In yesterday’s post on the GEODSS optical space tracking system, I frequently referred to the visual magnitudes of space objects and to the object sizes corresponding to those brightnesses. This post briefly discusses brightness and visual magnitudes and the standard procedure for assigning a size to an object of a given magnitude.
The primary sensors used for detecting and tracking objects in orbit do not directly measure the size of these objects. Instead they measure either the brightness of the sunlight reflected off an object or its radar cross section (RCS). However, it is desirable to have a simple, standardized method for converting the measured brightness or RCS of an object to a size. This conversion can be used both to provide some very basic information about the object as well as to make estimates of how small an object a given sensor can detect. It cannot, however, be used to predict with any precision the brightness of a space object at any given moment, as this can vary considerably with factors such as the object’s orientation.
The U.S. Space Surveillance network primarily tracks deep space objects (those with orbital periods greater than 225 minutes) using optical sensors that detect reflected sunlight, although radars also make important contributions. The observed brightness of a space object depends on many factors besides its size, such as its orientation, its surface composition and the viewing geometry.
The obvious choice for a reference object in converting brightness to size is a sphere of uniform surface composition, since it is independent of orientation. The size derived from using such a sphere can vary greatly from the actual size of the object, but it is still useful to have an approximate standard size model. If more accuracy and detail about the size of a given object are needed, additional measurements can sometimes be made, for example by using radars.
The brightness of objects in the visible band is typically described in terms of their apparent visual magnitude mv. The magnitude scale is logarithmic, and is defined such that a difference of five magnitudes equals a factor of 100 in brightness, with the result that one magnitude equals a factor of 2.512. Thus we can relate the relative brightness of two objects to their magnitudes by: B1/B2 = 10[(m2-m1)/2.5].
The original zero magnitude was taken to be the star Vega – dimmer objects have positive magnitudes and brighter one have negative magnitudes. The sun has a magnitude of -26.73, the brightest star (Sirius) is -1.5, the faintest star visible to the eye is about +6, and the faintest object detectable by the Hubble Space Telescope is about +30.
The visual magnitude of an object in orbit around the earth can be written as:
mv = -26.7 – 2.5log(AρF(φ)) +5.0log(r),
where A is the object’s cross-sectional area, ρ is its reflection coefficient, r is its range (in the same units as A), φ is the object’s phase angle, and F is a function that depends on the object’s shape and orientation. (The -26.7 is the sun’s apparent visual magnitude.)
The phase angle φ is the angle between the observer and the sun measured from the space object, so that a phase angle of 0 degrees means that to the observer the space object and the sun are opposite each other in sky (generally the best circumstance for visual observation) and a phase angle of 180 degrees means the sun is directly behind the object as seen by the observer.
It is useful to consider the visual magnitude of two types of reference spheres, one a specular (mirror-like) reflector and the other a diffuse reflector. For a specular sphere, F(φ) = 1/(4π) = 0.08, and for a diffuse sphere, F(φ) = (2/(3π2))[(π – φ)cosφ + sinφ].
F(φ) for these two types of spheres is shown as a function of phase angle in Figure 1. For a diffuse sphere, a phase angle of zero gives the largest value of F(φ), which is about 0.21. Actual space objects will generally have phase functions much more complex than shown in Figure 1, and the phase functions can vary significantly over time.
Figure 1. F(φ) as a function of phase angle φ for diffuse and specular spheres.
For a typical geosynchronous range of 37,700 km (the shortest distance from Ithaca, NY to geosynchronous orbit), the magnitude can then be written as:
mv = 11.2 – 2.5log(AρF(φ)), where A is in square meters.
Consider a perfectly-diffuse-reflecting, one meter diameter sphere, at the optimal phase angle of zero. Then we get:
mv = 11.2 – 2.5log(0.21A) = 11.3 – 2.5log(0.165) = 11.3 + 2.0 = 13.2.
However, space objects are not perfect reflectors. A reflectivity (or albedo) of 0.1 is typically used in estimating the size of debris objects by optical measurements, although a reflectivity of 0.2 is more typical for intact spacecraft. For example, in order to achieve consistency between different observers, the participants in the debris measurement campaigns of the Inter-Agency Space Debris Coordination Committee have agreed to use a diffuse phase function with a phase angle of zero degrees and an albedo of 0.1 in reducing their data.
If a one meter diameter sphere in geostationary orbit had an albedo of 0.1, then its visual magnitude would be mv = 15.8. Figure 2 shows the relationship between sphere diameter and visual magnitude for both diffuse and specular spheres at a range of 37,700 km, with a reflectivity = 0.1, and a phase angle of 0 degrees.
Figure 2. Visual magnitudes of spheres at 37,700 km range, albedo = 0.1, phase angle = 0˚.
Detection Against a Background
The night sky is not perfectly dark, and so the visual detection from the ground of a space object must be done against this sky background. The sky background is frequently specified as a visual magnitude per square arc-second, that is, a flux equivalent to an object of the specified magnitude from every square arc-second of the field of view (1 arcsecond ≈ 5 x10-6 radians). For example, the capabilities of the Space Surveillance Network’s Ground-based Electro-Optical Deep Space Surveillance (GEODSS) System’s visible telescopes are typically assessed against a sky background mb of 19.5mv/arcsec2. A background of 19.5mv/arcsec2 can be regarded as a nominal background condition at a good site, with 21.0 mv/arcsec2 corresponding to a dark night sky and 17.5 mv/arcsec2 to a bright night sky.
At a site with no light pollution, more than two hours away from sunrise or sunset, and looking away from the galactic plane, the brightness of the night sky is 21.8-22.0 mv. At the solar maximum, the limiting magnitude will decrease by about 0.5 mv. At the Mauna Kea observatory site (one island over and about 5,000 feet higher than the 10,000 foot Maui GEODSS site), the sky background is fainter than 20.78 magnitudes/arc-second2 50% of the time and fainter than 21.37 magnitudes/arcsecond2 20% of the time for a random target.
The presence of the Moon can significantly affect the limiting magnitude, as shown in Figure 3, which shows the effect of a half and full Moon on an otherwise mb =19.5 night sky as function of the angular distance away from the Moon.
Figure 3. The effect of the Moon on night sky brightness. The angle between the Moon and the target being observed is in the plane containing the Moon, target, and zenith. The Moon is 30 degrees above the horizon, so an angle of 60 degrees corresponds to the zenith.
Above the atmosphere, there are still sources of background including the zodiacal light and individual stars, and the background will depend on the direction the telescope is looking. This background is sometimes described using the S10 scale, in which S10 = 1 corresponds to one tenth magnitude star per square degree (which equals 27.78 mag/arc-second2). A European study of space-based detection uses 200 S10 = 22.0 mag/arc-sec2 as the stellar background and two reports on the Space-Based Visible sensor use a figure for the celestial background of 2×10-10 w/cm2/sr, which also appears to correspond to about 22.0 mag/arc-sec2.
The magnitude of an object can be written as: m = -2.5log(F/F0), where F is the flux from the object and F0 is the flux from an object of magnitude zero. For visual magnitudes, the flux is typically measured at a wavelength of 5550 angstroms (the mV band, so that m = mV), for which F0 = 3.55×10-9 erg/s-cm2-angstrom. Thus the visual magnitude can also be written as: m = -2.5log(F) – 21.12, with F in erg/s-cm2-angstrom. The constant in the second term will vary slightly according to the wavelength band over which the measurement is taken. The value given here is for the V band, centered at 5500 angstroms.
 J. G. Williams and G. A. McCue, “An Analysis of Satellite Optical Characteristics Data,” Planetary and Space Science Vol. 14 (1966) pp. 839-847.
 Williams and McCue, “An Analysis of Satellite Optical Characteristics Data.”
 John Africano, Paul Kervin, Doyle Hall, Paul Sydney, John Ross, Tamara Payne, Steve Gregory, Kira Jorgensen, Kandy Jarvis, Tracy Parr-Thumm, Gene Stansbery, and Ed Barker, “Understanding Photometric Phase Angle Corrections,” Proceedings of the Fourth European Conference on Space Debris (ESA SP-587), Darmstadt, Germany, April 18-20, 2005.
 Since most optical telescopes used for space surveillance are located closer to the equator, a given object will be seen by them as slightly brighter (by as much as 0.1 magnitude).
 P.W. Kervin, J.L. Africano, P.F. Sidney, and D. Hall, “Small Satellite Characterization Techniques Applied to Orbital Debris,” Advances in Space Research, vol. 35 (2005), pp. 1214-1225.
 Africano, et. al.
 For example, in John R. Tower, Pradyumna K. Swain, Fu-Lung Hsueh, Robin M. Dawson, Peter A. Levine, Grazyna M. Meray, James T. Andrews, Verne L. Frantz, Mark S. Grygon, Michael A. Reale and Thomas M. Sudol, “Large Format backside Illuminated CCD Imager for Space Surveillance,” IEEE Transactions on Electron Devices, Vol. 50, No. 1 (January 2003), pp. 218-224.
 Douglas W. Streuber, Jr. and Elliot I. Bailis, “Imaging Charge-Coupled Devices for Deep-Space Surveillance,” Smart Sensors II, Proc. Soc. Photo-opt. Instr. Eng. 252 (1980), pp. 59-69.
 Kevin Krisciunas and Bradley E. Schaefer, “A Model of the Brightness of Moonlight,” Publications of the Astronomical Society of the Pacific, Vol. 103, pp. 1033-1039 (September, 1991).
 The effects of the moon were calculated using an expression given by Krisciunas and Schaefer.
 J. Bendisch, J.P. Hoffmann, R. Liebscher, and F. Rollenhagen, “Detection of Space Debris by the Use of Space Based Optical Sensors, Proceedings of the First European Conference on Space Debris, ESA SD-01, July 1993, pp. 91-97. C. P. Dyjak and D.C. Harrison, “Space-Based Visible Experiment,” Surveillance Technologies, Society of Photo-Optical Engineers, SPIE Vol. 1479 (1991), pp. 42-56; David C. Harrison and Joseph C. Chow, “The Space-Based Visible Sensor,” Johns Hopkins APL Technical Digest, Vol. 17, No. 2 (1996), pp. 226-234. The meaning of this stated flux is not entirely clear (to me), but assuming that the energy spectrum is the same as the sun, then:2×10-10 w/cm2/sr = 4.7 x 10-21 w/cm2/arcsec2. Solar flux at earth orbit = 0.137 w/cm2, so the flux per acr-second2 for the target is smaller than solar flux by 4.7 x 10-21/0.137 = 3.4×10-20 = 48.7 magnitudes. Thus target magnitude is 47.7 -26.7 = 22.0 mV.