Determination of the Size of an Astronomical Object Using Photon Counting Mode ^†

Abstract

A methodology for radiometric determination of the size of an astronomical object using photon counting mode is proposed in this study. An algorithm for applying the methodology is developed. The measurement accuracy and value of the characteristic coefficient of the photon counting mode are investigated. A numerical determination of the size of an astronomical object is developed with simulated experimental data and example measurement system parameters.

1. Introduction

The intense self-radiation of media and objects in the optical wavelength range [1,2,3] and the effective detection of weak optical signals [4,5] determine the application of optical radiometry in astronomical research [5,6,7].

Systems for astronomy and Free Space Optical Communication are characterized by large distances between the corresponding points, with high diffraction scattering of optical energy and energy losses due to the extinction of radiation in the atmospheric part of the transmission medium. This implies the registration of weak optical signals in the receiving parts of the systems. There is no current in the output circuits of the optical receivers. In these cases, the reception of the signals is carried out in the photon counting mode (PCM).

The nature of the PCM is that each of the values of the received optical flux is evaluated by the number of photons falling on the photocathode of the photoelectronic multiplier in the photon counting system (PCS) n for a time interval $t_r$ (time for one registration) [8].

Each value of the received signal at the output of the photoelectronic multiplier in (PCS) is formed by estimating the random number of single-electron pulses (SEPs) m that appear in a sufficiently long time interval $t_r$, where m is a Poisson random quantity. The duration $t_1$ of these single-electron pulses is less than the average $\mathsf{\Delta }t$ of the time interval between two contiguous ones.

Regarding PCM, it is necessary to note the important role of the inertia of the actual PCS. There are two types of inertia—the inertia from the first genus, characterized by non-prolonging recovery time, and the inertia from the second genus, characterized by prolonging recovery time [8].

The aim of this paper is to develop a methodology for radiometric determination of the size of space objects using photon counting mode (PCM) and an algorithm for its implementation. The measurement accuracy is investigated. The corresponding dependencies of the characteristic coefficient of photon counting mode (time axis duty cycle q) are analyzed. A numerical realization of the proposed algorithm, utilizing simulated experimental data and example measurement system parameters, has been presented. The algorithm is derived to measure the radius R of star A.

The black body is a model for a body that completely absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence, and does not reflect any of it. It can emit waves of any wavelength and visually has color. The radiation spectrum depends solely on its temperature. The sun, like other stars, is the best approximation for absolutely black bodies.

Planck’s Law of Radiation describes the spectral emittance $M_{\lambda }\left(\lambda ,T\right)$ of the black body into the half-space as a function of its temperature T and the observed wavelength $\lambda$.

2. Materials and Methods

2.1. Methodology Algorithm

1.

Using spectral radiometric measurement, we found the optical wavelength ${\lambda }_m$ for which the spectral emittance $M_{\lambda }\left(\lambda ,T\right)$ of the star’s own radiation has a maximum value. $T_A$ is the temperature of the star. Using the following Equation [1]

$${\left.{\displaystyle \frac{\partial M_{\lambda }\left(\lambda ,T_A\right)}{\partial \lambda }}\right|}_{\lambda ={\lambda }_m}=0$$

we obtain

$$T_A\left[K\right]={\displaystyle \frac{2898}{{\lambda }_m\left[\mu m\right]}}.$$

(1)

2.

Using an interference filter with the bandwidth $\mathsf{\Delta }\lambda$ in the surrounding area of the appropriate chosen wavelength ${\lambda }_0$, we obtain an experimental realization of the random number m of standard pulses that appear at the output of the photo counting system (PCS) during the registration time $t_r$. We assume m as an estimation of $〈m〉$. Because of PCS inertia, the resulting realization of  is less than the corresponding realization of the random number n. If the inertia of the PCS is characterized by a non-prolonging recovery time, we obtain [8]

$$〈m〉=〈n〉/\left(1+{\displaystyle \frac{t_s}{t_r}}〈n〉\right) , {\sigma }_m^2=〈m〉\left(1-{\displaystyle \frac{t_s}{t_r}}〈m〉\right)$$

(2)

where $t_s$ is the split time of the PCS.

3.

For $T=T_A$ and $\lambda ={\lambda }_0$, Plank’s law is

$$M_{\lambda }\left({\lambda }_0,T_A\right)\left[{\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2\mathsf{\mu }\mathrm{m}}}\right]=c_1{\left({\lambda }_0\left[\mathsf{\mu }\mathrm{m}\right]\right)}^{-5}{\left[exp\left({\displaystyle \frac{c_2}{{\lambda }_0\left[\mathsf{\mu }\mathrm{m}\right]T_A}}\right)-1\right]}^{-1},$$

(3)

where $c_1=3.74 ·{10}^8 \mathrm{W}{\mathsf{\mu }\mathrm{m}}^2/{\mathrm{m}}^2$ and $c_2=1.44 ·{10}^4 \mathsf{\mu }\mathrm{m}.\mathrm{K}$.

Following (3), we define the corresponding interference filter bandwidth Δλ spectral exitance.

$$\mathrm{M}\left({\lambda }_0,T_A,\Delta \lambda \right)=M_{\lambda }\left({\lambda }_0,T_A\right)\Delta \lambda .$$

(4)

Based on the energy budget of the entire optical radiometric system [7], we obtain

$$I={\displaystyle \frac{hc〈n〉}{\eta A{t}_r{\lambda }_0}}={\displaystyle \frac{hc〈k〉}{\eta A{\lambda }_0}}$$

(5)

where $I\left[\mathrm{W}/{\mathrm{m}}^2\right]$ is the intensity of star radiation in the aperture A of the receiving antenna, $\eta$ is the quantum efficiency of the photo multiplying tube photocathode, and $k=〈n〉/t_r$ is the single-electron pulse intensity at the output of the photo multiplying tube photocathode $h=6.626 · {10}^{-34} \mathrm{J}\mathrm{s}$, $c=3·{10}^8 \mathrm{m}/\mathrm{s}$.

Based on (2), for $〈n〉$ we write

$$〈n〉=〈m〉/\left(1-{\displaystyle \frac{t_s}{t_r}}〈m〉\right).$$

(6)

For the next stage of the analysis, it is convenient to introduce the connection between I and $\mathrm{M}\left({\lambda }_0,T_A,\mathsf{\Delta }\lambda \right)$. We use the correct relation for R << r (r is the distance between the sun and the star):

$$I=FL{\sigma }_{\alpha }/r^2,$$

(7)

where

$$L=M/\pi$$

(8)

is the radiance of star radiation, and

$$F=\pi R^2$$

(9)

is the area of the “visible” circular disk of the star with a diameter of 2R, and ${\tau }_{\alpha }$ is the transparency of the Earth’s atmosphere in the vertical direction.

Following Equations (5)–(9) for the star radius R, we obtain

$$R=\sqrt{hc〈k〉/\left(\eta A{\lambda }_{0}M\left({\lambda }_0,T_A,\mathsf{\Delta }\lambda \right){\sigma }_{\alpha }\right)}$$

(10)

2.2. Measurement Accuracy: Characteristic Coefficient of Photon Counting Mode

Since the value of $〈m〉$ is estimated with the experimentally obtained realization of m and is random, the value of $〈n〉$ in (6) is also random, i.e.,

$$n=m/\left(1-{\displaystyle \frac{t_s}{t_r}}m\right).$$

Consequently, when $k=〈n〉/t_r$, the random estimate is assumed to be

$$k=n/t_r.$$

(11)

Therefore, the result for R in (10) contains a random error and needs to be written as

$$R=\rho \sqrt{k}$$

(12)

where

$$\rho =r\sqrt{hc/\left(\eta A{\lambda }_{0}M{\sigma }_{\alpha }\right)}.$$

(13)

The relationship can be utilized as an estimate of the accuracy of the result for R, specifically about the overall outcome of the radiometric measurement:

$$SNR={\displaystyle \frac{〈R〉}{{\sigma }_R}},$$

(14)

where $〈R〉$ is estimated by R (10).

In order to define ${\sigma }_R$, using (2), we obtain

$${\sigma }_n=\sqrt{〈m〉}/\left(1-{\displaystyle \frac{t_s}{t_r}}〈m〉\right).$$

(15)

Following (11) and (15), we obtain

$${\sigma }_k={\displaystyle \frac{1}{t_r}}{\sigma }_n={\displaystyle \frac{1}{t_r}}\sqrt{〈m〉}/\left(1-{\displaystyle \frac{t_s}{t_r}}〈m〉\right).$$

(16)

Linearizing (12), we write

$${\sigma }_R={\left.{\sigma }_{Rk}{\displaystyle \frac{\partial R}{\partial k}}\right|}_{k=〈k〉}$$

and using (16), (11), we obtain

$${\sigma }_R={\displaystyle \frac{\rho }{2}}{\displaystyle \frac{{\sigma }_k}{\sqrt{〈k〉}}}=\rho {\left(2\sqrt{t_r}\sqrt{1-{\displaystyle \frac{t_s}{t_r}}〈m〉}\right)}^{-1}$$

(17)

Finally, for SNR, we obtain:

$$SNR=2\sqrt{{\displaystyle \frac{〈k〉t_r}{1+〈k〉t_s}}}.$$

(18)

In accordance with our analysis, we introduced the characteristic coefficient of photon counting mode, defined as time axis duty cycle q [8]:

$$q={\displaystyle \frac{t_1}{\mathsf{\Delta }\mathrm{t}}}={\displaystyle \frac{t_1}{t_r}}〈m〉={\displaystyle \frac{〈k〉t_1}{1+〈k〉t_s}},$$

(19)

where $t_1$ is the duration of the single-electron pulses.

We can assume that PCM is available when q < 0.5.

3. Results

In accordance with the aim of the present work, we apply the presented algorithm, assuming the following input data: $r=10$ parsecs (1 parsec is equal to about 3.26 light years or $3.084 . {10}^{16}$ m); ${\lambda }_0$=0.4 $\mathsf{\mu }\mathrm{m}$; $\mathsf{\Delta }\lambda =5 \mathrm{n}\mathrm{m}$; $\eta =0.3$; $A=1{ \mathrm{m}}^2$; $t_r=0.1s$; $t_s=15 \mathrm{n}\mathrm{s}$; $t_1=5 \mathrm{n}\mathrm{s}$.

We assume that as a result of the spectral radiometric measurement, we have obtained the maximum spectral density of the radiant exitance for ${\lambda }_m=0.25 \mathsf{\mu }\mathrm{m}$. Using (1), we find $T_A=\mathrm{11,592} \mathrm{K}$. For comparison, the temperature of Vega (α Lyrae) is on the order of ${10}^4\mathrm{K}$. Vega shines with white light and is located 8.13 parsecs (26.5 light years) away from the sun.

For $\mathsf{\Delta }\lambda =5 nm$ in the vicinity of ${\lambda }_0=0.4 \mathsf{\mu }\mathrm{m}$ and for $t_r=0.1 \mathrm{s}$, we register one realization of m. We consider the measurement result $m=2 . {10}^6$ and assume $m=〈m〉$. This value corresponds to a surface irradiance flux density $I=47.3 \mathrm{p}\mathrm{W}/{\mathrm{m}}^2$ in the aperture of the receiving antenna.

Using (3) with ${\lambda }_0=0.4 \mathsf{\mu }\mathrm{m}$ and $T_A=\mathrm{11,592}\mathrm{K}$, we calculate

$$M_{\lambda }\left({\lambda }_0,T_A\right)=1.713 \mathrm{G}\mathrm{W}/\left({\mathrm{m}}^2\mathsf{\mu }\mathrm{m}\right).$$

After substituting the result in (4) and taking into account $\mathsf{\Delta }\lambda =5 \mathrm{n}\mathrm{m}$;, we find

$$M_{\lambda }\left({\lambda }_0,T_A,\mathsf{\Delta }\lambda \right)=8.565 \mathrm{M}\mathrm{W}/{\mathrm{m}}^2.$$

Using $m=〈m〉=2 · {10}^6$, $t_r=0.1s$ and $t_s=15 \mathrm{n}\mathrm{s}$ in (6), we calculate $〈n〉=n=2.857 · {10}^6$, i.e., $〈k〉$=k=2.857$·{10}^7 {\mathrm{s}}^{-1}$.

A substitution of ${\tau }_{\alpha }=0.676$ [7] in (10) leads to

$$R=0.882 · {10}^9\mathrm{m}\mathrm{or} R=1.267.R_{SUN}$$

where $R_{SUN}$ is the radius of the SUN.

Fixing $\mathsf{\Delta }t=t_r/〈m〉=50ns$ and $q=t_1/\mathsf{\Delta }t=0.16$, we obtain q < 0.5, which confirms the presence of PCM.

We continue our research on the accuracy of radiometric measurement of R (i.e., SNR) and the characteristic coefficient of photon counting mode q.

Based on relations (18) and (19), we obtain functional dependencies on their defining parameters—$〈k〉$, $t_r$, $t_s$ and $t_1$.

The resulting dependence of SNR on $〈k〉$ for different values of $t_r$ and $t_s$ is graphically presented in Figure 1.

Figure 2 shows the dependence of q on $〈k〉$ for $t_s=10 \mathrm{n}\mathrm{s}$, 40 ns, 70 ns, 100 ns and ${\mathrm{t}}_{\mathrm{s}}=15\mathrm{n}\mathrm{s}$.

4. Discussion

Figure 1 shows that SNR increases with the increase in $〈k〉$ and $t_r$. The inertia of the PCS causes a “saturation” of SNR when $〈k〉$ increasing for a specific value of $t_r$. When $〈k〉\to \infty$, Equations (18) and (2) with (11) lead to $SNR\to 2\sqrt{t_r/t_s}$ and $m\to 2\sqrt{t_r/t_s}$. The transition reflects the fact that in each interval $t_s$ at the output of the PCS, exactly one standard pulse appears, which has a clear physical meaning $q\to t_1/t_s$.

Figure 2 depicts that q < 0.5 is fulfilled even though it has weak inertia. Therefore, PCM is available for $t_1\le 5 \mathrm{n}\mathrm{s}$, $t_s\le 100 \mathrm{n}\mathrm{s}$, $〈k〉\le {10}^{10} {\mathrm{s}}^{-1}$, and the performed analysis is valid in the indicated intervals of values.

The proposed method is applicable for radiometric determination of the radius of an astronomical object (a star), provided that its distance has been determined using the annual parallax method.