The influence of the Insight-HXMT/LE time response on timing analysis

The number of detected X-ray photons ($N$) in a given time interval $(t_{min},t_{max})$ obeys a Poisson distribution with a parameter $\lambda(t)$ (Emadzadeh & Speyer 2011), i.e.,

where $P(N=k)$ represents the possibility of detecting $k$ photons in the given time interval $(t_{min},t_{max})$; $\lambda(t)$ is called the time-varying rate or the intensity function in the unit of photons per second and $\int_{t_{min}}^{t_{max}}\lambda(t)dt=E[N]$, where $E[N]$ is the expectation of the variable $N$. Let $T_{0}$ represent the actual arrival time of an X-ray photon and $D$ the readout time. Therefore, the time $Z$ recorded by the LE/SCD is that

Here the $D$ is composed of $D_{0}$ and $D_{k}$, where $D_{0}$ equals to $10^{-5}$ second, which is the fixed delay caused by the 100 kHz working frequency of the readout mode of LE/SCDs, and $D_{k}$ is the time delay caused by the TRD effect.

The intensity function recorded by LE/SCDs is influenced by the TRD effect, which can be described as the convolution of the original intensity function $\lambda(t)$ and the TRD (Zhang et al. 2017). Considering the updated TRD and the fixed delay , the response intensity function $\lambda^{\prime}(t)$ is given by

where n=118 and $\lambda(t)$ is the original intensity function; $p_{k}$ is the $kth$ probability of delaying $10^{-5}k$ seconds. $p_{k}$ and the corresponding delay are shown in Fig. 2; $\delta$ is the Dirac delta function. Thus $h(t)$ is the impulse response function of LE/SCDs. The detection area of each SCD quadrant is a L-shaped strip, which is called the ’packed-pixel’ (Fig. 1). There are 120 ’packed-pixels’ in each quadrant. The 20 ’packed-pixels’ near the centre have a fixed read-out time. On the other hand, two ’packed-pixels’ close to the outskirt of SCDs are invalid, which are thus not considered. Therefore, with the working frequency of 100 kHz, all charges would be readout within 1.18 ms (Zhao et al. 2019). Because the 20 ’packed-pixels’ near the centre have a fixed delay and the area is smaller than the outside, the time resolution of LE/SCDs is often recorded as 0.98 ms (Chen et al. 2020). In this paper, all delay effects are taken into account. So the sequence number of formula (3) starts from 1 to 118. The LTR effect in the timing analysis can be studied by comparing $\lambda^{\prime}(t)$ with $\lambda(t)$.

The Monte Carlo (MC) simulation is widely used when dealing with difficult quantitative analysis. The general method is as follows:

1. 1.

   In the time interval $(t_{min},t_{max})$, the time set $\{t_{i}\}_{i=1}^{M}$ that represents arrival times of photons is sampled from the original intensity function $\lambda(t)$ by using the rejection sampling method. The time set $\{t_{i}\}_{i=1}^{M}$ satisfies $t_{1}\leq t_{2}\leq...\leq t_{i}\leq...\leq t_{M-1}\leq t_{M}$.The time set $\{t_{i}\}_{i=1}^{M}$ follows a given distribution, for examples, periodic signals, QPOs signals, etc.

2. 2.

   Add a random number to each value in the time set $\{t_{i}\}_{i=1}^{M}$ to get a new time set $\{u_{i}\}_{t=1}^{M}$, where $u_{i}=t_{i}+D$. We arrange $\{u_{i}\}_{t=1}^{M}$ in a ascending order and still denote it as $\{u_{i}\}_{t=1}^{M}$. Therefore, $\{u_{i}\}_{t=1}^{M}$ satisfies $t_{1}+10^{-5}\leq u_{1}\leq u_{2}\leq...\leq u_{i}\leq...\leq u_{M-1}\leq u_{M}\leq t_{M}+118\times 10^{-5}$.

3. 3.

   Extract two light curves from time sets $\{t_{i}\}_{i=1}^{M}$ and $\{u_{i}\}_{t=1}^{M}$, which represent signals before and after the LTR, respectively. Then the influence of the LTR on the timing analysis can be studied by comparing these two light curves.

In order to carry out the simulation more efficiently, a spectral-timing software package stingray for astrophysical X-ray data analysis is employed (Huppenkothen et al. 2019).

Let $P_{\lambda}(f)$ and $P_{\lambda^{\prime}}(f)$ represent PSDs of $\lambda(t)$ and $\lambda^{\prime}(t)$, respectively. Thus, their relation can be written as:

where $H(f)$ is the Fourier transform of the impulse response function $h(t)$ and $f$ represents the frequency. Considering the formula (3), $|H(f)|^{2}$ is given by

Obviously,

where $n$=118. Therefore, the $P_{\lambda^{\prime}}(f)$ is smaller than $P_{\lambda}(f)$ at any frequency and the attenuation factor is $|H(f)^{2}|$ (shown in Fig. 3).

In addition, we performed MC simulations using the method mentioned above. In practice, we assumed that the original light curve has an intensity of 1000 cts/s and an exposure of 100 s. We show the modulation of the PSD caused by the LTR in Fig. 3. We found that our simulations are well consistent with the theoretical estimation.

We used the MC method to simulate quasi-periodic oscillations (QPOs), which are typical signals in the astronomical timing analysis. In the PSD, QPOs commonly appear as a peak at a certain frequency with a finite width, which can be described as a Lorentzian function (Wang 2016)

where $\nu_{0}$ is the centroid frequency, $\omega$ is the full-width at half maximum (FWHM), and $A_{0}$ is the amplitude of the signal. The quality factor (define Q$\equiv\frac{\nu_{0}}{\omega}$) represents the significance of QPOs. By convention, signals with $Q>2$ are called QPOs while those with $Q\leq 2$ called the peaked noise (Wang 2016).

We simulated different centroid frequencies of QPOs. The detailed process is as follows:

• •

  Consider Lorentzian-shape PSDs described by the formula (7) with specific parameters. In practice, we performed simulations assuming variable centroid frequencies between 10 Hz and 600 Hz with a resolution of 10 Hz. We also set $A_{0}\equiv 1$ and $\omega\equiv\frac{\nu_{0}}{10}$. Therefore, the Q factor of QPOs signals we simulated always equals to 10.

• •

  Simulate a light curve for a given PSD using the algorithm proposed by Timmer & König (1995). The light curve duration and root mean square (RMS) were set at 500 s and 0.5, respectively. The time resolution of the light curve was set at 1/4 reciprocal of the centroid frequency of the corresponding QPOs.

• •

  Using the light curve as a intensity function, the time set $\{t_{i}\}_{i=1}^{M}$ before the LTR and the time set $\{u_{i}\}_{i=1}^{M}$ after the LTR were obtained by using the method mentioned in subsection 2.2.

• •

  Generate PSDs using the time sets $\{t_{i}\}_{i=1}^{M}$ and $\{u_{i}\}_{i=1}^{M}$. In order to suppress the Poisson noise, each light curve was divided into 10-second segments to calculate PSDs independently, and then an averaged PSD could be obtained.

  Finally, we got resulting parameters of QPOs after the LTR, and compared them with the input models.

Fig. 4 shows an example. The PSD calculated here is based on the Miyamoto normalization, which is a convenient method for calculating the RMS of QPOs (Miyamoto et al. 1992).

The influence of the LTR on QPOs parameters can be obtained as a function of the QPO frequency, which is defined as $\delta\equiv\frac{\beta}{\alpha}$. $\alpha$ represents the parameters of QPOs before the LTR, such as $\nu_{0}$, $\omega$, $A_{0}$ and RMS, and $\beta$ represents the corresponding values after the LTR. The $\delta$ is their ratio, which is denoted as $\delta_{\nu_{0}}$, $\delta_{\omega}$, $\delta_{A_{0}}$, $\delta_{QPOrms}$, respectively. Fig. 5 shows the changes of $\delta_{\nu_{0}}$, $\delta_{\omega}$, $\delta_{A_{0}}$ and $\delta_{QPOrms}$ with the QPO frequency. $\delta_{\nu_{0}}$ and $\delta_{\omega}$ are almost invariant, which indicates that they are not affected by the LTR.

On the other hand, $\delta_{A_{0}}$ of QPOs decreases with the increasing frequency, indicating that the LTR has a great influence on the peak value of QPOs. This is similar to the results obtained from subsection 3.1. We used a quadratic polynomial function $y=ax^{3}+bx^{2}+cx+d$ to fit $\delta_{A_{0}}$. The fitting parameters are $a=-1.05\times 10^{-9}$, $b=3.02\times 10^{-6}$, $c=-2.10\times 10^{-3}$, $d=0.954$, respectively. According to the simulation, the Q factor of QPOs does not change with the centroid frequency because both the FWHM $\omega$ and the centroid frequency $\nu_{0}$ of QPOs are unchanged after the LTR. Considering that the peak value of QPOs decreases as the frequency increasing, the actual data contains high frequency QPOs may miscalculate the Q factor because the significance of QPOs obtained is too low. The RMS results are shown in the panel d of Fig 5. It is similar to $\delta_{A_{0}}$, which decreases with the increase of the centroid frequency. We fitted the results with a function $y=ax^{2}+bx+c$. The fitting parameters are $a=1.21\times 10^{-6}$, $b=-1.11\times 10^{-3}$, $c=0.977$, respectively.

In the time domain, the time lag between two signals is usually calculated using the cross-correlation function (Li et al. 2012). The time displacement that maximizes the cross-correlation function is called the time delay between the two signals. In this section, we simulate two signals with a fixed time delay. Then we calculate their time lag after the LTR. In this way, we explore how the LTR influence the time lag.

For the sake of simplify, we chose a flat signal as the intensity function $\lambda(t)$. We sampled this intensity function to get the time set which represent arrival times of a batch of photons. Then we shifted the time sets with a constant $\tau$ to get a new time set. These two time sets have a fixed time lag $\tau$. Then cross-correlation function between light curves extracted from these two time sets were calculated, which is shown in Fig. 8. The time resolution of light curves was set at half of the time lag $\tau$. Two cross-correlation functions (before and after the LTR) are compared. It is clear that the smaller the time lag, the smaller the peak of the cross-correlation function. The LTR only reduces the values of the function and does not change the maximum position of it, so the calculated time lag actually will not be influenced.

However, when we consider light curves detected with LE and ME/HE, the LTR effect is expected to have an influence on the lag detection. In this case, two non-lag time sets were generated, one of which was responsed by the LE LTR. After that, the cross-correlation function was calculated between these two signals to obtain the time lag. This simulation shows that the resulting time lag is 1.18 millisecond. It is the maximum time delay of the LTR. This is reasonable because the intensity function is mainly delayed by 1.18 millisecond.

We used the cross-spectrum to estimate the time lag of two signals at different frequencies (Huppenkothen et al. 2019). Because we want to calculate the signal time lag at different frequencies, we assumed a set of sinusoidal signals to produce the the intensity function, and generated a time set. Then we shifted the time set to get a delayed time set. Finally, we extract two light curves from these two time sets. These two light curves were used to calculate the cross-spectrum to obtain the time lag. The result is similar to that of the cross-correction function. As shown in Fig. 9, after the LTR the time lag between two signals is not changed, despite the results become more diffuse. The time lag, in case that one signal responsed by LE while the other not, in the frequency domain is also similar to that in the time domain.