Visibility Predictions for Near-future Satellite Megaconstellations: Latitudes near 50° Will Experience the Worst Light Pollution

Now that we have a distribution of satellites in space, we need to determine the number of those satellites that are sunlit for a given observer at a given solar hour and time of year. To proceed, we use an initial coordinate system that places Earth's equator in the X–Y plane and the spin direction aligned in the Z direction. The satellites' Cartesian positions are then determined in this coordinate system from their orbital elements.

The observer's Cartesian coordinates on the Earth's surface are determined next, based on the desired latitude and solar hour of interest. We set midnight for an observer to be when they are on the +X-axis.

With the position of the observer now set relative to the satellites, we rotate all satellites and the observer about the Y axis by an angle consistent with the Sun's declination for the desired time of year, creating a new coordinate system ${\rm{X}}^{\prime} ,\,{\rm{Y}}={\rm{Y}}^{\prime} ,\,{\rm{Z}}^{\prime}$. The Sun is on the $-{\rm{X}}^{\prime}$ axis. To simplify the calculations, we take a satellite to be sunlit if its ${\rm{Y}}^{\prime} -{\rm{Z}}^{\prime}$ projection is greater than the radius of the Earth or if it is in the domain ${\rm{X}}^{\prime} \lt 0$. In other words, we do not include atmospheric refraction (if it was included, it would cause a slightly larger number of satellites to be partially sunlit on the edge of Earth's shadow).

Finally, we determine whether a given sunlit satellite is visible to the observer in question and, if so, make cuts based on the satellite's elevation for that observer. We proceed as follows: let the vector to the observer be denoted r _obs and the vector to a given satellite be r _sat, each with an origin at the geocentre. The vector from the observer to the satellite is thus s = r _sat − r _obs. The resulting zenith angle of the satellite for the observer is $\cos {\theta }_{z}={\boldsymbol{s}}\cdot {{\boldsymbol{r}}}_{\mathrm{sat}}/\left(| s| | {{\boldsymbol{r}}}_{\mathrm{sat}}| \right)$.

Figure 2 shows the on-sky density of sunlit satellites at different latitudes, different times of night, and different times of year. Many of the hundreds or even thousands of sunlit satellites in Figure 2 are close to the horizon, at higher airmasses than would be used for research observing. But there are still hundreds of sunlit satellites more than 30° above the horizon (Figure 3), where they could interfere with research astronomy.

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In the right panels of Figures 2 and 3, we highlight the number of sunlit satellites over the course of a night from four different observatory latitudes: observatories in Canada, which are all situated close to 50°N, observatories on the Canary Islands at 28°N, observatories in Hawaii at 20°N, and observatories in Chile which are all close to 30°S.

The first feature that is immediately obvious is that latitudes near 40°N and S have the most sunlit satellites, consistent with that found by, e.g., McDowell (2020). This is not surprising given the distribution of satellites in Figure 1. But due to Earth's shadow, there is a period of few-to-no sunlit satellites for ∼5–6 hr around midnight during the winter at these latitudes. However, during summer, there are more than a thousand sunlit satellites above the horizon all night long (see 50°N curves and yellow-filled contours in Figure 2, which represents ∼3000 sunlit satellites). Closer to the equator, where many research observatories are located, the seasonal variation is not so extreme, and there is a period of about 3 hr in the winter and near the equinoxes with few-to-no sunlit satellites. But there are still hundreds of sunlit satellites all night at these latitudes in each hemisphere's summer.

The distribution of sunlit satellites more than 30° above the horizon is similar (Figure 3). Mid-latitudes have a ∼5–6 hr period of no satellites centered around midnight in their winter and equinoxes. But in the summer, even from these mid-latitudes, there are dozens of sunlit satellites >30° above the horizon. This is more extreme at latitudes near 50°: there is no drop in the number of sunlit satellites >30° above the horizon, even at midnight, at 50°N or S latitude in their respective summers.

A satellite's brightness depends on its non-trivial bidirectional reflectance distribution function (BRDF), which takes into account the detailed shape of the satellite and its materials. The brightness at any given moment may depend sensitively on the satellite's exact orientation in space with respect to the incoming light and the observer. Satellites are known to vary in brightness as they move across the sky, continuously changing their orientation with respect to the observer. This evolving perspective can lead to sudden, and in the case of flares, dramatic changes in brightness. This brightness variability can be observed, in principle, even for a single satellite traversing the field of view of a telescope.

Satellite variability and flares have long been observed by astronomers, both by purposefully observing satellites and by the accidental passage of satellites through research images (e.g., Hainaut & Williams 2020; Mallama 2021a); naked-eye observations by many stargazers around the world also note this behavior. ⁶

In practice, the detailed BRDF is typically unknown and simplifying assumptions must be employed. Such assumptions will not capture sudden satellite variability, but can still be useful enough to simulate the overall trends and distributions of satellite brightness.

We use a diffuse Lambertian sphere reflection model (e.g., Pradhan et al. 2019) to estimate the distribution of visible magnitudes of these sunlit satellites:

$$\begin{eqnarray}&&\begin{array}{l}{m}_{v}={m}_{\mathrm{Sun}}-2.5\mathrm{log}\\ \quad \times \,\left(\displaystyle \frac{2}{3{\pi }^{2}}A\rho [(\pi -\phi )\cos \phi +\sin \phi ]\right)+5\mathrm{log}R\end{array}\end{eqnarray} \tag{ 1 }$$

where m_Sun is the apparent magnitude of the Sun (−26.77 in V-band, −26.47 in g'-band for AB magnitudes; Willmer 2018), A is the cross-sectional area of the satellite, ρ is the Bond albedo, ϕ is the solar phase angle between the observer and the satellite, and R is the distance between the observer and the satellite. We do not know the reflecting area or albedo of future satellites, and cross-sectional area will vary depending on viewing phase and satellite attitude. To further simplify our reflectance model, ρ and A are combined into one albedo-area variable, the effective area ζ = ρ A.

To calibrate our model, we compared our model satellite brightness distribution with the observed distribution of Starlink satellite magnitudes from Boley et al. (2021b), which used the Plaskett Telescope at the Dominion Astrophysical Observatory in Victoria, BC (latitude 48.5°N) to observe a set of Starlink satellites. While OneWeb satellites have so far proved fainter than Starlinks', they have approximately the same absolute magnitude (Mallama 2021a). Since Kuiper and Starnet/GW do not yet have any satellites in orbit, a set of Starlink observations is a reasonable starting point for our model calibration. We set up the model using the same general observing directional biases, matching the season, time of night of the observing run, and the latitude, and we only used Starlink satellite orbits near 550 km altitude, since that is the most prominent orbital shell occupied as of the time of observations (see Table 1). Such Starlink satellites have orbital inclinations of about 53°, meaning their orbits can pass through the sky at low zenith angles at the latitude of the Plaskett Telescope.

As discussed in Boley et al. (2021b), we first attempted to use only the Visorsat Starlinks in our calibration, in the hope that these brightness mitigations would be used on all future satellites. But upon further investigation, we found that although all Starlinks since the 2020 August 7 launch are reportedly equipped with visors, ⁷ there is no available information on how many of those satellites actually have the visors successfully deployed. The variation in brightness observed in Boley et al. (2021b) leads us to speculate that there may be inconsistencies among the visors, or that their effectiveness is very sensitive to the precise orientation between the satellite, Sun, and observer. We note that the latter case is hinted at by observations in Mallama (2021b).

With the above in mind, we fit our model to the entire distribution of observed Starlink satellites, both listed Visorsats and original-design Starlinks, which is shown in Figure 4. This may be a more realistic distribution of satellite brightnesses, since it includes both satellites with mitigation (Visorsats) and those without (original Starlinks). We also note that initial observations of Visorsats and original Starlinks by amateur astronomers (e.g., McDowell 2020) showed a wide range of brightnesses for both satellite types. Without regulatory obligations to make their satellites fainter, or widely adopted and followed industry standards, there will likely be a large range of reflectivities in future satellites.

The modeled albedo-area variable was changed until the resulting model distribution of Starlink satellites matched the observed distribution from Boley et al. (2021b), under similar observing directional and time biases (Figure 4). We also added a small scatter in brightness (Gaussian noise with σ = 0.5 magnitudes) to simulate the scatter observed in Boley et al. (2021b). This best fit corresponds to a satellite effective area ζ ≈ 0.8 m², which could realistically be matched by a reflecting satellite area of 4 m² with an albedo of 0.2, as an example.

The Plaskett measurements compare favourably with measurements from Mallama (2021a), but taken from 48.5°N rather than 39°–44°N, where satellites will remain bright throughout the night for the time of year of the observations (Section 3.2). The median observed $g^{\prime}$-magnitude in the Boley et al. (2021b) sample, which includes Starlink satellites with and without visors, is $g^{\prime} =5.7$, compared with a median of 5.9 measured by Mallama (2021a), which used an observing filter within 0.1 mag of V-band. For the Sun, $g^{\prime} -V=0.3$, so V ≈ 5.9 corresponds to $g^{\prime} \approx 6.2$. However, the observational biases between the samples are expected to be different. The results presented by a follow-up study (Mallama 2021b) find a median V of about 6.8 for Visorsats, after they have been adjusted to a range of 1000 km. Correcting this to 550 km yields a 550 km "absolute" V-magnitude of H_V ≈ 5.5 or H_g ≈ 5.8. This compares favourably to the Visorsat-only subsample from Boley et al. (2021b) which measured H_g = 5.7 (coincidentally the same value as the apparent magnitudes of their full sample).

We reiterate that there are many reflection effects that we are ignoring in our satellite brightness model, and that the brightness distribution is sensitively dependent on effective cross-sectional area and exact observing geometry. But based on comparison between this model and published observations of one satcon (Starlink), we find that the overall scale of the satellite brightness is well-represented.

We now apply this model calibration to the satellite distributions to calculate their apparent magnitudes as seen in the sky at different latitudes, different times of year, and different times of night, with the caveats discussed above. We also include realistic atmospheric extinction in this model (Kasten & Young 1989), which is significant for the lowest-elevation satellites. We calculate the visual magnitudes for all sunlit satellites in each field of view, but we pay particular attention to three magnitude cuts: those with g < 5, which are visible from a typical suburban environment; those with g < 6.5, which are naked-eye visible from a truly dark sky; and those with g < 7, which are bright enough to cause serious data loss for LSST (Tyson et al. 2020) and other research facilities. ^{8 , 9}

Figures 5–8 show the calculated all-sky satellite distributions and brightnesses for a selection of seasons and times of night for different latitudes on Earth. ¹⁰ The results are further summarized in Table 2, where the number of satellites above each magnitude cut is compared to the number of visible stars in the observer's sky. At latitudes near 50°N and S, where the light pollution is the worst, up to 8% of all point sources in the sky are satellites at certain times of night. Because the satellite orbits are symmetric about Earth's equator, the simulations can be applied to either hemisphere for the corresponding season. For example, the all-sky view for Hawaii (20°N) on the June solstice is equally applicable to São Paolo, Brazil (20°S) on the December solstice, but the all-sky image must be flipped north–south. The number of sunlit and visible satellites is given for each sky-field. These numbers are slightly randomized due to the pseudo-random satellite distribution and the small random scatter added in to the satellite brightnesses.

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There are notable patterns in the on-sky satellite brightness distributions. As expected, satellites get fainter close to the horizon, and are brightest as they pass overhead—this is due to a combination of effects: the phase angle, satellite range (distance between the satellite and the observer), and the satellite's instantaneous airmass. The satellite will be closest to the observer when it passes through the zenith. As the satellite moves toward the Sun from the zenith, its range increases and its phase angle also increases, causing the satellite to become faint quickly under the diffuse sphere model, although the results of Mallama (2021b) show that this behavior can be complex. Moreover, as the satellite moves away from the zenith and away from the Sun, the range is increasing (causing dimming) while the phase angle is decreasing (causing brightening), two competing effects that make overall on-sky brightness patterns difficult to predict.

The shadow of the Earth is apparent in many of the all-sky images, and is what blocks the closest overhead satellites from being sunlit close to the equator. In latitudes closer to the poles, because of geometry, the Earth's shadow does not block the satellites near the zenith, resulting in bright satellites being visible throughout the night.

In the locations close to the equator (0^◦, Figure 7, and Hawaii at 20°N, Figure 5), only faint satellites are visible, primarily close to the horizon. From Hawaii, and to a greater extent from the Vera Rubin Telescope site in Chile (30°S), there are a handful of bright satellites crossing the zenith close to sunrise and sunset in the summer. This could restrict the portions of the sky that are available for scientific studies in the summertime, when the weather is generally better. This effect of satcons on research astronomy can also be seen in Table 2, where satellites make up only a tiny fraction of all points sources in the sky at midnight in the range of g-magnitudes tested at 20°–30° latitudes.

At higher latitudes of 50°N (Figure 6), where many Canadian and European observing facilities are located, as well as a large fraction of the population of Canada, Europe, and northern-tier US states, the problem of light pollution from satellites is much more severe seasonally (and is equally severe at 50°S). In the winter, only a handful of satellites are visible late into the night and are faint, while near the equinox satellites are visible at midnight, but they are all faint. However, near the summer solstice, there are many bright satellites crossing the zenith all night long. The shadow of the Earth blocks many of the would-be brightly lit satellites close to the equinox, but this does not occur closer to the summer solstice. Table 2 quantifies this: at 50°N or S latitude, near the summer solstice, satellites will make up 6–7% of all visible point sources in the night sky at g < 5, g < 6.5, and g < 7 within a couple hours of sunrise or sunset; restated: as seen by the naked eye, about one in every 14 stars in the sky will actually be a satellite. Close to midnight, this fraction drops slightly to 5–6% but, as shown in Figure 6, the brightest satellites (g-magnitudes near 5–5.5) remain close to the zenith at all hours of the night.

It is nearly as severe of an effect at 40°N or S: 6–9% of point sources in the sky are going to be satellites for these magnitude cuts within a couple hours of sunrise and sunset near the summer solstice. At 40°N or S, the brightness of satellites drops off more steeply toward midnight and toward the equinox as compared with 50°N or S, so many fewer satellites are visible at midnight.

At even more northerly/southerly latitudes of 60°N or S, the fraction of satellites is 2–6% all night long near the summer solstice. The fraction drops quickly close to the equinox. Of course, close to the summer solstice at these high latitudes, there is very little darkness, so the summer light pollution by satellites might be much less noticeable than at latitudes of 50° or even 40°.

As a check of our calculations, and the most extreme case, we also include the view from the North Pole (Figure 8, equally applicable to the South Pole). Interestingly, there are never any satellites directly overhead: this is because satellites are not inclined by exactly 90°. Of course, the visibility pattern will be quite different at the North Pole than at other latitudes: the summer solstice satellites will not be visible at all due to 24 hr daylight, while the winter solstice satellites will be visible all day and night due to 24 hr darkness. Close to the winter solstice, only the highest-altitude satellites close to the zenith will be illuminated and visible (presumably 24 hr per day). For comparison, we also modeled the sky view at midnight, with the Sun at a declination of −18° (October 24), when there is 24 hr of astronomical twilight, and at −12° (November 12), when there is 24 hr of nautical twilight. As the Sun creeps to higher declinations, the bright satellites appear further and further south in the polar sky. While there are not any observing facilities at the North Pole, there are high-latitude facilities, and these predictions are equally applicable to Antarctic astronomical facilities.

These calculations highlight the unequally distributed light pollution that will result from satcons at different places on Earth: according to our simulations, 50°N and S latitudes will experience some of the worst effects of light pollution compared to anywhere else on Earth.

One recommendation from the SATCON1 report is that satellites should orbit at lower altitudes. While this makes them brighter, it also causes them to move through a given telescope field of view more quickly, smearing the reflected sunlight across more pixels in a shorter amount of time, decreasing the damage to science images. Our models here show that while this strategy works well for lower latitudes, where many research telescopes are located, it will lead to many bright satellites visible all night long at higher latitudes during times close to the summer solstice.

To more thoroughly explore the effects of satellite altitude, we build two toy models. Each model uses the same number of satellites and the orbits from Table 1, but all of the satellites have been moved to an orbital altitude of 550 km in one model and 1200 km in the other. Figure 9 shows these two satellite altitude toy models on the summer solstice, at nautical dusk and at midnight, and at many different latitudes, illustrating the patterns in light pollution across the Earth caused by satellites at different altitudes.

The number of sunlit satellites in the sky is much larger for the 1200 km model than the 550 km model, at all latitudes. The satellites in the 1200 km model are also on average much fainter, with significantly fewer satellites that are visible to the naked eye for all latitudes and at all times of night. However, while there are fewer bright satellites in the higher-altitude model, the rate of motion across the sky is slower, making each satellite visible in a given telescope's field of view for a longer period of time, as noted above. With this in mind, satcon operators will require more satellites at lower altitudes than at higher altitudes to provide comparable beam coverage, all other things being equal. This is not captured in this toy model, which uses the same number of satellites for the two altitude cases. Thus, the total number of satellites in the sky might not vary too much according to the chosen altitude, depending on the satcon designs. These trade-offs need to be handled with a high degree of care and continuously re-evaluated.

Another issue to consider is that while satellites are overall brighter in lower orbits, the number of sunlit and visible satellites drops very quickly after nautical dusk for the lower latitudes, giving a period of very few sunlit satellites for most of the night. However, this fast drop-off in low-orbit visible satellites does not happen at higher latitudes. For example, at 50°N, there continue to be hundreds of naked-eye visible satellites all night long near the summer solstice. The higher-orbit satellite model also produces many naked-eye visible satellites at high latitudes all night long in the summer, but not as many as the lower orbits.

There is no perfect solution—higher-altitude orbits may be better overall for naked-eye stargazers, but lower orbits are thought, based on the astronomy community's current understanding of satcon operation plans, to be better for large research facilities, which tend to be at lower latitudes.

The most obvious effect of satellites on optical research astronomy is the contamination of research images by bright satellite trails, appearing as lines cutting across the entire field of view of research or astrophotography images. ¹¹ Research astronomers are developing algorithms to process these streaks, such as automatically identifying, characterizing, and removing them from images (see, e.g., the SATCON2 Report). Such algorithms will help to mitigate some forms of data loss from science images and allow further image processing, but they cannot recover lost data.

If satellite trails are too bright, they can create cross-talk between CCD chips in large telescopes, most well-documented for the LSST telescope (Tyson et al. 2020). If satellite operators can reduce brightnesses below g-magnitude ∼7, then such cross-talks might be significantly reduced for some detectors, although this needs to be continuously evaluated. Further, if operators meet this brightness limit, it will fulfil the task of keeping most satellites below naked-eye visibility. SpaceX has stated that this is the goal for their Starlink satellites—a goal that has not yet been achieved (e.g., Mallama 2021a; Boley et al. 2021b).

The level of disruption to research astronomy will most strongly depend on satellite brightnesses and rates of motion. Consider Figure 3, which shows 400 satellites above 30^◦ elevation for at least a portion of each night. This corresponds to a sky area of π sr. The expectation that the satellite will pass through the field of view is then estimated as

$$\begin{eqnarray}&&E={RwtN}/{{\rm{\Omega }}}_{\mathrm{sky}},\end{eqnarray} \tag{ 2 }$$

where R is the typical sky motion of a satellite, w is the characteristic width of the detector (taken to be 1°, typical for a wide-field camera), t is the exposure time, and N is the number of satellites in the sky area Ω_sky. For R ≈ 2000'' s⁻¹ (typical for a satellite at ∼500 km orbital altitude) and an exposure time of t = 10 s, we find E ≈ 0.22. Assuming Poisson statistics, the probability of one or more satellites passing through the field of view of the square degree telescope is about 20% under the models explored here. For a three minute exposure, as is common for deep Kuiper Belt surveys (e.g., Bannister et al. 2018), a typical exposure will include four satellite trails, depending on satellite brightness. While most parts of the world will have some period of time each night with few-to-no sunlit satellites, telescopes at high latitudes will experience this rate of satellite contamination all summer.

This calculation assumes that the satellites are independent of each other, when actually, their motions across the sky are highly correlated. Our rough calculations here agree with the much more thorough analysis of the number of satellites per telescope field and the fraction of lost exposures for different telescope facilities that is presented in Williams et al. (2021).

This paper is focused on optical light pollution, but we note that radio astronomy has the potential to be strongly affected by the abundance of transmitting satellites, particularly if the electronics that are used allow for out-of-band emission or overtones falling into bands otherwise protected for astronomy (as happened with the Iridium satcon; see, e.g., Council 2010). Additional spectrum is further being sought, and the use of frequencies that are typically used in terrestrial applications might be explored for on-orbit infrastructure, further restricting the frequencies available for scientific research.

The SATCON2 report, available to the public in 2021 October ¹² , contains recommendations for how satcon operators can mitigate damage to radio astronomy.

Even when satellites are not sunlit, they can still potentially interfere with astronomical observations. They are too fast to cause a measurable change in, for example, an exoplanet transit observation. But occultation experiments like TAOS, which is using high-cadence observations with the goal of detecting occultations of stars by small bodies in the solar system (Zhang et al. 2013), could potentially experience a new source of noise.

A satellite occultation of a star lasts between about a tenth of to a full millisecond (Hainaut & Williams 2020), as LEO satellite on-sky motions are of the order 2000'' s⁻¹, as noted above. While this timescale is much shorter than what is expected for an occultation by a small body in the outer solar system, an observing cadence of a few frames per second would cause this short dip to be confused with the ∼0.1 s dip in brightness caused by a real occultation (Schlichting et al. 2012; Lehner et al. 2019). With 65,000 satellites, satellite occultations could become frequent enough, depending on a telescope's field of view, to produce a false-positive signal as high as is expected for distant solar system small bodies.

To approximately calculate this, consider the same setup as in Section 4.1, with 400 satellites passing through a sky area of π sr at any moment. At a range of 1000 km, a plausible angular size of a satellite is about 05. Taking the satellite's sky motion to be 2000'' s⁻¹, then the satellite population would sweep out an area equivalent to that entire section of the sky with a frequency of about 0.3 day⁻¹. During that time, if we are monitoring an average of 10,000 stars (as TAOS II plans to do; Lehner et al. 2019) in each field, then for 8 hr nights, we would expect around 1000 satellite occultation events per night, on average. This is much higher than the expected occultation frequency of stars by small trans-Neptunian objects (TNOs; <10⁻³ per star per year), and may become a limiting source of noise for this type of astronomy research.

As a result of adding more point sources, the total brightness of the night sky will increase. For an estimate, we calculate a summed flux density using V-band magnitudes of all stars in the Hipparcos catalog with V < 10, converting to Janskys using the AB magnitude zero-point, i.e., $m=-2.5{\mathrm{log}}_{10}({f}_{\nu }/3631\mathrm{Jy})$. Averaging over the entire sky gives a surface brightness of 13.7 Jy/sq. deg. We then repeat this calculation for our satellite models (using V-band estimates), including only the illuminated satellites and their corresponding brightness. For a latitude of 50° on the summer solstice at nautical twilight, satellite night sky brightness is about 0.5 Jy/sq. deg., about 4% of the starlight contribution.

This simplistic calculation ignores any reflection by diffuse particles or by the atmosphere. Diffuse emission caused by sunlight reflecting off the huge number of tiny pieces of debris in orbit ("space junk") has already been calculated to increase the overall brightness of the sky by 10% above pre-industrial levels (Kocifaj et al. 2021). The placement of such large amounts of infrastructure in space raises the issue of the further generation of space debris, particularly if there is an on-orbit accident such as a collision or explosion, compounding the brightness problem.