e<1 the trajectory is closed (ellipse), and E<0, e=l the trajectory is open(parabola), and E=0, e>1 the trajectory is open(hyperbola), and E>0. Equations 1, 2, and 3, together with the energy integral 4, provide most of relationships necessary to solve ngineering problems in orbital mech Types of Orbits Elliptic Orbits(e< 1) When the trajectory is elliptical, h=au(1-e)(see lecture D28). Then, the total specific energy simplifies to E==u/(2a), and the conservation of energy can be expressed as This expression shows that the energy(and the period) of an elliptical orbit depends only on the majo semi-axis. We also see that for a fixed a, the value of h determines the eccentricity. There are two limiting cases:e-1, which gives h-0, which in turn implies that the minor semi-axis of the ellipse b-0; and e=0 which corresponds to a circular orbit with h= vap. In the first case, the maximum value of the eccentricity is limited by the size of the planet, since, for sufficiently large values of e, the trajectory will Below we show three elliptical trajectories that have the same energy(same value of a), but different centricities e=0.5 e=0.9 Circular Orbits(e=0) This is a particular case of an elliptic orbit. The energy equation is given by equation 5. The radius is constant h2 v2r2 For orbits around the earth, u=gR, where g is the acceleration of gravity at the earth's surface, and R is the radius of the earth. Then -9Re < 1 the trajectory is closed (ellipse), and E < 0, e = 1 the trajectory is open (parabola), and E = 0, e > 1 the trajectory is open (hyperbola), and E > 0. Equations 1, 2, and 3, together with the energy integral 4, provide most of relationships necessary to solve basic engineering problems in orbital mechanics. Types of Orbits Elliptic Orbits (e < 1) When the trajectory is elliptical, h 2 = aµ(1−e 2 ) (see lecture D28). Then, the total specific energy simplifies to E = −µ/(2a), and the conservation of energy can be expressed as 1 2 v 2 − µ r = − µ 2a . (5) This expression shows that the energy (and the period) of an elliptical orbit depends only on the major semi-axis. We also see that for a fixed a, the value of h determines the eccentricity. There are two limiting cases: e → 1, which gives h → 0, which in turn implies that the minor semi-axis of the ellipse b → 0; and e = 0 which corresponds to a circular orbit with h = √ aµ. In the first case, the maximum value of the eccentricity is limited by the size of the planet, since, for sufficiently large values of e, the trajectory will collapse onto the planet’s surface. Below we show three elliptical trajectories that have the same energy (same value of a), but different eccentricities. Circular Orbits (e = 0) This is a particular case of an elliptic orbit. The energy equation is given by equation 5. The radius is constant r = h 2 µ = v 2 c r 2 µ . For orbits around the earth, µ = gR2 , where g is the acceleration of gravity at the earth’s surface, and R is the radius of the earth. Then, v 2 c = µ r = gR2 r , (6) 2