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OneWeb Satellite Brightness -- Characterized From 80,000 Visible Light Magnitudes

The mean apparent magnitude and the mean of magnitudes adjusted to a standard distance are reported. The illumination phase function for OneWeb satellites is determined and it differs strongly from that of VisorSat spacecraft. Brightness flares are characterized and the mean rate of magnitude variation during a pass is determined. Tools for planning observations that minimize interference from bright satellites are illustrated and discussed.


Introduction
The many bright satellites in low-earth-orbit are interfering with observational astronomy (Walker and Benvenuti 2022, Hall et al. 2021, Walker et al. 2020a, Walker et al. 2020b, Tyson et al. 2020, Otarola et al. 2020, Gallozzi et al. 2020, Hainaut and Williams 2020, McDowell 2020, Williams et al. 2021aand Williams et al. 2021b. To address this issue the International Astronomical Union (IAU 2022) recently announced the creation of its Centre for the Protection of Dark and Quiet Sky from Satellite Constellation Interference. An important aspect of such protection is characterization of the brightness of these spacecraft. Brightness information can enable researchers to plan their observations with reduced interference from satellites. This paper characterizes the magnitudes of the OneWeb constellation based on an analysis of 80,000 observations acquired by the MMT-9 observatory.
Section 2 describes the hardware, the observations and the database of the MMT-9 robotic observatory.
Section 3 reports the mean of OneWeb apparent magnitudes and their dispersion, as well as the corresponding values after adjustment to a standard distance of 1,000 km. Section 4 characterizes the magnitudes by fitting them to a Phase Function (PF) which reduces the uncertainty.
Section 5 describes the application of a physical model called the OneWeb Brightness Function (OBF).
The section begins by defining a satellite-centered coordinate system (SCCS) which corresponds to the axes of a OneWeb spacecraft. The SCCS is the frame of reference for the OBF. Then the computation of data needed for the OBF is addressed. The required data includes the positions of the Sun and the MMT-9 in the SCCS system. Next, evaluation of the parameter coefficients for the OBF is described. Finally, the ability of the OBF to improve magnitude predictions by reducing their uncertainties is illustrated.
Section 6 discusses the prediction of satellite magnitudes and illustrates the brightness distribution of OneWeb satellites across the sky. Section 7 characterizes OneWeb flares which are short duration brightness surges and addresses the rate of time varying brightness changes normally seen during a satellite pass. Section 8 lists the limitations of this study. Section 9 discusses the research reported here in the context of other similar studies. The conclusions from this analysis are summarized in Section 10.
Throughout the paper, comparisons are drawn between the brightness characteristics of OneWeb satellites and those of the large constellation of Starlink VisorSat spacecraft being launched by SpaceX.
The study of Starlink satellites (Mallama 2021) being used for comparison analyzed more than 100,000 MMT-9 magnitudes and is referred to herein as 'the Starlink paper'.

MMT-9 hardware and observations
Mini-MegaTORTORA (MMT-9) is located at coordinates 43.650 N and 41.431 E (Beskin et al. 2017). This automated observatory includes nine 71 mm diameter f/1.2 lenses and 2160 x 2560 sCMOS sensors. The detectors record the visible spectrum from blue through red. S. Karpov (private communication) derived a color transformation formula which indicates that MMT-9 magnitudes taken through the clear filter are within 0.1 magnitude of the Johnson-Cousins V band-pass for objects with solar-like color indices.
Equations to transform from MMT-9 magnitudes to V-band are provided in the Appendix to this paper.
The MMT-9 database (Karpov et al. 2015) contains photometry of satellite passes that are collected into track files. OneWeb track files containing over 80,000 magnitude records were downloaded for this study. The MMT-9 observations were recorded between February and December 2021 when satellites were more than 20 o above the horizon. The distribution of satellite azimuths relative to that of the Sun was reasonably well distributed as shown in Figure 1.

Magnitude statistics
Celestial observations are impacted directly by the apparent magnitudes of satellites, that is, the values of brightness as recorded by a sensor and corrected only for atmospheric extinction. In addition to the apparent magnitude, the magnitude at a standard distance is used in satellite data analysis. This value is computed by applying the inverse-square law of light to the apparent magnitude.
A distance of 1,000 km is usually chosen as the standard because it is convenient for comparisons between multiple satellite constellations at different altitudes above and below that height. The mean apparent brightness of OneWeb satellites is much less than that of VisorSats for which the corresponding average magnitude is 6.43. Meanwhile, at a common distance of 1000 km OneWeb spacecraft are slightly brighter than VisorSats which average 7.21. The VisorSat values are from the Starlink paper.

Phase function analysis
The previous section reported that standard deviations of OneWeb magnitudes around their mean values are quite large. Hence, these means are of limited use to astronomers who factor satellite brightness into their observing plans. Phase function (PF) analysis offers a potentially more powerful method for quantifying and predicting magnitudes as described below.
Information about satellite illumination based on the geometry of the spacecraft, the Sun and the observer (or sensor) is captured by the phase angle. That quantity is the arc length between the Sun and the observer as measured at the spacecraft. For phase angle zero the Sun and observer are aligned as seen from the satellite, and at 180 o they are in opposite directions.
Satellite brightness computations sometimes use a standard PF which corresponds to the fraction of a spherical object illuminated as a function of phase angle. This function is brightest at small phase angles when the side of the satellite facing the observer is mostly lit by the Sun and, conversely, it is faintest when the spacecraft is back-lit.
An improvement over the standard PF is an empirical PF which is derived by least-squares fitting of the observed magnitudes at a standard distance to their phase angles. Linear and quadratic equations are commonly used for fitting satellite magnitudes.
The empirical phase function for OneWeb satellites plotted in Figure 2 shows that the spacecraft become fainter with increase phase angle as expected. The linear and quadratic fits are almost identical.
The RMS of residuals around the mean linear and quadratic fit lines only reduce to 0.63 in each case, which compares to 0.66 with no fitting as reported in the previous section. After instrumental scatter is removed the RMS is 0.59. Table 3    While empirical PF analysis does not greatly improve the accuracy of magnitude prediction for OneWeb satellites it is useful to know that their behavior is much different from the standard PF and from the PF for VisorSats. This comparison is shown in Figure 3. The highly curved PF for VisorSat indicates strong forward and backward reflection of sunlight. This behavior is not seen with OneWeb satellites. In fact, the directional response of OneWeb is even weaker than that indicated by the standard PF.

OneWeb Brightness Function
A physical model for satellite illumination was described in the Starlink paper and here it is applied to OneWeb spacecraft. In this section, a satellite-centered coordinate system (SCCS) corresponding to the general shape and orientation of a satellite is defined. The SCCS serves as the frame of reference for the OneWeb Brightness Function (OBF). Then the computation of data needed for the OBF is detailed. This  Coefficients of the three OBF parameters (solar elevation, MMT-9 elevation and MMT-9 azimuth) were determined by fitting them to the 1,000-km magnitudes. The fitting process was iterated 100 times and the resulting coefficients are listed in Table 4. Subsections of the table correspond to 'all data' and to the first and second halves of 2021 taken separately. Values of the RMS between the observations and the OBF are also given in the Table. The RMS values during the second half of 2021 are smaller than those in the first half. When instrumental scatter is removed from the later data the RMS reduces to 0.44 which is less than that of the empirical PF.  The solar elevation function for OneWeb is plotted in Figure 5 which shows that brightness increases as elevation becomes more negative. This geometry corresponds to sunlight impacting the nadir-facing side of the spacecraft at a more perpendicular angle. Brightening can be expected in this case because the nadir side also faces the observer.
The MMT-9 azimuth function is illustrated in Figure 6. This relationship resembles that of the PF (see

Magnitude prediction
Astronomers can avoid the brightest satellites by selecting favorable sky areas and optimal times of night for acquiring observations. OneWeb PF and OBF models used to characterize satellite brightness can also predict 1000-km magnitudes. Those 1000-km magnitudes can, in turn, can be converted to apparent magnitudes by applying the inverse-square law of light. The inputs needed to compute model brightness are the azimuths and elevations of the Sun and of the spacecraft in the local horizon reference frame as well as the satellite distance.
Examples of brightness predictions for OneWeb satellites based on the PF and OBF are shown as all-sky maps in Figures 8 and 9 Model magnitude predictions are also useful for generating brightness statistics and understanding changes that occur with time of night. Figure 11 plots the percentage of sky occupied by OneWeb satellites brighter than three threshold magnitudes. There are separate sets of curves for the whole sky and for that part above 30 o elevation. The graph indicates that interference from OneWeb spacecraft brighter than magnitude 9 and above 30 o elevation peaks when the Sun is between 20 o and 30 o below the horizon. If the Sun is nearer the horizon than 20 o the satellites tend to be fainter and if the Sun is further from the horizon than 30 o many of the spacecraft are eclipsed by the Earth's shadow. Figure 11. The percentage of sky area with OneWeb magnitudes brighter than magnitudes 8, 9 and 10, is plotted as a function of solar elevation. The solid lines are for the sky above 30 o elevation while the dashed lines correspond to the whole sky. The data were generated with the OBF model.
The high 1,200 km altitude of OneWeb satellites makes them fainter the VisorSats at 550 km, which is more favorable for astronomical observers. However, the lower VisorSat altitudes have the advantage that they are eclipsed more than OneWeb spacecraft. Figures 12 and 13 illustrate the different sizes of the eclipse regions when the Sun is at -30 o elevation. Figures 12 and 13. The eclipse region for VisorSats (right) includes nearly the whole sky when the Sun is 30 o below the horizon. However, that region covers less than half of the sky for OneWeb satellites (left) at the same solar elevation.

Brightness changes on short time scales
The intensity of trails from OneWeb satellites on individual astronomical images will change with time and with position on the sensor. Statistics of these variations may be useful in developing algorithms to reduce the impact of trails and to minimize the loss of science data. So, this section describes magnitude changes that occur on time-scales from seconds to tens of seconds.
The light curve graphs used for the analysis in this paper were inspected by eye for sudden brightness enhancements, called flares. Figure 14 shows an example of one of these surges. Flares are probably due to specular reflection of sunlight from polished surfaces on the satellites. The mean interval between flares that exceeded amplitude 0.5 in apparent magnitude was 721 seconds. This is much less frequent than VisorSat flares for which the mean interval is 129 seconds according to the Starlink paper. The percentage of the elapsed time in all the light curves spent in flares above 0.5 magnitudes was 0.9% for OneWeb as compared to 3.9% for VisorSat.
More gradual brightness variations during a pass were characterized by taking the means of apparent magnitudes in 10 second time intervals. The averaging was performed to reduce instrumental scatter.
Mean magnitudes in the adjacent time intervals were then differenced, and that result was divided by the time interval. The absolute value of the quotient was taken to be the rate of change per second.
Light curves containing flares were not used in this analysis.
The 10-second averages were also used to determine rates of magnitude change per degree of arc distance that the satellite traveled. This rate was determined by factoring in the satellite's azimuth and elevation at the mid-point of each interval.
The mean rates of OneWeb magnitude variation are given in Table 5  which indicates that the rates are highly variable. Nevertheless, the standard deviations of the means are very small, so the mean values are reliable. The Table also lists the corresponding statistics for VisorSat, taken from the Starlink paper. The VisorSat rate of change per second is nearly twice as large as that for OneWeb but the rate of change per degree is about that same size.

Limitations of this study
Several factors limit the general applicability and the accuracy of the results from this study. To begin with, the physical modeling of the spacecraft shape is rudimentary and there is no accounting for the light scattering properties of the surfaces.
Additionally, the satellite bodies and solar arrays are only nominally aligned with the SCCS frame of reference, while their actual attitude (yaw, pitch and roll) at the time of each observation is unknown.
Thus the SCCS angles may not be exactly aligned with the OBF parameters corresponding to the physical spacecraft. As a result, the SCCS and OBF do not constitute a formal bi-directional reflectance model.
A further limitation is that the data are all from one location. If spacecraft attitude depends upon geographic latitude and longitude then the magnitudes acquired at the MMT-9 site may not be representative of other regions.
Finally, the observations only span 11 months and there is some evidence for brightness change during that interval. Thus, the observed magnitude of OneWeb satellites could evolve in the future.

Discussion
This section puts the analysis reported here into the context of other photometric research involving OneWeb satellites. The first such study (Mallama, 2020) was a small-scale investigation of 639 MMT-9 magnitudes that were acquired shortly after the first satellites reached their operational altitude. The mean 1000-km magnitude from that examination was 7.18 as compared to 7.05 reported here. That paper did not report statistics pertaining to the satellite phase function, flaring frequency or rates of brightness change. for the dim magnitudes resulting from their all-sky observing strategy. While their paper does not report an average satellite elevation, the geometry of a hemisphere gives some indication. Half the surface area of that geometric shape is above 30 o and the other half is below. This suggests a value for the mean elevation of an all-sky survey. By contrast, the mean satellite elevation for the photometry reported in this paper is considerably higher at 57 o .

Conclusions
More than 80,000 OneWeb magnitudes recorded by the MMT-9 robotic observatory were analyzed.
These visible light values represent brightness from the red part of the spectrum through the blue, and they are within about 0.1 magnitude of the V-band.
The mean apparent magnitude for OneWeb satellites is 7.85 with a standard deviation of 0.68 after the estimated instrumental scatter is removed. The mean of magnitudes adjusted to a range of 1,000 km is 7.05 with a standard deviation of 0.62 when adjusted for the scatter. The formal uncertainties of these means are less than 0.01 magnitudes. There is some evidence that the satellites became fainter during the second half of year 2021.
The phase function (PF) for OneWeb satellites indicates that brightness is only weakly related to this illumination angle. The RMS residuals for a linear fit and a quadratic fit are the same. The RMS to the PF for all of 2021 with instrumental scatter removed is 0.59 and for the latter half of the year alone it is 0.49.
A OneWeb Brightness Function (OBF) was tailored to the shape and orientation of the satellites. The