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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 . If shadow locations are known over , 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 position for lunar equatorial regions necessitated use of a Moon-centered celestial coordinate system. A Lunar-Celestial Equatorial Coordinate System is analogous to the Equatorial coordinates in celestial geometry. Rather than considering the 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 as seen from the Moon. This enables local sun 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 = of an object on the in Earth’s Equatorial Coordinate (EC) frame Az = solar δ = solar 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 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 ;

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 system that is still used today, derived from the Moon’s center of rotation and relationships between the lunar rotational axis/, 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 . 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 .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 , or zero-reference direction plane for any angular measurements in the - 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 , 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.

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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 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., ). 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. This corresponds δ ϒ the point of zero longitude and latitude in the existing α longitude/latitude grid used to map the lunar surface, and in LCE coordinates becomes analogous to the vernal equinox in Earth Equatorial Coordinates. The tilt of the h Moon’s spin axis relative to its orbit about the Earth is approximately 6.68º, and the about the Figure 2. Solar declination and hour angle defined Earth is fixed at an angle of 5.14º relative to the Earth-Sun in the Earth Equatorial reference frame. ecliptic plane. This means that the axial tilt of the Moon relative to the Earth-Sun ecliptic is fixed at 1.54º11 and, as a consequence, the ecliptic plane of the Sun relative to the Moon (or the path of the Sun across the sky as seen from the Moon) is inclined at 1.54º relative to celestial equator in the LCE system. Knowing this fixed, Moon-Sun ecliptic plane, the declination of the Sun relative to the Moon, δm, may be determined at a given time (Figure 3). The lunar , as for any celestial body, may be defined as the time it takes for the Moon to complete one full rotation about its axis, or for the Sun to complete a full North light/dark cycle in its apparent path relative to the Moon (a Celestial “solar day” on the Moon). Due to , this occurs when the Moon completes one full orbit about the Earth, Pole with the lunar near-side fixed to face the Earth such that, Path of relative to a point on the Moon’s surface, the Sun still rises Earth in the and sets in the . “Noon” becomes the Path of Sun precise point of full moon. Thus, the hour angle on the lunar surface remains analogous to the hour angle as δ determined for a point on the Earth, with angles before m noon defined as negative, and those after noon defined as positive. Hour angle at varying of lunar day is depicted using a top-down view in Figure 4, in which the shaded blue and grey regions on the disk of the Moon represent the lunar near-side and far-side, respectively. Now that the solar declination and hour angle have been defined and may be found in a lunar reference frame, the elevation angle of the Sun, El, at a given time can be Figure 3. Solar declination angle defined in the calculated relative to a point on the surface of the Moon LCE reference frame. using Eq. (1), in which “φ” is the local latitude of the surface location.

sin(El)  cos(h)cos( m )  sin( m )sin() (1)

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To Earth Before Noon, h is negative h h To Sun To Earth To Earth To Sun To Sun To Sun h h

To Earth After Noon, h is positive

– local point on lunar surface

Figure 4. Hour angle for a single surface point throughout the lunar day (pre-dawn to after-dark).

Knowing the solar elevation angle, the , Az, may be calculated using Eq. (2).

sin( )cos()  cos(h)cos( )sin() cos(Az)  m m (2) cos(El)

III. Shadowing Calculations and Applications Once the local solar azimuth and elevation angle are known at a given time for a point on the lunar surface, the shadowing of that point and at that time may be determined through use of local topography and trigonometric relationships. This may be accomplished by considering an elevation model of the local area large enough to capture surface topography relevant to shading, though still sufficiently constrained such that planetary curvature is not a significant factor and a cylindrically projected (“flat”) elevation map is an acceptable model. For example, for a point in an equatorial region (i.e., below 60º local latitude), the azimuth angle is measured clockwise with due north equal to zero. High resolution elevation models and lunar surface maps are available through data captured by the Lunar Reconnaissance Orbiter (LRO) and Lunar Orbiter Laser Altimeter (LOLA) instrument. Elevation angle has a range from a minimum of zero when the sun is at the horizon to a maximum of 90º when the sun reaches its zenith directly overhead; if El is ever calculated as negative, than the sun below the horizon and not visible, N indicating that it is currently night at this location. y Assuming a surface point in the northern hemisphere above 1.54º local latitude, the Sun will travel from an azimuth of near 90º to near 270º throughout the lunar day. By dividing the elevation map of the local area into eight segments using standard quadrants and map vertices, as shown in Figure 5, the path of the solar Surface Point Az = 145° x azimuth may be depicted as moving through regions one to four as defined in the figure. Also shown in 4 1 Figure 5, a direct line between the Sun and the surface R point may be projected onto the map of the relevant 3 2 region of the lunar surface at any given point in the (x, y) Sun’s path (in this figure, at Az = 145º). Following the designations in the figure and taking the surface location point as the origin, the projected line is described in terms of the depicted x and y directions Figure 5. Direct line (blue) from a surface point to the are as follows (Eq. 3a-d): Sun as projected onto an x-y map plane for Az = 145°.

Region 1: y  x tan(  Az) (3a)

Region 2: x  y tan(  Az) (3b)

Region 3: x  y tan(Az   ) (3c) 4 International Conference on Environmental Systems

Region 4: y  x tan(Az   ) (3d)

Where x and y represent the distances, in the respective directions, between the local point and a given point along the projected line, and Az (typically defined in degrees) is measured in radians. With the projected line between Sun and surface point known, the “sun height,” ds, at any given distance, R, along the line from the local point can be determined from the solar elevation angle using Eq. 4.

ds  R tan(El) (4)

For every point in the elevation map along the projected line of the Sun, the sun height can be compared against the surface elevation, d, at that point. If at any point along the projected line the surface elevation is found to be greater than the sun height, such that the condition d ≥ ds is met, than the Sun is being blocked by a surface feature, and the local point on the lunar surface is in shadow at that particular time/Sun position. By repeating this process over the full range of Az, the shadowing conditions of the lunar surface point become known over the duration of the lunar day. Similarly, it is possible to determine the shadowing characteristics of a total region on the lunar surface at a given time of lunar day by repeating the shadowing point analysis for all points within the desired range of an elevation map. An example of the result of such an analysis is shown in Figure 6, which depicts a section of the Hadley Rille region with all shadowed areas as black for a hypothetical solar elevation angle of 20° and azimuth angle of 180°. These shadowing calculations may be applied to a specific EVA path or region of exploration interest in order to inform proper mission planning. Knowledge of shadows over time for a desired traverse path, be it intended for a human or robotic explorer, enables use of basic thermal analysis to determine the explorer temperature and thermal characteristics. For human astronauts, this provides information such as energy required in a heating system, or energy from water sublimated (and thus mass of water sublimated) required to maintain internal space suit temperature at a desired level for astronaut comfort and survival. With energy depletion known, suit power levels over time may also be anticipated for a given portable life support system and battery type. Also calculable from the duration a human spends in sun or shadow is the external space suit Figure 6. Northern end of Hadley Rille (edge of Mare temperature, which can determine whether a region is Imbrium) with shadows shaded for Sun El = 20° and safe for human exploration. For example, the Az = 110°. The Apennine Front and St. George Extravehicular Mobility Unit (EMU) helmet designs Crater are shadowed in the lower right corner. specify that the safe temperature range for use, and to avoid helmet failure, is between -50°C and 125°C13. Performing thermal analysis from shadows near the southern end of Hadley Rille shows that astronauts traveling on a north-south path just one day after lunar would spend sufficient time in the shadow of adjacent mountains to face external suit temperatures near -55°C, making travel along such a path at that time outside safe parameters for human survivability. For robotic exploration vehicles, not only can shadowing determination provide thermal information for any potentially sensitive components with limited allowable temperature range, but provide a more detailed first-order analysis of power availability over time. If an explorer utilizes a solar or photovoltaic power system, knowledge of the vehicle’s time spent in sunlight will enable creation of a complete profile of power system charging over time. Furthermore, shadow locations over the region of operational interest may inform the explorer’s movements to avoid areas where prolonged lack of sunlight exposure could lead to power system depletion – and potentially loss of mission. Other applications for the shadow analysis include determination of optimal locations for placement of fixed installations, such as light reflectors, solar-powered scientific devices or stations, and identification of areas of potential interest due to prolonged time spent in shadow.

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IV. Conclusion Through use of modern, accurate understanding of relative locations for solar system objects, the LCE coordinate system has been derived and employed to determine shadowing characteristics for specific locations on the lunar surface. Presence of shadows over a region of interest are determined on a point-by-point basis, taking into account relative Sun position from all perspectives in the area. Ease of calculation can potentially provide rapid insight on surface shadowing for use in design of early mission concepts and mission planning. Understanding of these shadowing characteristics have implications on design of any future lunar systems, determining thermal characteristics, potential power availability, and related survivability metrics for lunar explorers. This extends to informing decisions on path planning for traverses to be made by both humans and robotic vehicles. The process shown here was only presented for equatorial regions on the Moon, for which a cylindrically projected map of the lunar surface can apply with some accuracy. For future use, this method in the LCE frame may be adjusted and applied to alternate map projections for conversion to local solar azimuth and elevation descriptors, allowing determination of shadowing in the lunar polar regions in order to inform exploration of mission concepts investigating scientific interests in those areas.10

Acknowledgments This work was made possible through collaboration with Planetary Geology Department at Brown University, as well as support from the Skolkovo Institute of Technology in Russia. This research is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1148903.

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