text{-next}$ to measure the PA, so $\phi_j$ is the angle between $(k\times k^{\prime})$ and meridian plane ( $(k\times k^{\prime})$ -next). Expectedly, the variation of P, and $\chi$ as a function of $\theta_s$ is same for all scattered meridian planes. In Table 2, we have noted the results for $\theta_i$ = 0, 15, 30, 45, 60, 75, and 90 degree along with $f_{\theta_s}$ (a total no. of scattered emergent photons with $\theta_s$ on a given meridian plane). For example, the emergent photons along the z-axis ( $\theta_s$ =0) are unpolarised for any opening angle $\theta_i$ , as in this case for all incident photons the scattering angle is $\theta = \theta_i$ , and the electric vector of k ^′ is isotropically distributed. In general, for a given $\theta_i$ , the emergent photons escape maximally at $\theta_s$ $\sim$ $\theta_i$ , for example, for $\theta_i$ = 15 $^{\circ}$ the maximum emergent photons escape with $\theta_s$ $\sim$ 25 $^{\circ}$ and have PD $\sim$ 0.06.

Table 2. The PD and PA of the emergent photons for case IV along with $f_{\theta_s}$ (total no. of the emergent photons for a given $\theta_s$ ). The results are presented for seven different opening angles of cone (on which incident photon lies), $\theta_i$ = 0, 15, 30, 45, 60, 75, and 90 degree.

^*the corresponding value is not reliable due to low statistics.

Note: for $\theta_i = \theta_s$ , one has small P (in range of (0.002-0.08)) and maximum $f_{\theta_s}$ .

We compare the above results with the results of case III. Since the given k ^′ can be arisen from the incident meridian ( $\phi_s+\phi_{is}$ )-plane with $\phi_{is}$ $\equiv$ [− $\pi$ , $\pi$ ] of case III, or particularly n different situations of case III in range of $\phi_i$ $\equiv$ $[0,2\pi]$ . As the P and $\chi$ of case III are uniquely determined for a given $\phi_{is}$ , the resultant P and $\chi$ of case IV for a given $\theta_s$ can be obtained by weighted averaging method, as defined in equation (18). For present case this equation is rewritten as:

(19) \begin{equation} P \cos\!(2(\xi+\chi)) + 1 = \sum_{j=1}^{n} f_j \left [p_j\cos\!(2(\xi+\chi_j))+1\right], \end{equation}

here, $f_{j}$ is the fraction of emergent scattered photons at $\theta_s$ to total that arisen due to the $j\mathrm{th}$ incident meridian $(\phi_s+\phi_{is})$ -plane, and $p_j$ , and $\chi_j$ are corresponding PD and PA, respectively. We find that the results are similar in both ways.

6.6. Case V: General case

Finally, we consider a general case where there is no restriction on k. That is, the scattered photon k ^′ is arising from all directions of k. We compute the P as a function of $\theta_s$ for any meridian $\phi_s$ -plane, We find P $\approx$ 11, 0 $\%$ for $\theta_s$ = 90, 0 degree, respectively. The $\chi$ is $\sim$ 90 degree for all $\theta_s$ . Since we are measuring $\chi$ with respect to $(k\times k^{\prime})$ -next, it signifies that $k^{\prime}_{e}$ is parallel to the (x,y)-plane. Clearly, the all scattering planes (which generate the k ^′ on $\phi_s$ -plane) are not a $\phi_s$ -plane, but we notice that on averaged the scattering plane is mostly a scattered meridian plane; thus, the results $\chi$ $\sim$ 90 $^{\circ}$ confirm that the polarisation of single scattered unpolarised (incident) photons is perpendicular to the scattering plane. The results are shown in Table 3.

Table 3. The PD of the emergent photons for general case V from any given meridian plane after single scattering of the randomly oriented unpolarised incident photons (or, the re-estimation of laws of darkening of Chandrasekhar (Reference Chandrasekhar1946) with having a general Klein–Nishina cross section, equation (1)).

Like case IV, the results of case V are verified by a weighted averaging method using the result of case IV. For convenience in Table 2 we have also listed the $f_{\theta_s}$ along with the P and $\chi$ . Thus, $f_j = \frac{f_{\theta_s}}{\Sigma_{j=1}^n f_{\theta_s}}$ for a given $\theta_s$ and $\theta_i$ (here, $i=j$ ), and with having $n = 7$ we obtain the results of case V approximately using Table 2 and equation (19).

The results are qualitatively agreed with almost century old calculations of Chandrasekhar (Reference Chandrasekhar1946) (see also, Chandrasekhar Reference Chandrasekhar1960). Chandrasekhar (Reference Chandrasekhar1946) had solved the radiative transfer equations, which is governed by the Thomson scattering by free electron, for the intensities of two states of polarisation, one $I_l$ is along the meridian plane and other one $I_r$ is perpendicular to it, with no incident radiation. In defining the source function for $I_l$ or $I_r$ , the considered cross section for the polarisation in perpendicular to and parallel to the scattering plane is either for unpolarised photons, or combinations of two polarised photons with polarisation angle $\phi$ and $\phi + \pi/2$ , see equations (2) and (3) of their paper and table 2 for the results (and for refined results see table XXIV in Chandrasekhar Reference Chandrasekhar1960). So, the laws of darkening for the PD of the emergent photon is only for the single scattering event in Thomson regime for unpolarised incident photons. In the next section, we compute the laws of darkening for multi scattering events.

For all cases, we have estimated the PD and PA by two another methods, (i) we perform second scattering and construct the modulation curve for either $\theta = 0$ or $\theta \equiv [0,25^{\circ}]$ to estimate the P as discussed in Section 2.3, (ii) we construct the modulation curve using the probability $p(\eta)$ (of equation (17)) and estimate the P, as discussed in Section 4.1. We find that in all cases the results match qualitatively with these two methods.

7.1. Multi-scattering

To verify the MC calculations for multi scattering we consider the case I with $\theta_i \equiv [0,\pi]$ , as in this case the scattered photon has PD $\sim$ 0.33 in all directions of $\theta_s$ . We first compute (say, first method) the P value of 2nd times scattered photon for this case. Next, we consider (say, 2nd method) a partially polarised incident photon with P = 0.33, $\chi$ = 90 degree and $(k \times k^{\prime})$ lies either on the meridian plane or perpendicular to the meridian plane (and here, for 0.66 fraction of unpolarised photons, the $(k \times k^{\prime})$ lies randomly) and estimate the P after single scattering. We find that for both methods the result qualitatively agrees, with P $\sim 0.1-0.12$ on a given meridian plane.

Interestingly, we find that for all cases, I $-$ V, after 3-4 numbers of scattering, the maximum P value is reduced to $\sim$ $0.02 - 0.05$ on any meridian plane. Here, again we consider a Thomson regime and almost rest electron. Therefore, for all cases and average scattering number $>$ 4, the emergent scattered photon is mainly unpolarised with averaged maximum $P \sim 0.035$ . Here, we like to point out that if these emergent photons again freshly scatter in optically thin corona with average scattering number $\sim$ 1 then the P $\&$ $\chi$ will be described like case V.

7.2. Energy dependency of polarisation

The prime focus is here to study the polarisation properties for XRBs, and XRBs frequently transit from soft spectral state to hard state and vice versa. To understand and to explore the energy dependency of polarisation for Comptonised photons, we consider two different steady spectral states with unpolarised seed photons. The first is a soft spectrum (Model 1) with low electron medium temperature $kT_e$ $= 2.5$ keV (see, e.g. Kumar & Misra Reference Kumar and Misra2014). In view of neutron star NS low-mass XRBs, we consider two different seed photon source (black body) temperatures corresponding to Hot-seed and Cold-seed photon model (see, e.g. Kumar & Misra Reference Kumar and Misra2016a, references therein) which are $kT_b$ $=$ 1.5 and 0.7 keV, respectively. We refer Model 1a with temperature $kT_e =$ 2.5 keV and $kT_b =$ 1.5 keV and for Model 1b $kT_e =$ 2.5 keV and $kT_b =$ 0.7 keV. Since, for consistency we consider only a spherical corona geometry, but these two models are defined based on the seed photon source geometry (e.g. Lin et al. Reference Lin, Remillard and Homan2007); thus, our study does not give physical insight of the model but only provides the dependency of PD and PA on $kT_b$ variations. The second is a hard spectrum (Model 2) with high electron medium temperature $kT_e =$ 100 keV and $kT_b =$ 0.3 keV. To explore the general results we take two optical depth values in such a way that the corresponding average scattering number $\langle N_{sc}\rangle$ $\approx$ 1.1 and 5.0. Thus, conclusively to explore the energy dependency of PD and PA, we take mainly three Models 1a, 1b, and 2 with two values of $\langle N_{sc}\rangle$ $\approx$ 1.1 and 5.0. PD and PA have been computed for four values of $\theta_s$ = 30 $^{\circ}$ , 45 $^{\circ}$ , 60 $^{\circ}$ , 75 $^{\circ}$ . In considered spherical geometry, if one assumes, the z-axis is along the radio-jet direction, then the (x,y)-plane will mimic the accretion disc and so the angle $\theta_s$ is equivalent to the disc inclination angle i (or the angle between the line of sight and the normal to the disc plane). Thus, we also study the variation of PD and PA with disc inclination angle i. And now, in present convention, PA = 90 $^{\circ}$ signifies that the electric vector is parallel to the disc plane.

The general results for Model 1a and 1b are shown in Fig. 9. The upper and lower panels are for $\langle N_{sc}\rangle$ $\approx$ 1.1 and 5, respectively. In the left panel, the seed photon flux is shown by dashed curve and Comptonized photon flux by solid line. As expected for $\langle N_{sc}\rangle$ $\approx$ 1.1 the PD as a function of $\theta_s$ (or, i) is slightly lower than the respective values listed in Table 3. The PD values for given i are almost constant over the energy bin, also qualitatively independent from the seed photon source temperature $kT_b$ . Due to the low photons statistics, the PD and PA are computed upto photon energy 8 and 16 keV for Model 1a and 1b, respectively, and we also notice that their values are fluctuated around this energy bin. Since for unpolarised incident photons and for single scattering, it is expected that the electric vector of Comptonized photons will lie normal to the scattering plane. And for $\langle N_{sc}\rangle$ $\approx$ 1.1 we find almost the same, here for all i values, the PA lies from 89 $^{\circ}$ to 95 $^{\circ}$ .

Figure 9. Comptonized photons density distribution, degree of polarisation and angle of polarisation are shown in left, middle, and right panels, respectively, when the seed photons (shown by dashed line in left panel) are unpolarised. The upper and lower panels are for the average scattering number $\approx$ 1.1 and 5, respectively. PD and PA are computed for four $\theta_s$ (or, alternatively the disc inclination angle i) values = 30, 45, 60, and 75 degree, which are shown by dashed-dotted, solid, dotted, and dashed lines, respectively. PA is computed on perpendicular plane to the escaped photon directions, not on the sky plane, here PA = 90 $^{\circ}$ signifies that the electric vector parallel to the (x,y)-plane or accretion disc, see the text for details. The energy bins are 0.8, 1, 2, 4, 6, 8, 10, 14, 18, 22, 26, and 30 keV.

In model 1, the spectral parameters are in Thomson regime, so the distribution of scattering angle will be $\propto (1+ \cos^2\theta)$ . In considered spherical corona geometry, the Comptonized photons after any number of scattering follow the same scattering angle distribution of Thomson regime. But the Comptonized photons which escape the medium (corona) do not follow it for $\langle N_{sc}\rangle >$ $\sim$ 1.1. The deviation from Thomson cross section for escaped Comptonized photons can be understood as. Supposed, before escaping the medium the scattered photon is at scattering site whose distance is 0.7L from the centre, then to escape the medium in forward direction the photon has to travel collision free path of length 0.3L but in case of backward direction the collision free path length is 1.7L. With having the exponential distribution for the collision free path it is more likely, on average (as the probability for travelling the scattered photon in forward or backward direction is same), that the photon will escape the medium in a forward direction in comparison to the backward direction. In addition, the trend for escaping the photons in forward direction increases with $\langle N_{sc}\rangle$ , as the mean free path for photons decreases with increasing optical depth. For a given medium, it is also expected that after some value of $\langle N_{sc}\rangle$ the scattering angle distribution will get saturated, we find, the saturation occurs around $\langle N_{sc}\rangle$ = 25. The results are shown in Fig. 10.

Figure 10. The $\theta$ -angle distribution of escaped Comptonized photons in spherical corona after experienced of average scattering number $\langle N_{sc}\rangle$ . The blue, red, orange, cyan, magenta, and green curves are for $\langle N_{sc}\rangle$ = 1.1, 2.0, 3.0, 5.0, 26.7, and 44.1, respectively. The Klein–Nishina cross section in Thomson regime is shown by black dotted curve. Note, the Comptonized photons which are inside the medium always follows Klein–Nishina cross section for any given $\langle N_{sc}\rangle$ .

For $\langle N_{sc}\rangle$ $\approx$ 5 (in lower panel of Fig. 9), the PD and PA are calculated upto photon energy 14 and 22 keV for Model 1b and 1a, respectively. Like, $\langle N_{sc}\rangle$ $\approx$ 1.1, the PD and PA are independent of the k $T_b$ . PD is almost constant over the photon energies ( $<$ 10 keV). The fluctuation in PD and PA values above 10 keV is due to the low photons statistics. PA values range from 80 $^{\circ}$ to 120 $^{\circ}$ when the i ranges from 30 $^{\circ}$ to 75 $^{\circ}$ . We observe that the $\theta$ -angle distribution for escaped Comptonized photons for a given i has an extra small hump (by a factor $\sim$ 1.1 $-$ 1.2 from the corresponding values of cyan curve of Fig. 10) around $\theta = i$ , which leads to a maximum value of PD for i = 45 $^{\circ}$ . Here, we find the maximum value of PD $\sim 0.025$ for i =45 $^{\circ}$ . In general, in the soft state of XRBs the optical depth is relatively high, to characterise this we consider Model 1a with $\langle N_{sc}\rangle$ $\approx$ 26.7. The results are shown in Fig. 11. Here, the magnitude of PD values is similar to the case of $\langle N_{sc}\rangle$ $\approx$ 5, only the dependency of PD value on i has changed. The range of PA values is wider now and for i = 75 $^{\circ}$ PA is $\sim$ 150 $^{\circ}$ .

Figure 11. For spectral parameter Model 1a and the average scattering number $\approx$ 26.7. The rests are same as Fig. 9.

In Fig. 12, the general results for Model 2 have been shown. The upper and lower panels are $\langle N_{SC}\rangle$ $\approx$ 1.1 and 5. For $\langle N_{SC}\rangle$ $\approx$ 1.1, the PD and PA have been computed upto 30 keV. Like Model 1, PD is constant over photons energy ( $<$ 10 keV), and PA is $\sim$ 90 $^{\circ}$ and independent to the photons energy ( $<$ 6 keV). The magnitude of PD is slightly lower in comparison to the Model 1, for example, for i = 75 $^{\circ}$ PD = 0.07 and 0.05 for Model 1 and 2, respectively. Also, for above 6 keV photons energy the PA ranges 95 $^{\circ}$ - $\sim$ 100 $^{\circ}$ . Here, the decrement in PD values and the deviation of PA from 90 $^{\circ}$ is mainly due to the multi scattering. Since, in Model 2 due to the large $kT_e$ = 100 keV and low $kT_b$ = 0.3 keV, the photons of energy $>$ 2 keV on average experienced a large scattering number from the averaged value 1.1. For example, for 2–10 keV photons the averaged scattering number varies from 1.1 to 1.5, while for 10–70 keV it varies from 1.5 to 2.5. For $\langle N_{SC}\rangle$ $\approx$ 5, as expected the PD value is slightly lower in comparison to the corresponding value of Model 1. The increasing behaviour of PD for low photons energy ( $<$ 10 keV) is mainly corresponded to the $\theta$ -angle distribution for escaped photons, as the photons with energy less than 7 keV have average scattering number less than 5. The PA values range from 80 $^{\circ}$ to 150 $^{\circ}$ .

Figure 12. For spectral parameter Model 2 and the upper and lower panels are for average scattering number $\approx$ 1.1 and 5.0, respectively. The energy bins are 0.8, 1, 2, 4, 6, 8, 10, 14, 18, 22, 26, 30, 40, 50, 100, and 200 keV. The rests are same as Fig. 9.

For completeness, we have also computed the PD and PA for the Wien spectrum of Model 1 and 2, the results are shown in Fig. 13. We know that the Wien spectrum does not depend on the seed photon spectrum but only depends on the electron medium temperature. In addition, for low $kT_e$ (or non-relativistic electrons) one needs a large scattering number to generate the Wien spectrum, while for large $kT_e$ (relativistic electron) one need comparatively small scattering number (e.g. Kumar & Kushwaha Reference Kumar and Kushwaha2021). We compute the Wien spectrum for Model 1 and 2 with $\langle N_{SC}\rangle$ = 170 and 45, respectively. For Model 1, the dependency of PD on i, and the range of PD are similar to the case of Model 1a with $\langle N_{SC}\rangle$ = 26.7. While for Model 2 the magnitude of PD is around 0.01 upto photon energy 1 000 keV. Since, in the case of Model 2, in last, the seed photons are mainly $\sim$ 200 keV photons. From equation (7) (or, see Fig. 2) we know that the PD value for 200 keV photons is comparatively smaller than the seed photon of energy 10 keV (or $<$ 10 keV), as a result we obtain a smaller value comparison to the Model 1. Conclusively, The dependency of PD on i of Wien spectra for unpolarised seed photons does not behave like case V, see Table 3. This also indicates that the emergent photons from thin accretion disc (in which, the emergent spectrum is a black body due to the large optical depth by Thomson scattering Shakura & Sunyaev Reference Shakura and Sunyaev1973) would be mainly unpolarised with maximum PD $\sim$ 0.03 for i $\sim$ 45 $^{\circ}$ .

Figure 13. For Wien spectra of Model 1a and 2 with average scattering number $\approx$ 170 and 45, respectively. The energy bins for Model 1a are 0.8, 1, 2, 4, 6, 8, 10, 14, 18, 22, and 26 keV and for Model 2 are 30, 40, 50, 60, 80, 100, 200, 400, 600, 1 000, and 5 000 keV. The rests are same as Fig. 9.

7.3. Geometry dependency

With having spherical corona we compute PD and PA for two extreme sets of spectral parameters. Particularly for large optical depth, the PD values are always less than 0.03 for photons of energy $<$ 10 keV. Recently IXPE has measured PD for many XRBs sources, and for a few sources the estimated PD is greater than the 0.03 in 2–8 keV energy band (e.g. Krawczynski et al. Reference Krawczynski2022; Jayasurya et al. Reference Jayasurya, Agrawal and Chatterjee2023). Authors explained the comparatively large PD mainly in terms of different possible geometry for scattering medium in the disc. In Section 6, we had noticed that the PD also depends on incident photon direction, see the Table 2 for Case IV; also see, for example, Case I for the variation of PD as a function of $\theta$ . Therefore, the dependency of PD on the geometry can be happened in terms of either distribution of incident photon direction or distribution of $\theta$ -angle for the escaped Comptonized photons, or both. To understand the dependency of PD on geometry for simplicity we ad-hocly exclude the some range of $\theta$ of the Comptonized escaped photons in estimation of PD without bothering about the spectra. However, in Appendix B we argue that in Thomson limit the Comptonized spectrum is independent of $\theta$ , and thus the spectra would not get changed in these cases. The results are shown in Fig. 14. The left and right panels are for the Model 1a and 2, respectively. For Model 1a, we consider two sets of $\theta$ range, [30,180 $^{\circ}$ ] and [0,150 $^{\circ}$ ], while [45,180 $^{\circ}$ ] and [0,135 $^{\circ}$ ] for Model 2. We find that PD values after excluding the backward direction are smaller than the PD obtained by excluding the forward direction for both models. This is because of the $\theta$ -angle distribution of escaped photons, in which the maximum photons escape the medium with $\theta$ = 0, see Fig. 10. Hence, for considered geometry for both models the magnitude of PD is around 0.04 or above when we exclude the forward direction of escaped photon.

Figure 14. To understand the geometry affect in PD calculations, for simplicity we ad-hocly consider an asymmetric range of the scattering angle. The black and magenta curves are, in left panel for $\theta$ $\equiv$ [30,180 $^{\circ}$ ] and [0,150 $^{\circ}$ ] and in right panel for $\theta$ $\equiv$ [45,180 $^{\circ}$ ] and [0,135 $^{\circ}$ ], respectively. The left and right panels are for Model 1a with $\langle N_{sc}\rangle\approx 26.7$ and Model 2 with $\langle N_{sc}\rangle\approx 5.0$ , respectively. PD is computed for four i values = 30, 45, 60, and 75 degree, and shown by dashed-dotted, solid, dotted, and dashed lines, respectively. For left panel the energy bins are same as Fig. 9 while Fig. 12 for right panel.

7.4. Comparison with observations

In this section, we briefly discuss the polarisation properties of a few sources observed by IXPE. However, these explanation would be a purely qualitative, as we have a simple spherical corona geometry, and also the present calculations have not been implemented in the general relativity formalism. Moreover, the main motive of this exercise is to what extent one can learn in understanding of polarisation properties with having simple spherical corona geometry. We select five sources, in which two are black hole XRBs and three are neutron star XRBs.

• 4U 1630-47: IXPE had observed 4U 1630-47 from 23-Aug-2022 to 02-Sept-2022, when the source is in high soft spectral state. The polarisation properties have been analysed by three groups (Rawat et al. Reference Rawat, Garg and Méndez2023; Kushwaha et al. Reference Kushwaha, Jayasurya, Agrawal and Nandi2023; Ratheesh et al. Reference Ratheesh2023), and they found that PD is an energy-dependent which increases significantly with energy. In 2–8 keV band PD is $\sim$ 0.08 and PA in the sky plane is $\sim$ 18 $^{\circ}$ . They argue that the observations can explain in thin accretion disc with mildly relativistic outflowing medium.

  In the present scenario, the observed flux is described almost by Model 1a with $\langle N_{sc}\rangle$ = 1.1 (see left upper panel of Fig. 9). Therefore, PD is energy-independent and PD = 0.08 for i = 75 $^{\circ}$ . PA is always around 90 $^{\circ}$ , that is, the electric vector is parallel to the (x-y) plane or accretion disc. The energy dependency of PD can be arose by fresh scattering in optically thin ( $\langle N_{sc}\rangle$ = 1) medium (possibly, wind) where large fraction of high-energy photons experience scattering in comparison to the low-energy photons, which will increase the PD value to 0.1 for high-energy photons (see Table 3 and text).

• Cyg X-2: Farinelli et al. (Reference Farinelli2023) measure the polarisation properties of Cyg X-2 (a Z-source), in 2–8 keV band the PD = 0.018 and PA = 140 $^{\circ}$ where the polarisation is in the direction of radio jet (possibly on the sky plane). They argue that the observed PD cannot be explained with accretion disc geometry and suggest another geometry related to the neutron star surface. The observed flux of Cyg X-2 can be described nearly with Model 1a and $\langle N_{sc}\rangle$ =26.7. Thus for the inclination angle i $\approx$ 60 $^{\circ}$ , PD $\approx$ 0.015 and PA $\approx$ 120 $^{\circ}$ ; here, we remind that PA is measured in a perpendicular plane to the escaped photon direction. Hence, our calculation indicates that the observed PD can be explained, in general, with accretion disc + corona geometry.

  The similar range of PD ( $\approx$ 0.017) is measured for atoll source GX 9+9: by Chatterjee et al. (Reference Chatterjee, Agrawal, Jayasurya and Katoch2023). The reported PA is $\sim$ 63 $^{\circ}$ where the range of i is 40 $ -$ 60 $^{\circ}$ . The observed flux can be described here by Model 1a with $\langle N_{sc}\rangle$ lies in range 5–26. In our calculations, like Cyg X-2, the observed PD of GX 9+9 can be described in the accretion disc scenario. The calculated PA is around either (90–120 $^{\circ}$ ) or (60–90 $^{\circ}$ ) for a given range of i (here, 180 $^{\circ}$ differences in PA is due to the two possible definitions for modulation curve, see equation (16)).

• XTE J1701-462: IXPE had observed XTE J1701-462 during an outburst two times, Sept-2022 (epoch 1) and Oct-2022 (epoch 2). Jayasurya et al. (Reference Jayasurya, Agrawal and Chatterjee2023) measure a significant PD $\sim$ 0.045 for epoch 1 and a negligible PD $< $ 0.01 for epoch 2 in 2–8 keV band. The PA is $\sim$ 143 $^{\circ}$ , while the source has i close to 70 $^{\circ}$ . The observed flux can be described with Model 1a with $\langle N_{sc}\rangle$ $>$ 26 for epoch 1 while $\sim$ 25 for epoch 2 (see Fig. B1 for epoch 1 modeled flux). In present study with simplistic spherical geometry we can not explain the observed PD for epoch 1, one needs other geometry. However, if we exclude some fraction of forward directed escaped photons in this geometry, then we can explain the observed magnitude, see left panel of Fig. 14. Moreover the observed PD for epoch 2 can be explained in present study for $i \sim$ 70 $^{\circ}$ . The calculated PA for $i \sim$ 70 $^{\circ}$ is around 150 $^{\circ}$ . Therefore, in present observations, the polarisation properties of XTE J1701-462 indicates that the corona geometry changes in month scale. For a spectro-polarimatric study in details, we consider this source, see Appendix B.

• Cyg X-1: Krawczynski et al. (Reference Krawczynski2022) estimate the polarisation properties for hard state of Cyg X-1, the reported magnitude is PD $\approx$ 0.04 and PA $\approx$ 20 $^{\circ}$ in the plane of sky. They also find that the X-ray polarisation almost aligns with radio jet in the plane of sky, which is consistent with the previous results of Chauvin et al. (Reference Chauvin2018) using the PoGO+ balloon-borne polarimeter in 19–181 keV. Chauvin et al. (Reference Chauvin2018) had measured PD $\approx$ 0.045 and PA = 154 $^{\circ}$ in 19–181 keV. The observed flux can be described by Model 2 with $\langle N_{sc}\rangle$ $\approx$ 5, thus the estimated PD $<$ 0.025. Krawczynski et al. (Reference Krawczynski2022) found the PD $<$ 0.03 for a spherical lamp-post corona and non-spinning black hole. Therefore, our calculations are consistent with the result of Krawczynski et al. (Reference Krawczynski2022). Although, Krawczynski et al. (Reference Krawczynski2022) argue that the observed high PD can be explained with sandwich corona with i $\sim$ 45 $^{\circ}$ (see theirs Fig. 3).

  Clearly, like epoch 1 of XTE J1701-462, in present study with having spherical corona we can not explain the observed high PD. However, if we exclude the some fraction of forward escaped photons in spherical geometry then we can explain the observed high PD in broad band 2–181 keV (see right panel of Fig. 14).