which shows that the velocity of a circular orbit is inversely proportional to the radius. We now consider two particular orbits of interest 1)r=R This corresponds to a hypothetical satellite orbiting the earth at a zero altitude above the earths Irface. The orbits velocity is gR=7910m/s and the period, from equation 3, is 84.4 This period is called the "Schuler"period, and it is the minimum period that any free fight object can have in orbit around the earth 2)Synchronous Orbits These are orbits whose period is the same as the earth's rotational period( 24 h). In addition, if the orbit is in the equatorial plane, the orbit is said to be geostationary because the satellite will stay fixed relative to an observer on the earth. Using equation 3 =42042km≈6.6B, hich corresponds to an altitude above the surface of 5.6R Example Elliptical Orbits Consider a satellite launched from an altitude d above the earth's surface, with velocity vc=VH/(R+d) If the direction of the velocity is orthogonal to the position vector, the trajectory will clearly be a circular orbit of radius R +d. However, if the velocity is in any other direction, the trajectory will be an ellipse of semi-major axis equal to R+d. The characteristics of the ellipse can easily be determined as follows: knowing r and u, we can determine h; using equation 4, we can determine e; and from the trajectory equation 1,we can determine 0, and hence the orientation of the ellipse. Parabolic Orbits(e= 1) From equation 1, we see that r-+ oo for 0-T. From the energy integral, with E=0, we have that 2-2=0=2which shows that the velocity of a circular orbit is inversely proportional to the radius. We now consider two particular orbits of interest: 1) r = R This corresponds to a hypothetical satellite orbiting the earth at a zero altitude above the earth’s surface. The orbit’s velocity is vc = p gR = 7910 m/s, and the period, from equation 3, is T = 2π p gR2 R 3/2 = 2π s R g = 84.4 min . This period is called the “Schuler” period, and it is the minimum period that any free flight object can have in orbit around the earth. 2) Synchronous Orbits These are orbits whose period is the same as the earth’s rotational period ( 24 h). In addition, if the orbit is in the equatorial plane, the orbit is said to be geostationary because the satellite will stay fixed relative to an observer on the earth. Using equation 3, a =  T 2 gR2 4π 2 1/3 = 42042 km ≈ 6.6R, which corresponds to an altitude above the surface of 5.6R. Example Elliptical Orbits Consider a satellite launched from an altitude d above the earth’s surface, with velocity vc = p µ/(R + d). If the direction of the velocity is orthogonal to the position vector, the trajectory will clearly be a circular orbit of radius R + d. However, if the velocity is in any other direction, the trajectory will be an ellipse of semi-major axis equal to R+d. The characteristics of the ellipse can easily be determined as follows: knowing r and v, we can determine h; using equation 4, we can determine e; and from the trajectory equation 1, we can determine θ, and hence the orientation of the ellipse. Parabolic Orbits (e = 1) From equation 1, we see that r → ∞ for θ → π. From the energy integral, with E = 0, we have that, 1 2 v 2 e − µ r = 0, v2 e = 2µ r . (7) 3