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Radiation Solar radiation drives the atmosphere. The amount of radiation the Earth’s surface and atmosphere receives is dependent on: The angle at which the sun’s rays strike the Earth’s surface at various latitudes. – Latitude – Elevation-Zenith angle – Declination angle — Time of year — Refraction of radiation — Altitude of site above sea level The time of day& year. Length of day. Depleting effects of the atmosphere. Depletion by particles between Sun and Earth. The planets Planets orbits Kepler’s Laws 1. The orbits of the planets are ellipses with the Sun at one focus of the ellipse. Most have nearly circular orbits. d1 d2 constant a = 1/2 longest axis = semi-major axis b = 1/2 shortest axis = semi-minor axis c = distance of foci from center Eccentricity, e = c/a Always less than 1. Earth: e = 0.0167 2. The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse. 3. The ratio of the squares of the revolutionary periods for two planets is 2 3 P1 R1 equal to the ratio of the 2 3 cubes of their semi-major axes. P R2 2 If period is given in Earth years and distance in Astronomical units, AU, then: 2 3 P1 R1 2 3 and, P R 2 3 1 1 1year 1AU If period is given in Earth days and distance in gigameters, Gm, then: 2 3 P1 R1 2 3 365.25463days 149.59787Gm P12 133410.9447R1 3 3347928.929 3 12 P1 .1996216379R 3 So, the time for one orbit is given by: where, Y aR 2 Y = period of planet in years, a = 0.1996days/Gm-3/2 R = Distance planet is from Sun in Gm A Gigameter = 106 km = 109 meters v M 0.0333988 sin(M) 0.0003486 sin (2 M) 0.0000050 sin(3 M) Earth Satellites Also move in elliptical orbits which are nearly circular. Consider a satellite moving in a circular orbit. Dv dv Acceleration is: a lim D t 0 Dt dt In diagram, Dv points toward the orbit center, so acceleration also points toward orbit center. Called centripital “center seeking” acceleration, ac. The vectors, v, vo, and Dv form a triangle geometrically similar to triangle ABC. The angle Dq is equal to the angle between CA and CB. Then, Dv DL v r v Or, Dv DL r Dv v DL Then, a lim lim D t 0 Dt Dt 0 r Dt DL v Since, Dlim t 0 Dt Then, satellites moving in a circular orbit v 2 are moving with an acceleration: ac r which is directed toward the center of their orbit. The force giving the satellite the acceleration it has (directed toward the center of its orbit is gravity). mme The force of gravity is given by: Fg G 2 r 2 msatell iteme v Then, since F=ma, G msatell ite 2 r r 11 2 2 6.672x10 N m kg G = Gravitational Constant 11 6.672x10 m s kg 3 2 Let torbit be the time to complete one orbit of the earth. The distance that satellite travels in one orbit is 2pr. Since, distance rate time 2 pr Then, 2 pr v t orbit and, v t orbi t Substituting into the force equation gives: m m 4p 2 r 2 G satell ite e 2 r msatell ite r (torbit ) 2 2 m 4 p r Rearranging gives: G e 2 2 r (torbi t) torbi t 2 4p r Gme 2 3 3 2 torbit 2 pr Gme For a satellite moving in a circular orbit. Season Effects Amount of incoming solar radiation (insolation) received at the Earth’s surface is dependent on: Angle at which sun’s rays strike the Earth. Time of day Depleting effects of atmosphere. Length of day. Determining Elevation angle of Sun at noon. Must know solar declination angle. C (d d ) d = Julian date of year. r s r cos dy dr = 173 fr = 23.45o dy = 365 or 366 C = 360 or 2p Graphically: Equation: sin y sin f sin s C t cosf cos s cos UTC l e td f = latitude, le = longitude, tUTC = Universal Coordinated Time, td = 24 hours, y = elevation angle, s = solar declination angle. Azimuth angle: Angle of Sun’s position relative to north is: cosa sin s sin f cosz cosf sin z where, a = azimuth angle, s = solar declination angle, f = latitude, z = zenith angle = (C/4)-y C = 360 or 2p Sunrise, Sunset & Twilight Geometric sunrise/sunset: Center of sun has zero elevation angle. Apparent sunrise/sunset: Top of sun crosses horizon as viewed by observer. Center at elevation angle of -0.833o. Civil twilight: Sun center no lower than -6o. Military twilight: Sun center no lower than 12o. Astronomical twilight: Sun center no lower than -18o. - Corresponding times: tUTC sin f sin sin y t d s l e arccos C cos f cos s where, td = 24 hours (length of day) le = longitude f = latitude, s = solar declination angle, y = elevation angle C = 360 or 2p Flux Flux density: Rate of transfer of a quantity across a unit area (perpendicular to the flow) per unit time. Mass flux: kg/m2 s Heat flux: J/m2 s or W/m2 Kinematics: The branch of mechanics dealing with the description of the motion of bodies or fluids without reference to the forces producing the motion. kg Kinematic mass flux F air 2 m m s wind speed = kg s m3 J o 2 F K m m s Kinematic heat flux = air C p kg J s 3 o m kg K = temperature wind speed Propagation of radiation Speed of light varies with the medium through which it passes. In a vacuum: co = 299, 792, 458m/s co Wavelength: l v, in units meters/cycle or mm, where, v = frequency, in units of Hz = cycles/s, and mm = 1 x 10-6 m. Wavenumber: , number of waves per meter, = 1/l cycles/m. Circular (angular) frequency = w radians/s = 2pv Emission Any object with temperature above absolute zero emits energy. Blackbody: a perfect emitter. Emits the maximum possible radiation for its temperature.