Chapter 5 Satellite orbits Remote Sensing of Ocean Color Instructor: Dr. Cheng-Chien LiuCheng-Chien Liu Department of Earth Science National Cheng-Kung University Last updated: 7 April 2003 Source:

5.1 Introduction  Orbital mechanics Early studies  Tyco Brahe, Copernicus, Keppler and Galileo Three laws of planetary motion (Keppler)  Orbits of all the planets are ellipses with the sun at one of the focal points  Any planet moves by such a way that radius-vector from the sun to the planet covers in equal time intervals the equal areas  T 2 /a 3 =const, where T is the period of the planet revolution, a is the length of the bigger axis of the orbit

Fig Fig Historic satellite ocean color sensors Source:

5.1 Introduction (cont.)  Orbital mechanics (cont.) The universal law of gravity (Newton 1666)  F = GmM/r 2  Artificial earth orbiting satellite Sputnik (USSR 1957)  Perturbing forces  F = GmM/r 2 is not enough Earth  non-spherical shape Ocean tides, atmospheric drag, Sun, … Osculating orbit

5.1 Introduction (cont.)  Orbit prediction model Why need?  Control and receiving data  GPS Ephemeris = fn(t)  Help us to determine the orbit based on the goals of satellite mission

5.2 Terminology and coordinate systems  Coordinate system Geodetic Geocentric  using this system  Origin: center of mass  Axis 1: ON (Origin to North Pole)  Axis 2: OA (Aries)  Axis 3:  ON &&  OA  Aries: Aries is a point in space the direction of which is determined by the intersection of the equatorial plane of the Earth (ie the plane that contains the equator) and the orbital plane defined by the Earth’s annual motion around the Sun ( see Figure 1.1) such that a line from the Sun’s centre to the Earth along this direction points toward Aries at the time of the Vernal Equinox. This latter event occurs at a date near March 20 each year when the Sun’s apparent position, as it moves from South to North in the sky, crosses the equatorial plane. (Lynch 2000)

Fig Fig Shown are two orientations of the Earth with respect to the Sun for the months of January and July. In July, at location A, the Sun appears directly overhead of point in the northern hemisphere. In January, at location B, the Sun appears directly overhead of this point in the southern hemisphere. Accordingly, as the Earth progresses in its orbit about the Sun in the solar ecliptic plane, it will reach a point at about March 20, when the Sun will be directly overhead of a point on the Earth’s equator as the Sun makes this transition “from southern to northern latitudes”. This particular time, the Vernal Equinor, defines a geometry such that a line from the Earth’s centre C through the line of intersection of the solar ecliptic plane and the plane of the Earth’s equator defines the direction of Aries. Source:

5.2 Terminology and coordinate systems (cont.)  A simple elliptical orbit R = a(1-e 2 ) / (1+cos  )  e: orbit eccentricity  a: ellipse semi-major axis   : the true anomaly measured from perigee P  Apogee A Period T = 2  (a 3/2 /G 1/2 M 1/2 ) Perigee P Apogee A Ascending Node N

Fig Fig A simple elliptical orbit Source:

Fig Fig Illustration of a satellite orbit Source:

5.2 Terminology and coordinate systems (cont.)  The sub-satellite track Instantaneous location of the satellite in the orbital plane r (CS) The orbit’s inclination i  The angle between the plane of the satellite and the Earth’s equatorial plane measured anticlockwise from the later. The right ascention   The angle measured eastward from the direction of Aries to N The argument of Perigee   The angle between P and N True anomaly (mean anomaly)   The angle between CP and CS (measured from CP)

Fig Fig Earth’s orbit around Sun Source:

5.3 Sun synchronous (Polar) orbits  Prograde or retrograde i < 90 0 : prograde i > 90 0 : retrograde  most earth observations satellites  Make observations at the same time each day over the whole annual cycle  d  /dt > 0 Only pass between the latitude +i and -i  Sun synchronous d  /dt = 2  /  1 0 per day = 1.99 x rad s -1

5.3 Sun synchronous (Polar) orbits (cont.)  Sun synchronous (cont.) NOAA/AVHRR  H = 1450 km  a = 7828 km  Assume e = 0  cos i =  i =  T = 2  [GM/r 3 ] -1/2 =114.9 min  Displacement during one orbit  114.9/24/60*360 0  29 0  29/360*40074  3200 km  Normally two on orbit at any time. One is deployed with a morning ascending node and the other an afternoon ascending node

5.3 Sun synchronous (Polar) orbits (cont.)  Sun synchronous (cont.) NASA/MODIS  H = 705 km  a = 7083 km  Assume e = 0  cos i = ?  i = ?  T = 2  [GM/r 3 ] -1/2 =? min  Displacement during one orbit  ?/24/60*360 0  ? 0  ?/360*40074  ? Km

5.4 Geosynchronous (geostationary) orbits  T = 24hr  a = 35,843 km

Fig Fig Illustration of a geostationary orbit Source:

Fig Fig How does Terra take a global snapshot Source:

5.5 Computer-based modeling of satellite orbits