Recent arXiv:astro-ph Picks: December 2024

Here are some recent submissions on astro-ph that I found to be especially interesting. Text excerpts below are quoted directly from the articles. My comments are in italics.

Short-Term Evolution and Risks of Debris Cloud Stemming from Collisions in Geostationary Orbit
https://arxiv.org/abs/2412.13586

The geostationary orbit is a popular orbit for communication, meteorological, and navigation satellites due to its apparent motionless. Nearly all geostationary satellites are positioned in a circular orbit with a radius of 42,164 km, making this region particularly vulnerable to space traffic accidents due to the high concentration of objects and the absence of natural debris-clearing mechanisms. The growing population in geostationary region raises concerns about the potential risks posed by fragments stemming from explosions and collisions, particularly following the breakup of Intelsat-33e, which remained operational in geostationary orbit until October 19, 2024.

A breakup event generates a large number of fragments of varying sizes. In the geostationary region, only fragments larger than 1 meter are routinely tracked by the Space Surveillance Network, as the sensitivity of ground-based sensors decreases significantly with distance. However, small, non-trackable fragments can still cause catastrophic damage to spacecraft. The collision velocity of spacecraft in geostationary orbit can reach up to 4 km/s, while micro-meteoroids may hit at speeds of up to 72 km/s.

The impact of a debris cloud is inherently global as it disperses around the entire Earth.

By 2024, over 1,000 objects have been observed near the geostationary orbit (GEO). Nearly all objects exhibit inclinations of less than 15 degrees, with the majority having inclinations of less than 1 degree. Once a fragmentation event occurs, the GEO objects will be exposed to considerable risks, as they are densely clustered along a single ring above the Equator.

More about Intelsat 33e and its breakup:
https://en.wikipedia.org/wiki/Intelsat_33e


Sun-like stars produce superflares roughly once per century
https://arxiv.org/abs/2412.12265

Stellar superflares are energetic outbursts of electromagnetic radiation, similar to solar flares but releasing more energy, up to 1036 erg on main sequence stars. It is unknown whether the Sun can generate superflares, and if so, how often they might occur. We used photometry from the Kepler space observatory to investigate superflares on other stars with Sun-like fundamental parameters. We identified 2889 superflares on 2527 Sun-like stars, out of 56450 observed. This detection rate indicates that superflares with energies >1034 erg occur roughly once per century on stars with Sun-like temperature and variability. The resulting stellar superflare frequency-energy distribution is consistent with an extrapolation of the Sun’s flare distribution to higher energies, so we suggest that both are generated by the same physical mechanism.

Solar flares are sudden local bursts of bright electromagnetic emission from the Sun, which release a large amount of energy within a short interval of time. The increase in short-wavelength solar radiation during flares influences the Earth’s upper atmosphere and ionosphere, sometimes causing radio blackouts and ionosphere density changes. Solar flares are frequently accompanied by the expulsion of large volumes of plasma, known as coronal mass ejections (CMEs), which accelerate charged particles to high energies. When these solar energetic particles (SEPs) reach Earth, they cause radiation hazards to spacecraft, aircraft and humans. Extreme SEP events can produce isotopes, called cosmogenic isotopes, which form when high-energy particles interact with the Earth’s atmosphere. These isotopes are then recorded in natural archives, such as tree rings and ice cores. The total amount of energy released by each flare varies by many orders of magnitude, as determined by a complex interplay between the physical mechanisms of particle acceleration and plasma heating in the Sun’s
atmosphere.

Solar flares have been observed for less than two centuries. Although thousands of them have been detected and measured, only about a dozen are known to have exceeded a bolometric (integrated over all wavelengths) energy of 1032 erg. Among them was the Carrington Event on 1 September 1859, which was accompanied by a CME that had the strongest recorded impact on Earth. Modern estimates of the Carrington Event’s total bolometric energy are 4 × 1032 to 6 × 1032 erg.

It is unknown whether the Sun can unleash flares with even higher energies, often referred to as superflares, and if so, how frequently that could happen. The period of direct solar observations is too short to reach any firm conclusions. There are two indirect methods to investigate the potential for more intense flares on the Sun. One method uses extreme SEP events recorded in cosmogenic isotope data, which have been used to quantify the occurrence rate of strong CMEs reaching Earth over the past few millennia. There are five confirmed (and three candidate) extreme SEP events that are known to have occurred in the last 10,000 yr, implying a mean occurrence rate of ∼ 10−3 yr−1. However, the relationship between SEPs and flares is poorly understood, especially for the stronger events.

A second method is to study superflares on stars similar to the Sun. If the properties of the observed stars sufficiently match the Sun, the superflare occurrence rate on those stars can be used to estimate the rate on the Sun.

We found that Sun-like stars produce superflares with bolometric energies > 1034 erg roughly once per century. That is more than an order of magnitude more energetic than any solar flare recorded during the space age, about sixty years. Between 1996 and 2012 twelve solar flares had bolometric energies > 1032 erg, but none were > 1033 erg. The most powerful solar flare recorded occurred on 28 October 2003, with an estimated bolometric energy of 7 × 1032 erg, which exceeds estimates for the Carrington Event (4 × 1032 to 6 × 1032 erg).

We cannot exclude the possibility that there is an inherent difference between flaring and non-flaring stars that was not accounted for by our selection criteria. If so, the flaring stars in the Kepler observations would not be representative of the Sun. Approximately 30% of flaring stars are known to have a binary companion. Flares in those systems might originate on the companion star or be triggered by tidal interactions. If instead our sample of Sun-like stars is representative of the Sun’s future behavior, it is substantially more likely to produce a superflare than was previously thought.

More about the Carrington Event:
https://en.wikipedia.org/wiki/Carrington_Event


ChronoFlow: A Data-Driven Model for Gyrochronology
https://arxiv.org/abs/2412.12244

Gyrochronology is a technique for constraining stellar ages using rotation periods, which change over a star’s main sequence lifetime due to magnetic braking. This technique shows promise for main sequence FGKM stars, where other methods are imprecise. However, models have historically struggled to capture the observed rotational dispersion in stellar populations. To properly understand this complexity, we have assembled the largest standardized data catalog of rotators in open clusters to date, consisting of ~7,400 stars across 30 open clusters/associations spanning ages of 1.5 Myr to 4 Gyr.

Stars in open clusters are all about the same age, so this is highly useful in training models that use stellar rotation periods to determine stellar age.
https://en.wikipedia.org/wiki/Gyrochronology


On The Lunar Origin of Near-Earth Asteroid 2024 PT5
https://arxiv.org/abs/2412.10264

The Near-Earth Asteroid (NEA) 2024 PT5 is on an Earth-like orbit which remained in Earth’s immediate vicinity for several months at the end of 2024. PT5’s orbit is challenging to populate with asteroids originating from the Main Belt and is more commonly associated with rocket bodies mistakenly identified as natural objects or with debris ejected from impacts on the Moon. We obtained visible and near-infrared reflectance spectra of PT5 with the Lowell Discovery Telescope and NASA Infrared Telescope Facility on 2024 August 16. The combined reflectance spectrum matches lunar samples but does not match any known asteroid types—it is pyroxene-rich while asteroids of comparable spectral redness are olivine-rich. Moreover, the amount of solar radiation pressure observed on the PT5 trajectory is orders of magnitude lower than what would be expected for an artificial object. We therefore conclude that 2024 PT5 is ejecta from an impact on the Moon, thus making PT5 the second NEA suggested to be sourced from the surface of the Moon. While one object might be an outlier, two suggest that there is an underlying population to be characterized. Long-term predictions of the position of 2024 PT5 are challenging due to the slow Earth encounters characteristic of objects in these orbits. A population of near-Earth objects which are sourced by the Moon would be important to characterize for understanding how impacts work on our nearest neighbor and for identifying the source regions of asteroids and meteorites from this under-studied population of objects on very Earth-like orbits.

Perhaps the most significant conclusion to finding a second near-Earth object with an apparently Moon-like surface composition is the realization of lunar ejecta as a genuine population of objects. The Quasi-Satellite Kamo‘oalewa has a slightly redder spectrum than 2024 PT5, but the higher quality of our data at longer wavelengths (the Quasi-Satellite was significantly dimmer, so only photometry was obtained beyond ≈ 1.25μm) makes a discussion of how different the two spectra are only qualitative. At the very least, the two lunar NEOs do not look identical. Sharkey et al. (2021) argued that the red spectrum of Kamo‘oalewa was partially due to space weathering – an exposure time of a few million years was likely sufficient to explain its surface properties and was similar to its approximate dynamical lifetime and even the age of the crater that Jiao et al. (2024) suggested it came from, Giordano Bruno. If correct, perhaps 2024 PT5 has a somewhat younger surface than the larger Kamo‘oalewa. In any case, PT5 is smaller than Kamo‘oalewa and thus the craters that are energetic enough to produce an object its size are more common – a more recent ejection age, and thus a ‘younger’ surface might be preferred from that argument as well. (Granted, smaller fragments would be more common than larger ones in cratering events of any size as well.) Further work to study these two objects and to find more lunar-like NEOs will be needed to ascertain the origin of these differences and how they can be related to the circumstances of their creation. At any rate, the smaller size of PT5 means that we are approaching being able to study the impactors and outcomes from the kinds of small impacts seen regularly by the Lunar Reconaissance Orbiter.

For more information about 2024 PT5 and Kamo‘oalewa:
https://en.wikipedia.org/wiki/2024_PT5
https://en.wikipedia.org/wiki/469219_Kamo%CA%BBoalewa


Call to Protect the Dark and Quiet Sky from Harmful Interference by Satellite Constellations
https://arxiv.org/abs/2412.08244

The growing number of satellite constellations in low Earth orbit (LEO) enhances global communications and Earth observation, and support of space commerce is a high priority of many governments. At the same time, the proliferation of satellites in LEO has negative effects on astronomical observations and research, and the preservation of the dark and quiet sky. These satellite constellations reflect sunlight onto optical telescopes, and their radio emission impacts radio observatories, jeopardising our access to essential scientific discoveries through astronomy. The changing visual appearance of the sky also impacts our cultural heritage and environment. Both ground-based observatories and space-based telescopes in LEO are affected, and there are no places on Earth that can escape the effects of satellite constellations given their global nature. The minimally disturbed dark and radio-quiet sky1 is crucial for conducting fundamental research in astronomy and important public services such as planetary defence, technology development, and high-precision geolocation.

Some aspects of satellite deployment and operation are regulated by States and intergovernmental organisations. While regulatory agencies in some States have started to require operators to coordinate with their national astronomy agencies over impacts, mitigation of the impact of space objects on astronomical activities is not sufficiently regulated.

1We refer to the radio-quiet sky as simply the ‘quiet sky’

To address this issue, the CPS [International Astronomical Union (IAU) Centre for the Protection of the Dark and Quiet Sky from Satellite Constellation Interference (CPS)] urges States and the international community to:

1) Safeguard access to the dark and quiet sky and prevent catastrophic
loss of high quality observations.

2) Increase financial support for astronomy to offset and compensate the impacts on observatory operations and implement mitigation measures at observatories and in software.

3) Encourage and support satellite operators and industry to collaborate with the astronomy community to develop, share and adopt best practices in interference mitigation, leading to widely adopted standards and guidelines.

4) Provide incentive measures for the space industry to develop the required technology to minimise negative impacts. Support the establishment of test labs for brightness and basic research into alternate less reflective materials
and reduction of unwanted radiation in the radio regime for spacecraft manufacturing.

5) In the longer term, establish regulations and conditions of authorization and supervision based on practical experience as well as the general provisions of international law and main principles of environmental law to codify industry best practices that mitigate the negative impacts on astronomical observations. Satellites in LEO should be designed and operated in ways that minimise adverse effects on astronomy and the dark and quiet sky.

6) Continue to support finding solutions to space sustainability issues, including the problem of increasing space debris leading to a brighter sky. Minimising the production of space debris will also benefit the field of astronomy and all sky observers worldwide.

The elephant in the room—not specifically mentioned in this report—is that countries and companies should be sharing satellite constellations as much as possible to minimize the number of satellite constellations in orbit. This is analogous to the co-location often required for terrestrial communication towers. Our current satellite constellation predicament illustrates yet another reason why we need a binding set of international laws that apply to all nations and are enforced by a global authority. The sooner we have this the better, as our cultural survival—if not our physical survival—may depend upon it.


A New Method to Derive an Empirical Lower Limit on the Mass Density of a UFO
https://arxiv.org/abs/2412.12142

I derive a lower limit on the mass of an Unidentified Flying Object (UFO) based on measurements of its speed and acceleration, as well as the infrared luminosity of the airglow around it. If the object’s radial velocity can be neglected, the mass limit is independent of distance. Measuring the distance and angular size of the object allows to infer its minimum mass density. The Galileo Project will be collecting the necessary data on millions of objects in the sky over the coming year.

Any object moving through air radiates excess heat in the form of infrared airglow luminosity, L. The airglow luminosity is a fraction of the total power dissipated by the object’s speed, v, times the frictional force of air acting on the object. The radiative efficiency depends on the specific shape of the object and the turbulence and thermodynamic conditions in the atmosphere around it. If the object accelerates, then this friction force must be smaller than the force provided by the engine which propels the object. The net force equals the object’s mass, M, times its acceleration, a.

In conclusion, one gets an unavoidable lower limit on the mass of an accelerating object. The object’s mass must be larger than the infrared luminosity from heated air around it, divided by the product of the object’s acceleration and speed.

This limit provides an elegant way to constrain the minimum mass of Unidentified Flying Objects (UFOs), also labeled as Unidentified Anomalous Phenomena (UAPs). To turn the inequality into an equality, one needs to know the detailed object shape and atmospheric conditions around the object.

The first Galileo Project Observatory at Harvard University collects data on ∼ 105 objects in the sky every month. A comprehensive description of its commissioning data on ∼ 5 × 105 objects was provided in a recent paper (Dominé et al. 2024). The data includes infrared images captured by an all-sky Dalek array of eight uncooled infrared cameras placed on half a sphere.

Within the coming month, the Galileo Project’s research team plans to employ multiple Daleks separated by a few miles, in order to measure distances to objects through the method of triangulation.

If the measured velocity and acceleration of a technological object are outside the flight characteristics and performance envelopes of drones or airplanes, then the object would be classified by the Galileo Project’s research team as an outlier. In such a case, it would be interesting to calculate the minimum mass density of the object. If the result exceeds normal solid densities, then the object would qualify as anomalous, a UAP. Infrared emission by the object would be a source of confusion, unless the object is resolved and the emission from it can be separated from the heated air around it.

All flying objects made by humans have a volume-averaged mass density ⟨ρ⟩ which is orders of magnitude below 22.6 g cm−3, the density of Osmium – which is the densest metal known on Earth. A UFO with a higher mass density than Osmium would have to carry exotic material, not found on Earth.

By summer 2025, there will be three Galileo Project observatories operating in three different states within the U.S. and collecting data on a few million objects per year. With new quantitative data on infrared luminosities, velocities and accelerations of technological objects, it would be possible to check whether there are any UFOs denser than Osmium.

I admire the author, Avi Loeb, Harvard astrophysics professor, for his creative approaches to interesting problems outside the mainstream that many of his colleagues tend to avoid. Lately, he’s been focusing a lot on technosignatures, and I imagine he has a keen interest in the recent spate of unexplained nighttime drone sightings in New Jersey and elsewhere. For more about Loeb and the Galileo Project:
https://en.wikipedia.org/wiki/Avi_Loeb
https://en.wikipedia.org/wiki/The_Galileo_Project


Beyond CCDs: Characterization of sCMOS detectors for optical astronomy
https://arxiv.org/abs/2409.16449

Modern scientific complementary metal-oxide semiconductor (sCMOS) detectors provide a highly competitive alternative to charge-coupled devices (CCDs), the latter of which have historically been dominant in optical imaging. sCMOS boast comparable performances to CCDs with faster frame rates, lower read noise, and a higher dynamic range. Furthermore, their lower production costs are shifting the industry to abandon CCD support and production in favour of CMOS, making their characterization urgent. In this work, we characterized a variety of high-end commercially available sCMOS detectors to gauge the state of this technology in the context of applications in optical astronomy. We evaluated a range of sCMOS detectors, including larger pixel models such as the Teledyne Prime 95B and the Andor Sona-11, which are similar to CCDs in pixel size and suitable for wide-field astronomy. Additionally, we assessed smaller pixel detectors like the Ximea xiJ and Andor Sona-6, which are better suited for deep-sky imaging. Furthermore, high-sensitivity quantitative sCMOS detectors such as the Hamamatsu Orca-Quest C15550-20UP, capable of resolving individual photoelectrons, were also tested. In-lab testing showed low levels of dark current, read noise, faulty pixels, and fixed pattern noise, as well as linearity levels above 98% across all detectors. The Orca-Quest had particularly low noise levels with a dark current of 0.0067±0.0003 e/s (at −20C with air cooling) and a read noise of 0.37±0.09 e using its standard readout mode. Our tests revealed that the latest generation of sCMOS detectors excels in optical imaging performance, offering a more accessible alternative to CCDs for future optical astronomy instruments.

The Hamamatsu Orca-Quest CP15550-20UP, simply called Orca-Quest, is advertised as being a quantitative CMOS detector with extremely low noise levels and photoelectron counting capabilities. It features a custom 9.4-megapixel sensor with 4.6 × 4.6 μm pixels. The Orca-Quest has two scan modes that were characterized: standard and ultra-quiet. The ultra-quiet mode has a much lower frame rate at 5 frames per second (fps) compared to the standard mode’s 120 fps, which allows for much lower read noise. Also characterized was the ‘photon number resolving’ readout mode which claims to report the integer number of incident photoelectrons based on a proprietary calibrated algorithm using the ultra-quiet scan. The Orca-Quest has a detector-imposed temperature lock at −20C when air-cooled. The standard and ultra-quiet modes are 16-bit, with a saturation limit of 65536 ADU while the photon number resolving mode has a saturation limit of only 200 ADU. The Orca-Quest boasts a peak quantum efficiency of 85%.

Unlike CCDs, which use a single global amplifier with a shift register, sCMOS pixels have individual readout electronics, requiring each pixel to be tested as an independent detector. Historically, this led to high fixed pattern noise in CMOS detectors, but we found negligible fixed pattern noise in almost all the detectors we analyzed pixel-wise.

Satellites and More – 2020 #1

Edmund Weiss (1837-1917) and many astronomers since have called asteroids “vermin of the sky”, but on October 4, 1957 another “species” of sky vermin made its debut: artificial satellites.  In the process of video recording stars for possible asteroid occultations, I frequently see satellites passing through my 17 × 11 arcminute field of view.

I’ve put together a video montage of satellites I serendipitously recorded during the first half of 2020.  Many of the satellites move across the field as “dashes” because of the longer integration times I need to use for some of my asteroid occultation work. A table of these events is shown below the video. The range is the distance between observer and satellite at the time of observation. North is up and east is to the left.

North is up and east is to the left; field size 17′ x 11′

Interestingly, three of the above satellites (7,9,18) are in retrograde orbits, that is their orbital inclination is > 90˚ and their east-west component of motion is towards the west instead of the east. However, I was surprised to find that two of the prograde orbiting satellites (5,6) appear to be orbiting retrograde! Both have orbital inclinations close to 90˚ (82.6˚ and 87.5˚, respectively), and both were in the western sky at northern declinations at the time of observation. But another satellite (8) with an orbital inclination of 82.5˚ at a southern declination in the southern sky at the time of observation exhibited the expected “barely” prograde motion. I suspect the ~0.5 km/s rotation of the Earth towards the east might have something to do with this “apparent retrograde” motion, but I was unable to find any reference that describes this situation.

Satellite #12 has an interesting story. It is the Inertial Upper Stage (IUS) used to launch USA-48 (Magnum), a classified DoD payload, from the Space Shuttle Discovery (STS-33).

In addition to these 18 satellites, I observed 7 geosynchronous satellites, shown below.

This non-operational Soviet communications satellite is a “tumbler”, meaning its changing orientation causes variation in its brightness, as shown below.

This non-operational communications satellite is also a tumbler, as seen in this light curve from a portion of the video.

SGDC-1 is a Brazilian geostationary communications satellite stationed over longitude 75˚ W, and in this video is followed by Star One C3 which will replace Brasilsat B3, also located over longitude 75˚ W.
Star One C3, a geostationary television satellite led by SGDC-1 and followed by GOES-16.
GOES-16, a geostationary weather satellite that is the primary weather satellite for the U.S., is stationed over longitude 75.2˚ W. Star One C3 precedes it in this video.
Intelsat 16 is a geostationary television satellite stationed over longitude 76˚ W currently.

There were four satellites I was unable to identify, shown in the video below. They were either classified satellites or, more likely, small pieces of space debris that only government agencies or contractors are keeping track of.

Unidentifiable satellites

Occasionally, I record other phenomena of interest. Meteors during this period are described here, and you will find a couple of other curiosities below.

An aircraft with flashing lights passed near the field containing UCAC4 376-101735 between 10:06:44 and 10:06:47 UT on 16 Apr 2020.
High energy particles zap the imaging chip from time to time, and here is one of the more interesting ones during the period, recorded on 9 May 2020 from 9:09:18 – 9:09:20 UT in the field of UCAC4 397-127754.

References
Hughes, D. W. & Marsden, B. G. 2007, J. Astron. Hist. Heritage, 10, 21

Geostationary Satellite Declinations

A few years ago, I was doing some telescope sweeping of the meridian sky around declination -6˚ when, to my surprise and delight, a 10th- or 11th-magnitude slow-moving object entered my field of view. As it slowly traversed eastward through the field, I remembered the declination I was pointed to and realized that it must be a geostationary, or at least a geosynchronous, satellite. Centering the moving object and then turning off the telescope’s clock drive confirmed my suspicions. The object was a geosynchronous satellite because it appeared to lay motionless while all the stars in the field drifted toward the west. Serendipity is the spice of life!

Satellites stationed in orbits that are always directly above the Earth’s equator and that have an orbital period of 23h 56m 04.0905s (one sidereal day) have the interesting property of remaining stationary as seen from any point on the surface of the Earth. This property of geostationary satellites, as they are called, is used to great advantage by many communications and weather satellites. There are currently at least 554 satellites in geosynchronous orbits. They are stationed all around the Earth at various longitudes.

At what altitude do geostationary satellites orbit the Earth? It is well above human-occupied spacecraft like the International Space Station which currently orbits 260 miles above the Earth’s surface. Geosynchronous orbit lies some 22,236 miles above the Earth’s equator. This is quite a ways out, as the entire Earth subtends an angle of only 17° 12′ at this distance—about the same as the angular distance between Capella (α Aur) and Elnath (β Tau).

Looking at it another way, geostationary satellites orbit at an altitude that is 2.8 Earth diameters above the equator. Since the Moon orbits at a distance that ranges between 27.4 and 31.4 Earth diameters above the Earth’s surface, geosynchronous orbit is about 1/10 of the way to the Moon.

If you have a telescope, know where to point it, and turn tracking off, you can see a geostationary satellite as a stationary point of light while the stars drift by due to the Earth’s rotation. At our latitude here in southern Wisconsin (43° N), the area where you want to search for geostationary satellites (near the meridian) is around declination -6° 37′. Remember, declination tells you how many degrees above or below the celestial equator an object is, and the numbers range from -90° to +90°, the south celestial pole and north celestial pole, respectively. The celestial equator has a declination of 0°.

For any latitude1, the declination you want to search is given by

\delta _{gs}=\textup{tan}^{-1}\left [ 6.611\textup{ csc }\phi - \textup{cot }\phi \right ]-90^{\circ}

where δgs is the declination of the geostationary satellite in degrees
     and ϕ is your latitude in degrees

Since most calculators don’t have the cosecant (csc) or cotangent (cot) functions, this formula can be rewritten in a slightly more complicated form as

\delta _{gs}=\textup{tan}^{-1}\left [ \frac{6.611}{\textup{sin }\phi }-\frac{1}{\textup{tan }\phi } \right ]-90^{\circ}

Why aren’t the satellites right on the celestial equator (δ = 0°)? They would be if they were millions of miles away or if we were located on the Earth’s equator, but at our northern latitude trigonometric parallax causes us to see the satellites somewhat below the celestial equator, relative to the distant stars.

What if the geostationary satellite is situated east or west of your meridian? How do you calculate its declination then? As you might expect, because the range (observer-to-satellite distance) is greater the further from the meridian the satellite is, the less the parallax is, and therefore the closer the declination is to the equator, though not by a lot. The declination is also symmetric about the meridian, east and west: a geostationary satellite one hour east of the meridian will have the same declination as another geostationary satellite one hour west of the meridian.

If you know the longitude of the geostationary satellite (for example, the GOES-16 weather satellite is stationed above 75.2˚ W longitude), you can calculate its declination (and right ascension) using the following two-step process.

\textup{h}=\textup{tan}^{-1}\left [ \frac{\textup{sin }\Delta\lambda  }{\textup{cos }\Delta \lambda-0.15126\textup{ cos }\phi  } \right ]

where h is the hour angle in degrees
     and Δλ = λsat − λobs , the difference between the satellite and observer
          longitudes, in degrees
     and ϕ is the latitude of the observer in degrees

\delta _{gs}=\textup{tan}^{-1}\left [ \frac{-0.15126\textup{ sin }\phi \textup{ sin h}}{\textup{sin }\Delta \lambda } \right ]

To determine the right ascension of the geostationary satellite, add the value of h to your local sidereal time (the right ascension of objects on your meridian). Make sure you convert h to hours before adding it to your LST.

What if you want to calculate the geostationary declination at a particular hour angle? That is a bit trickier. I could not figure out how to manipulate the equation for h above so that Δλ = f (h,φ). Instead, I rewrote the equation as

\sin \Delta \lambda =\tan h\cdot \left ( \cos \Delta \lambda -0.15126\cos \phi  \right )

and using h as a starting value for Δλ, substituted it into the cos Δλ expression, calculated sin Δλ, took the arcsine to get a new value of Δλ, then substituted that back into the cos Δλ expression, and iterated. Fortunately, the value of Δλ converges very fast. Once you have Δλ, you can use the two-step process we used earlier to determine the declination of the geostationary satellite for a particular hour angle.

Please note that the value of the hour angle h we use here is positive east of the meridian and negative west of the meridian. This is opposite from the normal astronomical sense.

Here is a simple SAS program illustrating how to do all these calculations using a computer.

And here is the output from that program.

1 For latitudes south of the equator, add 180° to get your meridian geostationary declination. The equation goes singular at the equator (φ=0°) and at the poles (φ=90° N and 90° S) since we’re dividing by sin φ = 0 at the equator and tan φ is undefined at the poles. However, as you asymptotically get closer and closer to latitude 0° (0.0001° and -0.0001°, for example) you find that the meridian geostationary declination approaches δ = 0°. Likewise, as you asymptotically approach latitude 90° N and 90° S, you’ll find that the meridian geostationary declination approaches -8°36′ and +8°36′, respectively. Of course, in both cases the geostationary satellites always remain below your horizon. How far north or south in latitude would you have to go, then, to find that geostationary satellites on your meridian are on your horizon due south or due north, respectively? Through a little algebraic manipulation of the first equation above and utilizing some simple trigonometric identities, one finds that at latitudes 81°18′ N and 81°18′ S, geostationary satellites on your meridian would be on the horizon. North or south of there, respectively, you would not be able to see them because the Earth would be in the way.

References

Gérard Maral, Michel Bousquet, Zhili Sun. Satellite Communications Systems: Systems, Techniques and Technology, Fifth Edition. Wiley, 2009. See section 8.3.6.3 Polar mounting.

Satellite, Meteor, and Aircraft Crossings 2019

Edmund Weiss (1837-1917) and many astronomers since have called asteroids “vermin of the sky”, but on October 4, 1957 another “species” of sky vermin made its debut: artificial satellites.  In the process of video recording stars for possible asteroid occultations, I frequently see satellites passing through my ~¼° field of view.

I’ve put together a video montage of satellites I serendipitously recorded between March 31, 2019 and July 12, 2019.  Many of the satellite crossings are moving across the fields as “dashes” because of the longer integration times I need to use for some of my asteroid occultation work. A table of these events is shown below the video. The range is the distance between observer and satellite at the time of observation.

Satellites in higher orbits take longer to cross the field. When possible, I’ve included graphs of brightness as a function of time for these slower-moving satellites after each individual video and corresponding table. When you watch the videos of geostationary satellites, you are actually seeing the rotation of the Earth as the line between you and the satellite sweeps across the stars as the Earth rotates!

Uncertain of identification
A tumbler with sun glints!
A high-amplitude tumbler! Satellite is no longer operational.

I caught one meteor on 4 Jan 2019 between 5:32:57 and 5:32:59 UT. Field location was UCAC4 419-017279. I’m pretty sure the meteor was a Quadrantid!

And two aircraft crossed my field: on 7 Dec 2018 1:40:05 – 1:40:13 UT (UCAC4 563-026131) and 26 Jun 2019 5:02:07 – 5:02:10 UT (UCAC4 291-144196).

And high energy particles (natural radioactivity or cosmic rays) “zing” my CCD/CMOS detector every once in a while. Here are a few examples: 5 Jan 2019 3:46:00 – 3:46:02 UT (UCAC4 473-001074); 20 Apr 2019 3:41:46 – 3:41:47 UT (UCAC4 501-062663); 30 Jun 2019 7:37:31 – 7:37:33 (UCAC4 354-179484) and 7:47:41 – 7:46:44 (TYC 6243-00130-1).

References
Hughes, D. W. & Marsden, B. G. 2007, J. Astron. Hist. Heritage, 10, 21