Numerical simulation of image formation in an optical device in the problem of space monitoring

Space monitoring systems are being developed for space object and debris observation. There are the following problems in this area: optimizing the operation of existing optoelectronic devices, locating new devices, analyzing the monitoring results of outer space. It is necessary to use modeling in order to solve these problems. In this regard, the authors aimed to investigate the process of image formation on a photodetector under specified observation conditions. A mathematical model of image formation was constructed and a software implementation of this model was developed. Synthetic images of the night sky and images of the trajectories of space objects under given conditions were formed. The obtained results were compared with real images from telescopes.


Introduction
Space monitoring is the systematic observation and tracking of objects in outer space.This includes both natural celestial bodies, such as planets, stars, and galaxies, as well as manmade objects like satellites and space debris.Space monitoring plays a crucial role in various fields, including astronomy, astrophysics, aerospace engineering, and national security.It allows scientists to gather data about the universe, study celestial phenomena, track space objects for collision avoidance, and ensure the safety and sustainability of space activities.The number of space objects and debris in near-Earth space is rapidly increasing therefore the tracking problem is becoming more and more important.It is necessary to evaluate their coordinates in order to maintain catalogs and analyze possible dangerous situations.Space monitoring systems consisting of optoelectronic devices (OED) are used for their continuous observation.Due to the rapid development of such systems the following problems appear in this area: • optimization of the workflow of existing optoelectronic devices; • optimal placement of new OED; • analysis of the space monitoring effectiveness; Modeling of the monitoring process allows to solve these problems without large material and time costs.This innovative process involves the use of computer models and advanced algorithms to recreate the vastness of outer space and simulate the behavior of celestial bodies and phenomena.By mimicking real-life conditions and interactions, space monitoring simulation provides a valuable tool for understanding and predicting the complexities of space missions and observations.One of the key aspects of the monitoring system modeling is image formation,

Input intensity field formation
The first step is to form the intensity input field that consists of space objects such as satellites, stars and debris.A list of all artificial objects to be observed is contained in an input catalog of the model.The trajectory of each object should be calculated before field formation.The following models are used in ballistic calculations: • the Earth gravitational potential: EGM2008 using interpolation and tides; • the dynamic model of solar radiation; • atmospheric resistance and solar pressure by midsection area; • atmosphere model: GOST R 25645.166-2004;• ephemerides GPL DE for third-body gravity; The calculations are made by Everhart integration method up to 15 order of accuracy with a modified integration error estimation method.The Tycho-2 catalog is used to obtain the list of stars observed and annual aberrations are considered to clarify the coordinates.Then objects that are visible through a telescope are determined considering input OED viewing angle.Intensities of visible objects should be calculated.Magnitudes of potentially observable stars can be obtained from catalogue.The apparent magnitude m can be converted into intensity with the Pogson equation: where I 0 and m 0 -intensity and magnitude of the reference object (e.g.Sun).
To simulate the propagation of light reflected from a space object, the latter is assumed by the diffuse reflective sphere of radius R. In this case, the apparent intensity of the space object can be calculated by the formula: where I Sun -Sun reference intensity, JP (ξ) -diffuse reflection function, ξ -the angle between observer direction and direction to the Sun from the object, ⃗ r obs -observer position, ⃗ r -object position.
Environmental illuminances also strongly influence the input intensity field, therefore, sky, Sun and Moon illuminance models are considered [1].

Field transformation through the atmosphere
Light beams are refracted by the atmosphere and therefore the object visible coordinates should be offset by a refraction angle ε.Let h real be the height of radiating object above the horizon in degrees.Then where ε is measured in arc minutes [2].
Once the input intensity field is obtained, one need to construct a mathematical model of field transformation.When the light passes through the optical system, the intensity field is transformed as follows [3] where I(η x , η y ), I ′ (η ′ x , η ′ y ) -input and output intensities, respectively, h(η x , η y ) -point spread function (PSF) (point source image).Thus, the PSF is the main characteristic of an optical system.In the problem there are the following field transformations: through the Earth atmosphere and through the optical system of OED.The empirical model -the Gaussian blur function -is used for atmosphere PSF [3]: where r is the angular radius, d is the size of the scattering spot in angular units, the average value of which is 5".The integral (4) has the form of convolution and hence can be calculated using Fourier transform:

Field transformation through the optical system
In order to find the PSF of an optical system it is necessary to obtain a source point image.Since space objects observed are far from the pupil the wavefront firstly transformed according to Fresnel integral in the far (Fraunhofer) zone.This transformation turns into Fourier transform in the canonical coordinates (canonical coordinates are described in Appendix A): The input wavefront U (ρ x , ρ y ) when passing through the optical system is transformed as follows: where U ′ (ρ ′ x , ρ ′ y ) -output wavefront, function τ (ρ x , ρ y ) determines light energy transmission, δ(ρ x , ρ y ) -aperture function that limits the scope.The aperture function equals to circle delta function in canonical coordinates: δ(ρ 2 x + ρ 2 y < 1).The change of field phase is determined by the function of wave aberration W (ρ x , ρ y ).The wave aberration is usually represented as a decomposition over Zernike polynomials in polar coordinates: where R m n (ρ) -radial Zernike polynomials.Such representation is convenient for basic types of aberration: spherical, astigmatism, coma.
Diffraction field transform to the plane image after optical system can be described as follows: The wavefront of point source is delta function: U (η x , η y ) = δ(η x , η y ).Hence where -pupillary function of the optical system and h(ρ ′ x , ρ ′ y ) -PSF of the optical system.

Image generation on a photodetector
The number of photons should be calculated for each pixel of a matrix detector from the output intensity field in order to generate a matrix image.The conversion from intensity to the number of photons can be carried out according to the following formula: where I is the pixel radiation intensity, t exp is the exposure time, S is the pixel area, λ is the wavelength of the radiation.
Noise is a significant factor the number of photons in a pixel depends on.Two kinds of noise are considered: thermal and readout.The former is generated by the thermal agitation of the charge, the latter is connected with the image readout process.Both types can be simulated by a normal distribution and the result noise can be calculated as follows: where N therm , N read -variances of thermal and readout noise, respectively, N noise -a random variable with normal distribution, the zero expectation and variance of N therm + N read

Table 1 .
Space image generation is a complex process consisting of 4 parts.Every part has specified input data and physical models used.A brief description is provided in the table1The process of space image generation