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1 - Light and Magnitude

Published online by Cambridge University Press:  01 February 2017

Thomas H. Burbine
Affiliation:
Mount Holyoke College, Massachusetts

Summary

Type
Chapter
Information
Asteroids
Astronomical and Geological Bodies
, pp. 1 - 28
Publisher: Cambridge University Press
Print publication year: 2016

1.1 Light

Except in extremely rare circumstances, minor planets are too faint to be seen with the naked eye. So to study a minor planet, you need a telescope. A telescope gathers and focuses light from the body, allowing objects too faint to be seen with the eye to be observed and studied. The telescope needs a detector to capture the light.

1.1.1 Electromagnetic Spectrum

Light usually refers to the visible portion of the electromagnetic spectrum that can be seen with the naked eye. All parts of the electromagnetic spectrum have been used to study asteroids; however, the visible is the one that is most often used since the radiation from the Sun peaks and the atmosphere is relatively transparent in this wavelength region. The light that allows us to see asteroids originated from the Sun. Light from the Sun is reflected by the asteroid and then detected by the observer.

The regions (Table 1.1 and Figure 1.1) of the electromagnetic spectrum (in order of decreasing wavelength) are radio, microwave, infrared, visible, ultraviolet, X-rays, and gamma rays. Visible light is usually broken up (in order of decreasing wavelength) into red, orange, yellow, green, blue, indigo, and violet. This sequence is often remembered as ROYGBIV. In a vacuum, light travels at a speed (c) of 299792458 m/s (usually written as 3 × 108 m/s).

Table 1.1 The wavelength regions for different parts of the electromagnetic spectrum

Region Wavelength range (meters)
radio > 1
microwave 1 × 10−3 − 1
infrared 7.0 × 10−7 − 1 × 10−3
visible 4.0 × 10−7 − 7.0 × 10−7
ultraviolet 1 × 10−8 − 4.0 × 10−7
X-ray 1 × 10−11 − 1 × 10−8
gamma ray < 1 × 10−11

Figure 1.1 The wavelengths (m) and frequencies (Hz) of different regions of the electromagnetic spectrum. The wavelength regions that penetrate the Earth’s atmosphere, a list of different bodies with similar sizes to the wavelengths of each region, and the temperatures of different bodies emitting black body radiation primarily at that wavelength.

Credit: NASA.

Light can be characterized as both a wave and a particle. Particles of light are called photons and are massless with no charge. Since light also acts like a wave, its properties can also be characterized by both its frequency (ν) and wavelength (λ). The units of frequency are usually given as hertz (Hz) (cycles/second) while the units of wavelength vary [e.g., meters (m), microns (μm) (1 × 10−6 m), nanometers (nm) (1 × 10−9 m), angstroms (Å) (1 × 10−10 m)]. The frequency and wavelength of a photon are inversely correlated and obey the formula c = νλ in a perfect vacuum. The energy of a photon is where h is Planck’s constant (6.626 × 10−34 Js). Radio waves have extremely long wavelengths, small frequencies, and small energies while gamma-ray photons have extremely small wavelengths, large frequencies, and large energies.

The wavelength, frequency, and energy of a photon are all interrelated, and knowing one of these values allows you to calculate the other two quantities. If you know the wavelength of a photon, you can calculate its frequency and energy. If you know its frequency, you can calculate its wavelength and energy. And if you know its energy, you can calculate its wavelength and frequency. So knowing the value of either the wavelength, frequency, or energy of the light used to study an asteroid, you are able to uniquely define its characteristics.

1.1.2 Atmosphere

All radiation does not pass equally through the Earth’s atmosphere. Visible light, short wavelength radio waves (including microwaves), and some infrared wavelengths pass relatively unimpeded through the atmosphere while X-rays, gamma rays, and some ultraviolet wavelengths are absorbed (Figure 1.2).

Figure 1.2 The transmittance of electromagnetic radiation through the atmosphere. Microwaves are included as part of the short wavelength radio waves. Atmospheric opacity is the amount of light absorbed by the atmosphere with 100% opacity indicating total absorption and 0% opacity indicating total transmission.

Credit: NASA.

There is a large atmospheric window (Figure 1.2) in the far ultraviolet, visible, and near-infrared (~0.3 to ~2.4 μm) where most wavelengths of light can pass relatively easily through the atmosphere (Gupta, Reference Gupta2003). From ~2.4 to ~3.5 μm, H2O, which is a relatively small constituent of the atmosphere, has a number of strong absorption bands that severely affect the transmission of these wavelengths of light through the atmosphere. Ultraviolet photons shortward of ~0.3 μm are significantly absorbed by the atmosphere (particularly by the ozone layer).

1.2 Black Bodies

The Sun, the source of the asteroid’s reflected light, is assumed to act like a black body. A black body is a theoretical object that absorbs all radiation that strikes it and emits radiation at all wavelengths. This emission is due to the thermal motion of charged particles in the material and is often called black body or thermal radiation.

Radiation from a star can be modeled as if the radiation was emitted from a black body. However, stars are not “perfect” black bodies. The temperature of a star will vary across its surface, so different regions will have different black body emission so the black body curve will not be the ideal black body curve for a body at one temperature. Also, atoms or ions in the atmosphere of the star absorb radiation being emitted. These absorption lines are apparent in measured stellar spectra (including the Sun) due to the absorption of light by different atoms.

Plots of the emitted intensity of radiation versus wavelength for theoretical black bodies have characteristic shapes that are just a function of temperature and wavelength (Figure 1.3). Using classical physics, the Rayleigh–Jeans law, proposed by Lord Rayleigh (1842–1919) and James Jeans (1877–1946), modeled black body radiation as proportional to T/λ4; however, this law inaccurately predicted that emitted radiation would continually increase with decreasing wavelength. The Planck function, proposed by Max Planck (1858–1947) using quantum theory, accurately predicts the spectral distribution of electromagnetic radiation from a black body and can be written as
Bλ(λ,T)=2hc2λ51ehc/λkT1 (1.1)

where the emitted intensity is given in terms of Wm−3sr−1 if the temperature is in kelvin (K) and the wavelength is in meters. The Boltzmann constant (k) is 1.381 × 10−23 JK−1. The Boltzmann constant relates the kinetic energy and temperature of a molecule in an ideal gas. A solid angle is a measure of the fraction of the surface of a sphere that a body covers as seen from the center of the sphere. A solid angle is given in steradians (sr) with a sphere subtending 4π steradians.

Figure 1.3 Black body curves for bodies at temperatures at 5000 K, 4000 K, and 3000 K plus the classical theory prediction for a body at 5000 K.

Credit: Dark Kule.
If the radiation is emitted isotropically (independent of direction), the Planck function can also be written (Delbó and Harris, Reference Delbó and Harris2002) as
B(λ,T)=2πhc2λ51ehc/λkT1 (1.2)

which has units of Wm−3. This formula gives the emitted flux from a black body. A flux is the flow of some quantity per unit area.

The radiant flux (Fe) (or total power per unit area) emitted by the black body is a function of the fourth power of the temperature (T). This equation is called the Stefan–Boltzmann law and can be written as
Fe4πR2=σT4. (1.3)

The 4πR2 term is the total surface area of the body with a radius R. The Stefan–Boltzmann constant (σ) (5.67 × 10−8 Wm−2K−4) is a constant of proportionality that relates the intensity of the emitted radiation per unit area to the fourth power of the temperature. As the temperature increases, the amount of energy emitted per second will dramatically increase. If you double the surface temperature of a body, the amount of energy emitted per second by the body will increase by a factor of 16 times.

Asteroids reflect and also emit radiation. However, gray bodies, such as asteroids, do not emit all the energy that strikes them. The Stefan–Boltzmann Law for gray bodies is written as
Fe4πR2=εσT4. (1.4)

The emissivity(ε) term is the efficiency with which a body radiates thermal radiation. The emissivity is 1 for a “perfect” black body but will be less than 1 for any type of gray body. For asteroids, the emissivity is often assumed to be 0.9.

The peak of the black body curve (where the object emits most of its light) is inversely proportional to temperature with cooler black bodies peaking at longer wavelengths than hotter bodies (Figure 1.1), which is called Wien’s law. Wien’s law is given by the equation
λmax=2.898×103mKT. (1.5)

The calculated wavelength is in meters and the temperature is in kelvin. Radiation from the Sun peaks in the visible wavelength region while radiation from asteroids peaks in the infrared since they have much cooler surface temperatures.

Radiation detected from an asteroid is usually dominated by reflected light at wavelengths less than 2.5 µm while emitted thermal radiation usually dominates past 5 µm (Kim et al., Reference Kim, Lee, Nakagawa and Hasegawa2003). The temperature of an asteroid will be primarily be a function of its distance from the Sun and how the dark the surface is. Objects further from the Sun are heated by less solar radiation than those that are closer and will have cooler surface temperatures. The flux of solar radiation striking an asteroid follows an inverse square law where the flux is proportional to the inverse of the distance squared from the Sun. Darker surfaces also tend to absorb more radiation than lighter surfaces. So darker asteroids will tend to be hotter than brighter surfaces at the same distance from the Sun.

For asteroids, the transition between reflected and thermal radiation usually occurs between 2.5 and 5 µm and is a function of the surface temperature of the body. The exception are dark near-Earth asteroids, which have “thermal tails” between ~2 and ~2.5 µm. These thermal tails are due to these bodies having a measurable blackbody flux in the ~2–2.5 µm wavelength region due to their relatively high surface temperatures.

1.3 Albedo

Albedo is a quantity that defines how light or how dark a surface is. Values for albedos tend to range from 0 (perfectly absorbing) to 1 (perfectly reflecting). Albedo is the fraction of radiation reflected from a surface. However, there are many different ways to define the albedo of a surface.

Bond albedo (A) is the fraction of all the total radiation at all wavelengths and at all solar phase angles that is scattered from a surface. The solar phase angle (Figure 1.4) is the angle between the light incident on a body from the Sun and the light reflected from the body and detected by an observer. Since the Bond albedo accounts for all scattered light, it varies from 0 to 1. The visual Bond albedo (Av) only accounts for radiation scattered in the visible. The Bond albedo and the visual Bond albedo are often assumed to be similar in value since the Sun’s flux peaks in the visible.

Figure 1.4 Illustration of the phase angle for an observer on the Earth observing a planet or a minor planet.

The geometric albedo (p) is the ratio of the brightness of a body at zero phase angle relative to a theoretical flat and fully reflecting disk with the same cross section that reflects light diffusively (scatters light equally in all directions). Objects with diffuse reflectance are said to have a Lambertian reflectance. The geometric albedo can be greater than 1 if light from the surface is preferentially reflected backwards towards the observer. The visual geometric albedo (pv) is the geometric albedo that accounts for only visible light.

1.4 Temperature

The shape and intensity of the emitted radiation from an asteroid will be a function of its average surface temperature. This temperature is often called the effective temperature. The effective temperature of an asteroid can be estimated by assuming that it is a black body and then equating the emitted power and the absorbed power. The absorbed flux (Fa) in watts will be
Fa=(1A)πR2[S0(r2/AU2)] (1.6)

where A is the Bond albedo, S0 is the solar constant (~1366 W/m2), and r is the heliocentric distance in AU. The solar constant is the average solar flux at the top of the Earth’s atmosphere and is not actually a constant because it slightly varies over time. The S0(r2/AU2) term adjusts the solar flux for the asteroid’s distance from the Sun according to the inverse square law. The πR2 term is the cross-sectional area of the asteroid that is absorbing the radiation. One astronomical unit (AU) is approximately the mean of the average distance between the Earth and the Sun and is now defined by the IAU as 149597870700 meters.

If you equate the emitted flux with the absorbed flux, the resulting formula will be
4πR2εσT4=(1A)πR2S0[r2/(AU)2]. (1.7)

Solving for T4 produces the formula

T4=(1A)S04εσ[r2/(AU)2]. (1.8)

Then solving for the effective temperature produces the formula

T=((1A)S04εσ[r2/(AU)2])1/4 (1.9)

for the average temperature. So by increasing the albedo so there is less radiation absorbed, the average surface temperature of a body will decrease. Also if the distance from the Sun is increased for a body, its average surface temperature will become lower.

Example 1.1

A near-Earth asteroid has an effective surface temperature of ~270 K and was observed at a distance from the Sun of 1 AU. What is this object’s Bond albedo?

Solving for the Bond albedo using Equation 1.8 produces the formula
A=14εσT4[r2/(AU)2]S0. (1.10)

The emissivity (ε) is assumed to be 0.9. Substituting the values in the equation then produces the formula

A=14(0.9)(5.67×108)(270)4(1366) (1.11)

with all the units cancelling. The calculated Bond albedo will then be 0.21.

1.5 Telescopes

Telescopes gather light from astronomical bodies and focus it to produce an image. Telescopes that just use lenses to produce an image are called refracting telescopes, while those that use mirrors are called reflecting telescopes. The first telescopes were refractors and were pioneered by Galileo Galilei (1564–1642) starting in 1609. The Galilean telescope has a convex primary lens that collects the parallel rays of light from an object and focuses it and a concave eyepice that intercepts the light from the lens and renders the rays parallel again. This design was updated by Johannes Kepler (1571–1630) who used a convex lens as the eyepiece. This design was called the Keplerian telescope. Refracting telescopes were the primary research telescopes until the early twentieth century.

The size of a telescope is given by its aperture size, which is the diameter of its main optical component (lens or mirror). The larger the aperture, the brighter and sharper the image tends to be since more photons tend to be collected and focused. However as the size of the lens increases for refracting telescopes, the focal length also tends to increase, which increases the size of the telescope tube. The weight of the lens also increases, which increases the weight of the counterweight needed to balance the telescope.

Telescopes that are used today for research are almost all reflecting telescopes. Reflecting telescopes use a curved mirror (called the primary mirror) or a series of curved mirrors to focus an image. Isaac Newton (1642–1727) built the first reflecting telescope in 1668. Besides having smaller sizes, reflecting telescopes have a number of other advantages compared to refracting telescopes. It is easier to make a high-quality mirror than a lens. Large lenses are much heavier than large mirrors. Lenses are also affected by chromatic aberration where all wavelengths of light cannot be focused to the same point reflecting telescopes are not affected by chromatic aberration. There are a number of different types of designs for reflecting telescopes.

Mirrors can be spherical and have a curved shape that is either convex or concave. However unless corrected for, simple spherical mirrors suffer from spherical aberration where light reflected from the edge of the mirror is focused at a slightly different point than light reflecting from the center of the mirror. Mirrors can be made with hyperbolic or parabolic shapes to eliminate spherical aberration since they focus all light to a single point. Hyperbolic mirrors tend to be more difficult and expensive to make than parabolic mirrors. However, the disadvantage of parabolic mirrors is that they can distort point sources, which is called a coma since it resembles the tail of a comet.

The commonest type of reflecting telescope is the Cassegrain reflector. Invented in the 1600s, a Cassegrain reflector uses two mirrors: a primary concave mirror and a secondary convex mirror. There is an opening in the primary that allows light to pass to the eyepiece. The classical Cassegrain design has a parabolic primary mirror and a hyperbolic secondary mirror.

The most commonly used type of Cassegrain reflector for research is the Ritchey–Chrétien Telescope (RCT), which was invented in the early twentieth century by George Ritchey (1864–1945) and Henri Chrétien (1879–1956). A Ritchey–Chrétien reflector has a hyperbolic primary mirror and a hyperbolic secondary mirror. The advantage of using hyperbolic mirrors is that the images are relatively free of coma and spherical aberration. The Hubble Space Telescopes and the Keck Telescope all are Ritchey–Chrétien telescopes.

Catadioptric telescopes use both refraction (lens) and reflection (mirror) in their optical systems. Schmidt telescopes (often called Schmidt cameras) have a spherical primary mirror and a aspherical correcting lens. This type of telescope was invented by Bernhard Schmidt (1879–1935) in 1930. The main advantage of a Schmidt telescope is that it has a relatively wide field of view.

1.6 Detectors

Detectors are used to record an image produced by a telescope. An image of an astronomical body allows you to determine the position of a body relative to the stars. A series of these images allows you to determine whether the body is moving relative to the stars. This movement is called proper motion. If it is moving, the body lies within our Solar System. To identify a body as a planetary body, the same region of the sky must be observed over a period of time. Any object that moves relative to the stars over a time period of minutes, days, or hours is not a star but a “planet” or comet. The faster the body moves, the closer the object is to Earth. Besides discovering and determining the orbits of planetary bodies, images produced by a detector also can be used to determine how bright an object is in the sky and how this brightness (magnitude) changes over time.

An important quantity for judging how well a detector works is its quantum efficiency (QE). Quantum efficiency is how efficient a detector is at converting photons to measurable electrons. The higher the QE, the better the detector. The approximate QEs for a number of detectors are given in Table 1.2. As can be seen in the table, charge-coupled devices (CCDs) are the most efficient detectors that an astronomer can use.

Table 1.2 Quantum efficiencies for a number of detectors

Detector Quantum efficiency Reference
human eye ~10% Rose and Weimer (Reference Rose and Weimer1989)
photographic plate ~1–5% Irwin (Reference Irwin, Rodriquez Espinosa, Herrero and Sánchez1997)
photomultiplier tube ~20% Duerbeck (Reference Duerbeck and Roth2009)
charge-coupled device (CCD) ~90% Howell (Reference Howell2006)
infrared detector >70–80% Rieke (Reference Rieke2007)

1.6.1 Eyes

For the longest time, astronomical observations were only recorded by drawing on paper what you saw through the telescope. This changed in the middle of the nineteenth century when photographic plates started to be used. The first photograph of an astronomical object was of the Moon in 1840. The first asteroid photographed was (80) Sappho in 1896 by Isaac Roberts (1829–1904), while the first asteroid discovered by astrophotography was (323) Brucia in 1891 by Max Wolf (1863–1932). The first asteroid discovery survey is also attributed to Max Wolf who discovered over 200 objects using astrophotography.

1.6.2 Photographic Plates

Astrophotography works by exposing a photographic plate to light. Photographic plates are glass plates that are coated with a mixture of silver halide (e.g., AgBr) crystals in a thin layer of gelatin, which is called an emulsion. The gelatin keeps the grains well separated and fixed in place (Smith, Reference Smith1995). Plates are used since they do not wrinkle like film. Photography works due to a photochemical reaction between the incident light and the emulsion that the photons strike. When a photon strikes a silver halide grain, an electron is released. These electrons can meet some relatively rare silver ions that are also migrating through the crystal lattice of the grain (Smith, Reference Smith1995). So when these silver halide grains are exposed to light, free silver atoms can be produced. One produced silver atom will become ionized again; however, if several silver atoms come together, they can become a stable silver cluster (Appenzeller, Reference Appenzeller2012). These free silver atoms are much smaller the silver halide grains.

A chemical reducing agent is then used to develop the image. This developing agent uses these small silver ions as catalysts to convert the entire silver halide grain to silver. The developing needs to be stopped before all grains are turned into silver, which will ultimately happen at a much slower rate for silver halide grains that do not contain free silver. A fixing agent is then used to removed the unexposed silver halide. The image that is ultimately produced is a negative with the areas that appear dark being the regions on the plate that have absorbed photons. The brightest objects will tend to have the largest images on the photographic plate.

One advantage of plates over other types of detectors is that plates can have extremely large fields of views. Schmidt telescopes use photographic plates that could cover areas of ~6° by ~6° in the sky. One disadvantage is that most of the released electrons do not survive long enough to produce a silver atom since most are absorbed by either a positive hole or a halogen atom (Smith, Reference Smith1995). Therefore, photographic plates are not very efficient in detecting photons. Another disadvantage is that the strengths of images on a plate are not all linearly related to the number of photons that strike those particular areas.

1.6.3 Photomultiplier Tubes

In the 1940s, photomultiplier tubes started to be used to measure the brightnesses of celstial objects. Photons strike a photocathode (negatively charged electrode), which causes an electron to be dislodged due to the photelectric effect. The photoelectric effect is the ejection of electrons from a metal by the absorption of photons. These electrons are then accelerated down the tube using a seried of positively charged electrodes in a vacuum (usually called dynodes), which are the electron multiplier. The positive voltage increases the energy of the electrons and when the electrons strike a dynode, more electrons are released. Each dynode has a higher voltage. More and more electrons are produced during this process, which produces a current that is measured by the final electrode. Photomultipliers have a linear response since the signal is directly proportional to the measured flux of photons. Photomultiplier tubes have a much higher quantum efficiency than photographic plates. However, a single photomultplier tube can not be used to image a region of the sky since it can only measure the intensity of light from a single point source. To image a large region of the sky, an array of photomultiplier tubes must be used.

1.6.4 CCDs

In the 1980s, CCDs (charge-coupled devices) started to be used and became the primary way to record astronomical observations. CCDs are electronic detectors (Figure 1.5) that are broken up into millions of light-sensitive elements called pixels. Each pixel is composed of an epitaxial layer of silicon doped with a number of elements over a substrate of silicon. When photons strike the epitaxial layer, free electrons are produced through the photoelectric effect. For each pixel, a positively charged gate keeps the electrons from returning to the newly created electron holes. The number of electrons released is directly proportional to the number of photons that strike the pixel.

Figure 1.5 Image of a charge-coupled device (CCD).

Credit: NASA.

The information (charge or number of electrons) stored in each pixel needs to be read out by circuitry at the edges of the CCD. To transfer a charge to a neighboring pixel, the voltage of the neighboring pixel needs to be increased, which allows the charge to be transferred to that pixel. These charges are continually transferred until all the charges are read out. The measured charge for each pixel is then converted into a voltage, which can then be converted into counts (also called “Analog-to-Digital Units” or ADUs) measured by each pixel. The number of counts measured for each pixel is directly related to the number of electrons that were stored by each pixel. The electronics keep track of the counts for each pixel.

CCDs are much more sensitive to light (quantum efficiencies of ~90%) than photographic plates and photomultiplier tubes. Due to CCDs having a much higher QE, the exposure times needed to acquire similar images are much shorter using a CCD. CCDs also acquire images in a digital format that can be easily analyzed using computers. CCDs can be used over and over while photographic plates can only be used once. However, CCDs can only detect light at wavelengths as long as ~1.1 μm.

1.6.5 Infrared Detectors

Infrared detectors primarily work through photoconductivity (Rogalski, Reference Rogalski2002; Rieke, Reference Rieke2007). Photoconductivity is where the absorption of radiation increases the electrical conductivity of a material. Photons striking a semiconductor in an electric field free electrons that travel toward the electrodes and produce a current. Changes in the electrical current are then measured for each pixel of the array. The electrons are freed when the energy of the photons is equal to or larger than the binding energy of the electrons in the semiconductor crystal. Infrared detectors tend to be cooled to extremely low temperatures to reduce thermal electrical noise. InSb detectors are commonly used to detect infrared radiation between ~1 and ~5.6 μm. The semiconductors are made out of materials such as InSb and HgCdTe.

1.7 Observing Different Wavelengths of Light

To detect photons at specific wavelengths, filters (e.g., Bessell, Reference Bessell1990, Reference Bessell2005) or a spectrograph must be used. Filters transmit photons of light in a particular wavelength region (called a bandpass), while spectrographs disperse light into its component wavelengths. Filters tend to be used with photomultiplier tubes, CCDs, and infrared detectors, while spectrographs tend to be used with CCDs and infrared detectors.

1.7.1 Filters

Every filter has a bandwidth, which is the wavelength range in which the filter transmits light. Broadband filters have a bandwidth of less than 0.1 μm, intermediate band filters have a bandwidth between 0.007 and 0.04 μm, and narrow band filters have a bandwidth of less than 0.007 μm.

The UBV photometric system uses U (Ultraviolet) (mean wavelength of ~0.3663 μm), B (Blue) (~0.4361 μm), and V (Visual) (~0.5448 μm) filters. The UBV system is often called the Johnson or Johnson–Morgan system after the astronomers Harold Johnson (1921–1980) and William Morgan (1906–1994) who introduced this system (Johnson and Morgan, Reference Johnson and Morgan1953). The UBVRI photometric system adds R (Red) (~0.6407 μm) and I (Infrared) (~0.7980 μm) filters. The UBVRI system is often called the Johnson–Cousins system after Alan Cousins (1903–2001) who introduced the R and I filters. The JHK system uses filters at J (~1.2 μm), H (~1.6 μm), and K (~2.2 μm). These filter systems were developed for observing stars but have been used extensively to study asteroids.

1.7.2 Spectrographs

Spectrographs disperse light onto a detector using a prism, a diffraction grating, or a grism (combination of a grating and a prism). Prisms tend to be triangularly shaped transparent material. A diffraction grating is a series of equally spaced slits on an opaque screen. However, prisms and gratings can introduce chromatic aberration. A grating and prism in concert (grism) will eliminate chromatic aberration since all light will be focused to the same focal point. The spectrograph allows the intensity of light at different wavelengths to be measured by the detector.

1.8 Observatories

Observatories house the telescope and detector and can be located on the ground, in space, or in the air (plane or balloon). The observatory also houses the computers and electronics that control the telescope. Observatories range from simple sheds to elaborate facilities.

Locating an observatory on a mountain enhances the seeing since there is less atmosphere that the light passes through, which resulted in less distorted images. Since infrared photons tend to be absorbed by the atmosphere before they reach the ground (Figure 1.2), infrared observatories also tend to be located on mountains. Mountains also tend to have very dark skies to observe, which allows for excellent seeing. For example, Mauna Kea in Hawaii, the home of a number of observatories, including the W. M. Keck Observatory and the NASA (National Aeronautics and Space Administration) Infrared Telescope Facility (IRTF), is at a height of ~4200 meters above sea level. Palomar Observatory on Palomar Mountain in California is located at a height of ~1700 meters above sea level.

1.8.1 Space Telescopes

Telescopes that study asteroids are located in space for a variety of reasons. Space-based telescopes are not affected by atmospheric turbulence. Space-based telescopes can observe asteroids in the ultraviolet and infrared since the Earth’s atmosphere significantly absorbs ultraviolet and infrared light. X-ray and gamma rays are also absorbed by the Earth’s atmosphere; however, asteroids do not have high enough fluxes of X-rays and gamma rays to be observable by space-based telescopes. Only orbiting spacecraft can detect significant fluxes of X-ray and gamma-ray photons from asteroids.

1.8.2 Hubble Space Telescope

The most famous space telescope is the Hubble Space Telescope (HST) (Figure 1.6) (e.g., Baker, Reference Baker2015). Hubble has a 2.4-meter mirror and covers wavelengths from the ultraviolet to the near-infrared. Launched in 1990, HST has made significant minor planet discoveries (e.g., discovering four of Pluto’s moons) due to its unprecedented resolution. Initially, HST had five instruments. The initial instruments were a Wide Field and Planetary Camera (WFPC), Goddard High Resolution Spectrograph (GHRS), High Speed Photometer (HSP), Faint Object Camera (FOC), and the Faint Object Spectrograph (FOS). When first launched, its images were slightly out of focus and blurry due to its primary mirror’s curvature being slightly off. The primary mirror was too flat at its edges by 2.2 μm. In 1993, a servicing mission installed the Corrective Optics Space Telescope Axial Replacement (COSTAR) to correct for the primary mirror’s aberration for the GHRS, FOC, and FOS. To include this instrument on HST, the HSP needed to be removed. During the same mission, the WFPC was also replaced with Wide Field and Planetary Camera 2 (WFPC2), which had its own optical corrective optics. The Near Infrared Camera and Multi-Object Spectrometer (NICMOS) replaced the GHRS in 1997. NICMOS is currently not working due to problems with the cooling system. The WFPC2 was then replaced with the Wide Field Camera 3 (WFC3) in 2009. The WFC3 is used for imaging. The WFC3 has two CCDs that detect radiation in the ultraviolet and visible and also a separate infrared detector.

Figure 1.6 The Hubble Space Telescope as seen from the departing Space Shuttle Atlantis in 2009. The Hubble Space Telescope is 13.2 meters long.

Credit: NASA.

1.8.3 Wide-field Infrared Survey Explorer

The Wide-field Infrared Survey Explorer (WISE) (Figure 1.7) was launched in December 2009. WISE initially observed the sky using four bands (3.4, 4.6, 12, 22 μm) in the infrared. Each of the four infrared detectors contains one million pixels. The hydrogen coolant became depleted after 10 months, leaving only the 2.4 and 4.6 μm bands operable since the detector for the longer wavelengths could only obtain data at the original sensitivity at extremely low temperatures. After the hydrogen coolant was depleted, the mission was renamed NEOWISE for Near-Earth Object (NEO) WISE. NEOWISE is the asteroid detecting survey of the WISE telescope. NEOWISE was first extended for one month and then for an additional three months (October 2010 to February 2011). The WISE spacecraft transmitter was turned off, but the telescope was reactivated in September 2013 with NEOWISE observations restarting in December 2013.

Figure 1.7 The WISE spacecraft at Vandenberg Air Force Base. The WISE spacecraft is 2.85 meters long.

Credit: NASA.

One of the goals of the WISE/NEOWISE mission was to determine visual albedos and diameters of asteroids using infrared measurements (Sections 5.1.15.1.5). The WISE/NEOWISE mission has made ~2.3 million observations of ~159,000 objects, including ~34,500 discoveries. Over 200 near-Earth objects and over 20 comets were discovered.

1.9 Magnitude

The visual brightnesses of asteroids (and stars and galaxies) are usually given in terms of magnitude. The magnitude system is based on the ancient Greek system developed by Hipparchus (c.190–c.120 BC), which divided stars into six magnitude groups. The brightest stars were called first magnitude stars. The second brightest stars were called second magnitude stars. Stars that could barely be seen by the ancient Greeks were called sixth magnitude stars. When Galileo first used the telescope, he was able to see stars that were fainter than sixth magnitude objects. He called the brightest of these objects seventh magnitude stars.

In the 1800s, photometric measurements of artificial “stars” that matched real stars in brightness showed that first magnitude stars were approximately 100 times brighter than a sixth magnitude star. These artificial “stars” were light that was projected into a telescope’s field of view to match a real star in brightness. In 1856, Norman Pogson (1829–1891) defined that a 5 magnitude difference exactly equaled a 100 times difference in brightness. Therefore, a +1 magnitude star will be 2.512 times brighter than a +2 magnitude star since the fifth root of 100 is equal to 2.512. The formula for the ratio of the fluxes is then
FFref=2.512(mmref) (1.12)

where F is the measured flux (or brightness) of the object, Fref is the reference flux, m is the magnitude of your object, and mref is the reference magnitude, The reference flux is the flux from a star with a known magnitude, called a standard star. Flux is given in units of power (joules/second or watts) per unit area.

By taking the logarithm of both sides and rearranging the variables, the formula for calculating the magnitude (also called apparent magnitude) of an object becomes
mmref=2.5log10(FFref). (1.13)

So by determining the ratio of the flux of an unknown object with the flux of a star with a known magnitude, the magnitude of any object can be determined. The relative flux can be written as

FFref=100.4(mmref). (1.14)

The brightest objects have very negative magnitudes, while the faintest bodies have very positive magnitudes. The apparent magnitude of the Sun is −26.74 while the maximum apparent magnitude of Pluto is +13.65.

The reason that asteroids were first discovered with a telescope is that almost all asteroids are too faint to be seen with the naked eye, which only can see down to a magnitude of +6 in an extremely dark location. The most notable exception is (4) Vesta, which can be as bright as +5.1 at opposition; however, this apparent magnitude is still extremely difficult to see with the naked eye. Opposition is when the Earth and an object are approximately in a straight line as seen from the Sun. The object and the Sun are in “opposite” sides of the sky as seen from Earth. The body is roughly closest to the Earth during opposition and therefore should appear brightest. Near-Earth asteroid (99942) Apophis is predicted to have a magnitude of approximately +3 during its close approach in 2029 (Giorgini et al., Reference Giorgini, Benner, Ostro, Nolan and Busch2008).

Because the amount of light reaching a body is an inverse function of the distance (d) squared and the light must travel to a minor planet and then be reflected back to Earth, the brightness of a distant minor planet from Earth will vary approximately as the inverse function of the distance of the body from the Sun to the fourth power (Ortiz et al., Reference Ortiz, Moreno, Molina, Sanz and Gutiérrez2007; Jewitt, Reference Jewitt2010).

Example 1.2

If Pluto (Body 1) has an apparent magnitude of 13.7 at a distance of ~30 AU from the Sun, what would be the estimate of the apparent magnitude of a Pluto-like body (Body 2) at ~120 AU?

Since the brightness of a distant minor planet varies approximately as the inverse function of the distance of the body from the Sun to the fourth power, the ratio of the brightnesses of the two distant minor planets (F2F1) will be
F2F1(d1d2)4 (1.15)

where d1 is the distance of Body 1 to the Sun and d2 is the distance of Body 2 to the Sun. The formula for the relative brightnesses of Pluto at 30 AU and the Pluto-like body at 120 AU becomes

F2F1(30120)4=(14)4=1256. (1.16)

So the Pluto-like body will be approximately 256 times fainter than Pluto. The formula for the magnitude difference (Equation 1.13) will become

m2m1=2.5log10(F2F1) (1.17)

where F1 = Fref and m1 = mref. Substituting the values in the equation, the formula then becomes

m213.7=2.5log10(1256), (1.18)

which becomes

m213.7=6. (1.19)

The apparent magnitude of the Pluto-like body will then be +19.7.

1.9.1 Instrumental Magnitude

However when observing astronomical objects, what you actually measure is the counts from the object, not the actual flux. When you observe an object, what is first calculated is an instrumental magnitude (minst). The instrumental magnitude is not standardized to the magnitude of an astronomical body. The instrumental magnitude can then be calculated from the formula (Palmer and Davanhall, Reference Palmer and Davenhall2001)
minst=2.5log((i=1nCi)nCskyt×secondscounts)+offset (1.20)

where Ci is the counts in the ith pixel inside the aperture (a circular area), n is the number of pixels in the aperture, Csky is the average count in a background sky pixel, t is the integration time in seconds, and offset is an arbitrary constant. This constant is usually a relatively large positive number, which will then produce a positive instrumental magnitude.

If a star has a known magnitude and is not variable, the offset can be chosen so that it produces an instrumental magnitude that is the same as the actual magnitude. All asteroids have magnitudes that will vary since they are not perfectly spherical and do not have the same albedo. If the image has a number of stars with known magnitudes and are not variable, the offset will be an average value, with an uncertainty, that best determines the actual magnitude of all stars. This offset will be different for every image.

1.9.2 Color Index

The magnitude of a minor planet can be measured at different wavelengths by using different filters (e.g., Wood and Kuiper, Reference Wood and Kuiper1963). These observations are often presented as color indices by subtracting the magnitude measured using one filter from another. The magnitude at the longer wavelength is subtracted from the magnitude at the shorter wavelength to determine a color index (e.g., U−B, B−V). A positive color index indicates that the object is brighter at the longer wavelength, while a negative color index indicates that an object is fainter at the longer wavelength. Color indices for asteroids show a range values (e.g., Bowell and Lumme, Reference Bowell, Lumme and Gehrels1979; Hollis, Reference Hollis1994), indicating that different asteroids can reflect light differently than other asteroids.

1.9.3 Photometry

Photometry is the study of how the brightness of a surface depends on the illumination and viewing geometry. The importance (Li et al., Reference Li, Helfenstein, Buratti, Takir, Clark, Michel, DeMeo and Bottke2015) of photometric studies is that they give insight into the physical properties of the surface, allow corrections to be made so all observations are at the same viewing geometry, and allow for the prediction of asteroid magnitudes and reflectances at different phase angles. Different types of materials and different particles sizes will reflect light differently at different illumination and viewing geometries.

1.9.4 Magnitude Versus Phase Angle

The illumination of an asteroid at a phase angle of 0° is analogous to the illumination of the full Moon (Buchheim, Reference Buchheim2010). The illumination of the asteroid at a phase angle of 90° would be analogous to the illumination of the first or third quarter phase of the Moon. However because asteroids orbit so far from the Sun, main-belt asteroids usually do not reach phase angles much greater than 20–30° as observed from Earth (Buchheim, Reference Buchheim2010). However, near-Earth asteroids can be observed at much larger phase angles from Earth. Spacecrafts can also observe asteroids at larger phase angles than can be observed from Earth.

The magnitude (brightness) of an asteroid will be a function of phase angle. Asteroids are brightest at zero phase angle and becomes fainter at larger phase angles. Near zero phase angle, the asteroid becomes significantly brighter. This brightening at small phase angles for particulate (or rough) surfaces is called the opposition effect (or opposition surge). This name derives from the body being at opposition when this brightening occurs.

The opposition effect has been argued to be due to two effects (Hapke, Reference Hapke2002): shadow hiding and coherent backscatter. Shadow hiding is due to all shadows disappearing when an object is illuminated at zero phase angles. At all phase angles, shadows will be visible except at zero phase angle when each particle hides its own shadow. So as the phase angle gets close to zero, the brightness of an object will increase significantly. Coherent backscatter is due to multiply scattered light interfering constructively with each other as the light exits a medium near zero phase angle, causing a relative peak in brightness for observations near zero phase angle. The incident light must have a wavelength comparable to the particle size and be smaller than the distance between the scattering particles.

The reduced visual magnitude (V-magnitude) [H(α)] is the observed magnitude at a particular phase angle α if the body is at unit heliocentric (Sun–object) and geocentric (Earth–object) distances (Dymock, Reference Dymock2007). The reduced V-magnitude is the visual magnitude with the effect of distance removed. The reduced V-magnitude is only a function of phase angle. The reduced V-magnitude can be calculated using the formula
H(α)=Vobs(α)5log[rΔ(AU)2] (1.21)

where Vobs(α) is the observed V-magnitude, r is the heliocentric distance in AU, and Δ is the geocentric distance of the asteroid in AU. How the reduced V-magnitude changes for a body will only be a function of phase angle since the changes in magnitude due to changing distances between the Earth and the Sun to the asteroid have been removed.

Example 1.3

An asteroid is observed to have a visual magnitude of +20.6 at a phase angle of 25.7° at a heliocentric distance of 1.989 AU and a geocentric distance of 1.319 AU. What would be its reduced V-magnitude at the same phase angle?

Substituting these values into Equation 1.21 produces the formula
H(25.7°)=20.65log[(1.989 AU)(1.319 AU)(AU)2] (1.22)

Solving the equation results in a reduced V-magnitude at a phase angle of 25.7° of +18.5 for the asteroid.

1.9.5 H-G Magnitude System

To predict the magnitude of an asteroid as a function of phase angle, the H-G magnitude system was developed and then adopted in 1985 by the IAU (Bowell et al., Reference Bowell, Hapke, Domingue, Lumme, Peltoniemi, Harris, Binzel, Gehrels and Matthews1989). H is the absolute magnitude and G is the slope parameter.

Absolute magnitude is the apparent magnitude of an object at a set distance. For stars, the set distance is 10 parsecs (32.6 light years). For asteroids, the absolute magnitude (H) is the apparent magnitude of the asteroid if the object is 1 AU from both the Sun and the Earth with a phase angle of 0° (so it is fully illuminated). Zero phase angle is where the Sun illuminates the body directly behind the observer. However, this configuration for determining the absolute magnitude is only theoretical since an asteroid cannot be both 1 AU from the Sun and the Earth and have a phase angle of 0°.

The slope parameter describes how strongly the measured magnitude (brightness) of an asteroid depends on the phase angle. As G decreases, the slope of the phase curve becomes steeper. For G ≈ 0, the slope of the phase curve is very steep (changes dramatically with decreasing phase angle) while for G ≈ 1, the slope is very shallow (changes gradually with decreasing phase angle). The theoretical phase curves for two asteroids with two different slope parameters are shown in Figure 1.8.

Figure 1.8 Theoretical phase curves for two asteroids with different slope parameters (G = 0.10 and G = 0.30) but absolute magnitudes of +10 (Buchheim, Reference Buchheim2010). Note the steeper slope for the asteroid with a G = 0.10.

Credit: Robert Buchheim, Altimira Observatory.

The usual default value for G is 0.15 but G is known to vary significantly for different asteroids (Lagerkvist and Magnusson, Reference Lagerkvist and Magnusson1990). Higher-albedo bodies tend to have larger G values and, therefore, shallower phase curves. High-albedo bodies (pv = 0.46) have an average G of 0.43 while low-albedo (pv = 0.06) bodies have an average G of 0.12. However, less than 1% of known asteroids have measured G values (Vereš et al., Reference Vereš, Jedicke and Fitzsimmons2015). An accurate measurement of G requires a dense coverage of the phase curve over a large range in phase angle, which has not been done for most asteroids. Instead of G, it is often easier to measure the phase coefficient (β), which is the slope of the linear portion of the phase curve between 10° and 20° of phase angle (Dymock, Reference Dymock2007).

The absolute magnitude can be calculated from the formula
H=H(α)+2.5log[(1G)Φ1(α)+GΦ2(α)] (1.23)

where Φ1(α) and Φ2(α) are functions that describe the scattering off the surface (Dymock, Reference Dymock2007). These functions can be approximated as

Φi(α)=eAi(tanα2)Bi;i=1,2 (1.24)

where A1 = 3.33, B1 = 0.63, A2 = 1.87, and B2 = 1.22. So by calculating the reduced V-magnitude from Equation 1.21 from a measurement of the visual magnitude at a particular phase angle, the absolute magnitude can be determined using Equation 1.23.

If the absolute magnitude is already known (which is the case for most asteroids that have already been discovered), the effect of phase angle on the reduced visual magnitude can be determined by rearranging the components of Equation 1.23. The formula for calculating the reduced visual magnitude [H(α)] at solar phase angle α for a body with a particular H and G will be
H(α)=H2.5log[(1G)Φ1(α)+GΦ2(α)] (1.25)

This equation is valid for phase angles between 0° and 120° and for G values between 0 and 1. When calculating the values of the functions, the tangent is usually taken of the phase angle calculated in radians.

Example 1.4

For an asteroid with an H of +14, plot the phase curve from 0° to 30° for the reduced V-magnitude if the asteroid has a G of 0.15.

From Equation 1.24, the functions that describe the scattering off the surface are
Φ1(α)=e3.33(tanα2)0.63 (1.26)

and

Φ2(α)=e1.87(tanα2)1.22 (1.27)

Substituting all the values and formulas in Equation 1.25, the formula becomes

H(α)=142.5log[(10.15)e3.33(tanα2)0.63+(0.15)e1.87(tanα2)1.22] (1.28)

The phase angle in degrees can be converted to radians by the formula

α(radians)=π180°α(°) (1.29)

Substituting phase angles between 0° and 30° at 2° intervals results in the phase curve in Figure 1.9.

Figure 1.9 Phase curve for an asteroid with a slope parameter of 0.15 and an absolute magnitude of +14.

Even though it is called absolute, the absolute magnitude (and the slope parameter) for an asteroid can vary when observed at different oppositions (Dymock, Reference Dymock2007). Usually these values are averages of a number of observations at different oppositions. Since asteroids are not perfectly symmetrical and these objects are rotating, different aspects of the asteroid may be observed at different times in its orbit.

1.9.6 H-G1-G2 Magnitude System

A three parameter H-G1-G2 magnitude system (Muinonen et al., Reference Muinonen, Belskaya and Cellino2010) was adopted by the IAU in 2012. The absolute magnitude (H) is defined the same way in both this and the H-G system. The G1 and G2 parameters are new and are used with a number of functions to model the scattering off the surface and the opposition effect. This H-G1-G2 magnitude system better fits the phase curves of asteroids at low phase angles where backscattering occurs; however, this system is more complicated to implement. The H-G magnitude system is still used by many researchers (e.g., Reddy et al., Reference Reddy, Gary and Sanchez2015; Benishek and Pilcher, Reference Benishek and Pilcher2016) today due to its relative simplicity. For poorly sampled phase curves, a two parameter phase function (H-G12) was also developed by Muinonen et al. (Reference Muinonen, Belskaya and Cellino2010) since the G1 and G2 appear correlated for most asteroids.

1.10 Relationship between Albedos and Diameters

The observed magnitude of an asteroid will be a function of both the body’s diameter and albedo. A larger body will be brighter than a smaller one with the same albedo since it will have more surface area, which will reflect more light. A higher albedo body will be brighter than a lower albedo object of the same size since it will also reflect more light.

The diameter (D) of an asteroid in km can be estimated if the absolute magnitude (H) and the visual geometric albedo (pV) are known by using the relation (e.g., Fowler and Chillemi, Reference Fowler, Chillemi, Tedesco, Veeder, Fowler and Chillemi1992)
D=1329pv100.2H km. (1.30)

This formula is often used to roughly estimate a diameter by assuming an albedo since albedo information is not available for many asteroids.

Example 1.5

An asteroid has an absolute magnitude of +15 and a visual geometric albedo of 0.10. What is its estimated diameter?

Plugging in the values in Equation 1.30 produces the formula
D=13290.10100.2(15) km=4 km. (1.31)

Solving the formula gives the asteroid’s estimated diameter as approximately 4 km.

Questions

1) a) Compare the black body flux for a star with a surface temperature of 5800 K to the black body flux for an asteroid with an effective surface temperature of 250 K by drawing the relative shapes and intensities of both curves. Use wavelengths between ~0.1 and ~20 μm.

b) What are the differences between the two curves?

2) A near-Earth asteroid has an effective surface temperature of ~250 K and was observed at a distance from the Sun of 1.10 AU. Assume the emissivity is 0.9. What is this object’s Bond albedo?

3) Why are research observatories often located on mountains?

4) Give the advantages and disadvantages of photographic plates, photomultiplier tubes, and CCDs as detectors. Why is the CCD the primary type of detector that is used today to make visible astronomical observations?

5) If Pluto has an apparent magnitude of +13.7 at a distance of ~30 AU from the Sun, what would be the estimate of the apparent magnitude of a Pluto-like body at ~150 AU?

6) Show why a positive color index indicates that the object is brighter at the longer wavelength.

7) One asteroid has an absolute magnitude of +10. A second asteroid has an absolute magnitude of 17. Both have the same slope parameter (G). How many times brighter will the first asteroid be from the second asteroid at the same phase angle if they are both observed at the same distance from the Sun and the Earth?

8) An asteroid is observed to have a visual magnitude of +12.6 at a phase angle of 7.4° at a heliocentric distance of 2.638 AU and a geocentric distance of 3.538 AU. What would be its reduced V-magnitude at the same phase angle?

9) a) For an asteroid with an H of +14, plot the phase curve from 0° to 30° for the reduced V-magnitude if the body was a low-albedo object with a G of 0.12. Use the H-G magnitude system. When calculating the tangent, remember most computer programs want the phase angle inputted as radians and not as degrees.

b) For an asteroid with an H of +14, plot the phase curve from 0° to 30° for the reduced V-magnitude if the body was a high-albedo object with a G of 0.43. Use the H-G magnitude system.

10) a) An asteroid has an H magnitude of +15 and an visual geometric albedo of 0.20. What is this asteroid’s estimated diameter?

b) An asteroid has an H magnitude of +15 and an visual geometric albedo of 0.05. What is this asteroid’s estimated diameter?

Figure 0

Table 1.1 The wavelength regions for different parts of the electromagnetic spectrum

Figure 1

Figure 1.1 The wavelengths (m) and frequencies (Hz) of different regions of the electromagnetic spectrum. The wavelength regions that penetrate the Earth’s atmosphere, a list of different bodies with similar sizes to the wavelengths of each region, and the temperatures of different bodies emitting black body radiation primarily at that wavelength.

Credit: NASA.
Figure 2

Figure 1.2 The transmittance of electromagnetic radiation through the atmosphere. Microwaves are included as part of the short wavelength radio waves. Atmospheric opacity is the amount of light absorbed by the atmosphere with 100% opacity indicating total absorption and 0% opacity indicating total transmission.

Credit: NASA.
Figure 3

Figure 1.3 Black body curves for bodies at temperatures at 5000 K, 4000 K, and 3000 K plus the classical theory prediction for a body at 5000 K.

Credit: Dark Kule.
Figure 4

Figure 1.4 Illustration of the phase angle for an observer on the Earth observing a planet or a minor planet.

Figure 5

Table 1.2 Quantum efficiencies for a number of detectors

Figure 6

Figure 1.5 Image of a charge-coupled device (CCD).

Credit: NASA.
Figure 7

Figure 1.6 The Hubble Space Telescope as seen from the departing Space Shuttle Atlantis in 2009. The Hubble Space Telescope is 13.2 meters long.

Credit: NASA.
Figure 8

Figure 1.7 The WISE spacecraft at Vandenberg Air Force Base. The WISE spacecraft is 2.85 meters long.

Credit: NASA.
Figure 9

Figure 1.8 Theoretical phase curves for two asteroids with different slope parameters (G = 0.10 and G = 0.30) but absolute magnitudes of +10 (Buchheim, 2010). Note the steeper slope for the asteroid with a G = 0.10.

Credit: Robert Buchheim, Altimira Observatory.
Figure 10

Figure 1.9 Phase curve for an asteroid with a slope parameter of 0.15 and an absolute magnitude of +14.

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  • Light and Magnitude
  • Thomas H. Burbine, Mount Holyoke College, Massachusetts
  • Book: Asteroids
  • Online publication: 01 February 2017
  • Chapter DOI: https://doi.org/10.1017/9781316156582.002
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  • Light and Magnitude
  • Thomas H. Burbine, Mount Holyoke College, Massachusetts
  • Book: Asteroids
  • Online publication: 01 February 2017
  • Chapter DOI: https://doi.org/10.1017/9781316156582.002
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  • Light and Magnitude
  • Thomas H. Burbine, Mount Holyoke College, Massachusetts
  • Book: Asteroids
  • Online publication: 01 February 2017
  • Chapter DOI: https://doi.org/10.1017/9781316156582.002
Available formats
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