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Determination of Shadowing on the Lunar Surface Using a Lunar-Celestial Equatorial Coordinate System

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Determination of Shadowing on the Lunar Surface Using a Lunar-Celestial Equatorial Coordinate System 44th International Conference on Environmental Systems ICES-2014-93 13-17 July 2014, Tucson, Arizona Determination of Shadowing On the Lunar Surface Using a Lunar-Celestial Equatorial Coordinate System Sherrie A. Hall1 and Jeffrey A. Hoffman2 Massachusetts Institute of Technology, Cambridge, MA, 02139 Future human exploration of the lunar surface will depend on the ability to predict and model conditions for survivability during Extra-Vehicular Activity (EVAs). Estimating first-order astronaut metrics during EVAs, such as space suit temperature, transportation rover power requirements, suit heater power requirements, etc…, is necessary for the design of future space suits, transport vehicles, and other technologies involved in manned lunar surface operations. These metrics can be derived from knowledge of the locations of shadows cast by surrounding terrain for a given region of interest on the Moon. If shadow locations are known over time, they can also be used for estimation of thermal and power metrics for any technical, robotic, or solar-powered mission components on the lunar surface. Calculation of shadows based on exact sun position for lunar equatorial regions necessitated use of a Moon-centered celestial coordinate system. A Lunar-Celestial Equatorial Coordinate System is analogous to the Earth Equatorial coordinates in celestial geometry. Rather than considering the position of the Sun relative to the Moon by first comparing lunar orientation with the relative position of the Earth, this use of Moon- centered coordinates allows consideration of the path of the Sun in the sky as seen from the Moon. This enables local sun angles to be calculated in a horizontal coordinate system, just as they are for determining Sun position on the Earth. Thus, local Sun angles may be readily determined for a given time at a given point on the lunar surface, and trigonometric relationships can be used in conjunction with accurate surface elevation maps to determine whether that point is in shadow. Application of this technique can provide characterization of shadowing over any desired EVA path or exploration area of interest over time, making it possible to include thermal and power analysis of potential surface assets in a model. Nomenclature α = right ascension of an object on the celestial sphere in Earth’s Equatorial Coordinate (EC) frame Az = solar azimuth angle δ = solar declination in EC frame δm = Moon-centered solar declination in the Lunar-Celestial Equatorial (LCE) coordinate system d = lunar surface elevation at a given point ds = sun height at a given point relative to an observer’s fixed location on the lunar surface El = solar elevation angle h = hour angel of the sun φ = local latitude on the lunar surface R = distance (range) between an observer’s fixed location on the lunar surface and a given surface point x = distance in the x direction from an observer’s fixed location on the lunar surface to a given point y = distance in the y direction from an observer’s fixed location on the lunar surface to a given point ϒ = vernal equinox; first point of Aries 1 Doctoral Candidate, Department of Aeronautics and Astronautics, 70 Vassar St. Cambridge, MA 02139. 2 Professor of the Practice, Department of Aeronautics and Astronautics, 125 Massachusetts Ave. Cambridge, MA 02139. I. Introduction HE attempted development of coordinate systems for the Moon analogous to those of Earth that took place T during the Apollo era largely focused on finding a way to map the lunar surface, as well as describe the relative positions of the Earth and Moon in relation to each other. Many of these efforts stemmed from description and characterization of the Moon based on theory of lunar motion about the Earth.1,2 For actual lunar surface operations, much of the focus was on mapping of lunar surface features3 and creating a surface cartography system analogous to that of Earth. This resulted in the lunar latitude and longitude system that is still used today, derived from the Moon’s center of rotation and relationships between the lunar rotational axis/axial tilt, lunar polar axis, and orbital characteristics.4 Initial analyses from this era were based on limited data from orbital theory, ground observations, and approximate knowledge of lunar surface landmarks. Through studies of laser ranging data that originated with the placement of reflectors at fixed points on the lunar surface by the Apollo astronauts, the Moon’s position and rotational characteristics in relation to the Earth are now well defined.5-7 In general, accuracy and understanding of orbital parameters and relative positions of celestial bodies has improved over the years. Many Earth-centered, Sun-centered, and alternate celestial coordinate systems8 are now used to describe orbital paths are relative positions of solar-system objects in relation to each other, or to follow the paths of more distant objects. The NASA Jet Propulsion Lab (JPL) has even developed code and calculated accurate ephemerides for the planets, Sun, etc. for both past and future,9 based in the International Celestial Reference Frame (ICRF) – a celestial coordinate system centered at the barycenter of the solar system.8 With accurate modern knowledge of positions for celestial bodies, along with availability of position data in Cartesian form for objects relative to one another in space, it is now reasonable to construct a viable and accurate celestial coordinate system centered on Earth’s moon, and to use such a system to make calculations with a moon-referenced celestial sphere independent of the Earth-Moon orbital relationship. In the following discussion, a Lunar-Celestial Equatorial (LCE) Coordinate System, analogous to the Earth- centered Equatorial Coordinate System, is presented for application in determination of Sun position relative to a given point on the lunar surface. Knowledge of accurate Sun positions relative to the Moon allows calculation of the shadowing characteristics of regions on the lunar surface. Understanding of shadow locations over time is critical to future lunar research. Examination of shadows not only helps to inform scientific interests, such as investigation into the presence of ice in permanently shadowed regions near the lunar poles,10,11 but also has an impact on engineering design factors for any surface exploration efforts. For example, in the absence of an atmosphere, thermal properties of lunar surface assets, robotic or human, are entirely driven by the lunar daylight cycle, and sunlight availability determines feasibility of use (and sizing) of photovoltaic sources for surface-asset power. Some past efforts to describe lunar surface shadowing have involved using extensive databases of horizon appearance for approximate surface regions. This paper details the method for simple calculation of these surface shadowing characteristics by employing the LCE frame. II. Lunar-Celestial Equatorial Coordinates In the Earth-centered Local Horizontal reference frame,12 the position of the Sun, or any other celestial object in the sky, is determined relative to the observer’s point on the Earth’s surface. The exact solar position is specified by its local azimuth and elevation angles, as shown in Figure 1. For the Earth, these angles are derived through analysis of coordinates from the frequently-used Equatorial Coordinate System. In this system, the Earth’s equatorial over-head plane serves as the celestial equator, or zero-reference direction plane for any angular measurements in the north-south zenith direction.12 Thus, Earth latitude and longitude may be projected on to the celestial sphere in this reference frame. N While projected longitude can be useful for relative position and time calculations of celestial bodies, the reference direction used for measuring right ascension, α, of an object in the sky is the vernal equinox (ϒ), or point at El which the Sun’s path relative to Earth crosses the equator Az from south to north. The path of the Sun across the celestial sphere as seen from Earth, or solar ecliptic, is often referred to simply as the ecliptic in Earth-centered systems. As this paper focuses on the Moon, and in other Figure 1. Azimuth and Elevation for an object in work the path of the Moon from the Earth may be referred the sky in the Local Horizontal reference frame. 2 International Conference on Environmental Systems to as a “lunar ecliptic,” it should be noted to avoid confusion that any “ecliptic” plane referenced here will always indicate the path of the Sun, whether seen from the Earth or the Moon. In the Equatorial Coordinate System, the declination, δ, of a celestial object (such as the Sun) at a given point in time is defined as the vertical angle from the celestial equator to the object. This value is independent from any fixed location on the Earth’s surface. The hour angle, h, however, is defined relative to an observer’s particular position on the surface. Specifically, hour angle is measured from the observer’s longitudinal location to that of the Sun, such that hour angle is equal to zero when the sun is directly over-head (i.e., noon). For calculation purposes, hour angle is defined as negative prior to solar noon, and positive after noon. Solar declination and hour angle for the standard Earth-referenced equatorial frame are illustrated in Figure 2, in which the red star represents an observer’s local position on the Earth’s surface. A corresponding Lunar-Celestial Equatorial North Coordinate System has been identified to allow definition Celestial of parameters in a Moon-centered reference frame. In this Pole system, the lunar equatorial plane is used to define the celestial equator. With the moon tidally locked in its orbit, the most relevant fixed point of reference for Moon- Solar Ecliptic centered system becomes the point at which the Earth’s path in the sky crosses the lunar equator.
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