Extended Abstract
Introduction
Various religions, including Islam, Judaism, Hinduism, and Chinese, have utilized lunar calendars for chronology. Methods for forecasting the first sighting of the new lunar crescent existed as early as the Babylonians, and maybe earlier. The Babylonians reasoned that the lunar crescent can be seen with the naked eye under two conditions at sunset. First, the moon is older than 24 hours, and the moon's lag time is greater than 48 minutes. Fotheringham and Maunder developed standards for the seeing of the crescent moon at the beginning of the nineteenth century, and Bruin used his own criteria in 1977. Schaefer recently addressed crescent visibility extensively and integrated weather conditions into his work. Yallop then utilized the same database that Shaffer developed in 1997, but he overhauled some of the observation records extensively. Furthermore, many Muslim astronomers had developed their own criteria and published them in their literature. Despite the fact that different study organizations have created different criteria, there are still mistakes in the best time to forecast the crescent moon sighting. The use of old and conventional observations in modeling is one of these limitations, as is the use of non-uniform and heterogeneous observations. The Yallop criterion, for example, forecasts the visibility of the crescent moon for older crescents pessimistically. The Odeh criterion, on the other hand, forecasts young crescents with optimism. New Iranian criteria, such as the phase and altitude criteria (Mirsaeed criterion) and the triangular model (Iran criterion), have been presented in Iran. The goal of these criteria is to find the best timing between sunset and the first sighting of the crescent moon. Bruin, Schaefer, and Yallop have spent the last four decades developing the notion of the best moment. Because, after sunset, the sky darkens and the conditions for seeing the narrow crescent improve, while the moon approaches the horizon and the conditions for viewing the crescent moon worsen. Because the thickness of the atmosphere along the horizon is 3.7 times more than that of the zenith, the moonlight travels a greater distance than it did just a few minutes before. As a result, the sky towards the horizon is red or orange, and the crescent is not visible in this part of the sky.
 
Material and Methods
The objective of this study is to verify the rate of sky darkening in various regions and its influence on modeling the crescent visibility parameters of the moon, as well as to identify the best time to find out. To that end, 268 observational reports gathered from different divisions of Iran during the previous 20 years (2000-2021) were used to model the lunar crescent sighting. The proposed models are based not only on an examination of 20-year data to provide all effective tidal frequencies of the moon (the minimum period of moon’s notation motion is 18.61 years), but also on the use of sky-changing parameters such as local darkening rate and local sun occultation epoch time, the effect of the moon's distance from Earth, and the altitude of the moon from the horizon. The darkening rate of the sky factor was confirmed using various parameters and variables such as each point's geodetic latitude. Furthermore, unlike prior studies, the proposed models are developed using categorized observational reports with the least amount of error and can forecast the crescent sighting time in the presence of the sun (daylight time). The statistical correlation between the waiting time of each observation and the effective parameters in the lunar crescent visibility was studied in the first step. Following that, the parameters with the highest correlation values were chosen as the key quantities for modeling. After that, 17 alternative mathematical models with 2, 3, 4, and 5 parameters were implemented and tested, and the coefficients of the final two models (two and five parameter models) were determined using the least squares method as the suggested models.
 
Results
As a simple model, the two-parameter model can forecast crescent visibility with an average root-mean-square error (RMSE) of 4.7 minutes. The five-parameter model, on the other hand, was a more full and accurate model than the prior model, which was tested in two separate situations. They were evaluated over data for perigee distances of moon orbit (less than 375 thousand km) and observations for apogee distances of moon orbit (distance more than 390 thousand km) in the first and second cases, respectively. The findings of the 5-parameter model revealed that the first and second forms of the model had an average RMSE of 3.6 and 4.0 minutes to forecast the best time to see the crescent moon with the naked eye, respectively.
 
Conclusion 
The results revealed that the best period to observe the crescent moon is from 32 minutes after sunset to 12 minutes earlier than sunset owing to the angular separation of the moon from the sun (10 to 20 degrees) and the difference in the altitude of the moon from the sun (5 to 20 degrees). When a result, as the local darkening epoch time increases, so does the waiting epoch time. In other words, the lunar crescent appears earlier in the northern part of Iran than in the southern half.