For Earth-orbiting satellites, use RE = 6,378 km for the radius of the Earth and μ = 3.986(105) km3/s2 for the gravitational parameter. For problems involving other celestial bodies (the moon, Mars, etc.) see Appendix A for their respective radii and gravitational parameters.
Conceptual Problems
2.1 An Earth-orbiting satellite has the following position and velocity vectors expressed in polar coordinates:
r = 8,250ur km v = 1 2054ur+ 7 0263uθkm/s Determine the following:
a) Angular momentum (magnitude), h b) Specific energy,ξ
c) Semimajor axis, a d) Parameter, p e) Eccentricity, e
f) Perigee and apogee radii, rpand ra
g) Flight-path angle,γ, at this instant h) True anomaly,θ, at this instant.
2.2 Repeat Problem 2.1 for an Earth-orbiting satellite with the following position and velocity vectors expressed in polar coordinates:
r = 9,104urkm v = −0 7004ur+ 6 1422uθ km/s
2.3 Compute the eccentricity vector e using the Earth-orbiting satellite data in Problem 2.2 and show that its norm (magnitude) matches the eccentricity e as com-puted using the geometric parameters p and a.
2.4 An Earth-orbiting satellite has the following position and velocity vectors expressed in polar coordinates:
r = 1,2426ur km v = 4 78uθ km/s
Determine the following parameters when the satellite is 2,000 km above the Earth’s surface and approaching the Earth:
a) Orbital velocity b) Flight-path angle,γ c) True anomaly,θ.
2.5 An Earth-orbiting satellite has semimajor axis a = 9,180 km and eccentricity e= 0.12. Determine the radial position r, velocity v, and flight-path angleγ when the satellite is approaching Earth and 80 from perigee passage.
2.6 Develop an expression for eccentricity e in terms of specific energy ξ and angular momentum h.
2.7 At a particular instant in time, a tracking station determines that a space vehicle is at an altitude of 390.4 km with an inertial velocity of 9.7023 km/s and flight-path angle of 1.905 . Is this space vehicle in a closed orbit about the Earth or is it following an open-ended trajectory that will eventually“escape” Earth? Justify your answer.
2.8 An Earth-orbiting satellite has semimajor axis a = 8,230 km and eccentricity e= 0.12. Determine the satellite’s maximum altitude above the Earth’s surface.
2.9 Two satellites are being tracked by ground-based radar stations. Their altitusdes, inertial velocities, and flight-path angles at a particular instant in time are sum-marized in the following table:
Object Altitude (km) Velocity (km/s) Flight-path angle
Satellite A 1,769.7560 6.8164 4.9665
Satellite B 676.3674 7.8504 –6.8903
Are these two satellites in the same orbit? Explain your answer.
2.10 An Earth-orbiting satellite has the following position and velocity vectors expressed in polar coordinates:
r = 7,235ur km v = −0 204ur+ 8 832uθ km/s Determine the orbital period in minutes.
2.11 A tracking station determines that an Earth-observation satellite has perigee and apogee altitudes of 350 and 1,206 km, respectively. Determine the orbital period (in minutes) and the parameter p.
2.12 An Earth-orbiting satellite is at an altitude of 700 km with inertial velocity v= 7.3944 km/s and flight-path angleγ = 0. Is the satellite at perigee or apogee?
Justify your answer.
2.13 The ratio of apoapsis and periapsis radii for a particular satellite orbit is ra/rp= 1 6.
Determine the eccentricity of the orbit.
2.14 Derive an expression for the orbital period of a circular orbit in terms of its cir-cular velocity speed vc.
2.15 An Earth-observation satellite’s closest approach is at an altitude of 300 km. If the satellite returns to its perigee position every 2 h determine the apogee altitude and the orbital eccentricity.
2.16 A launch vehicle experiences a malfunction in its guidance system. At burnout of its upper rocket stage, the vehicle is at an altitude of 250 km with an inertial
velocity of 7.791 km/s and flight-path angle of 4.5 . Has the vehicle achieved a stable orbit? Explain your answer.
2.17 Figure P2.17 shows two satellites in Earth orbits: Satellite A is in a circular orbit with an altitude of 800 km, while Satellite B is in an elliptical orbit with a perigee altitude of 800 km. At the instant shown in Figure P2.17, Satellite B is passing through perigee while Satellite A lags behind Satellite B with an angular separa-tion of 60 . Determine the apogee altitude of the elliptical orbit so that Satellites A and B occupy the same radial position after one revolution of Satellite B (in other words, Satellites A and B perform a rendezvous maneuver when Satellite B returns to perigee after one orbital revolution).
2.18 Derive an expression for the time rate of true anomaly, θ, as a function of param-eter p, eccentricity e, and true anomalyθ.
2.19 A satellite is on a parabolic trajectory about the Earth. At 4,000 km above the Earth’s surface, it has a flight-path angle of 25 . Determine the velocity and true anomaly of the satellite at this instant.
2.20 A satellite is approaching Earth on a parabolic trajectory with a velocity of 5.423 km/s. If the projected perigee altitude of the parabolic trajectory is 800 km, determine the radius, flight-path angle, and true anomaly at this instant.
2.21 An interplanetary spacecraft fires an onboard rocket in order to depart a low-Earth orbit. At engine cutoff (at an altitude of 200 km), the spacecraft has an inertial velocity of 11.814 km/s and zero flight-path angle. Determine:
a) Eccentricity of the hyperbolic departure trajectory
b) Velocity when the spacecraft is at a radial distance of 400,000 km
c) Flight-path angle when the spacecraft is at a radial distance of 400,000 km d) True anomaly when the spacecraft is at a radial distance of 400,000 km
e) Hyperbolic excess speed, v+∞. 60ο
Satellite A
Satellite B
Satellite B orbit
Figure P2.17
2.22 An interplanetary spacecraft is departing Earth on a hyperbolic trajectory with eccentricity e = 1.4 and semimajor axis a =–16,900 km. Determine:
a) Perigee altitude
b) Radial and transverse velocity components at true anomalyθ = 100 c) True anomaly of the departure asymptote,θ+∞
d) Hyperbolic turning angle,δ.
2.23 In March 2016, a spacecraft launched in early 2014 is approaching Earth on a hyperbolic trajectory for a gravity assist maneuver. Its hyperbolic excess speed on the arrival asymptote is v−∞= 2.78 km/s and its projected perigee velocity is estimated by mission operators to be 10.9 km. Determine the perigee altitude and turning angleδ of the hyperbolic flyby.
MATLAB Problems
2.24 Write an M-file that will that will calculate the following characteristics of an Earth orbit (with the desired units):
Angular momentum, h (km2/s) Energy,ξ (km2/s2)
Semimajor axis, a (km) Parameter, p (km) Eccentricity, e Period, Tperiod(h)
Perigee and apogee radii, rpand ra(km) Flight-path angle,γ (deg)
True anomaly,θ (deg)
The inputs to the M-file are orbital radius r (in km), radial velocity vr(in km/s), and transverse velocity vθ (in km/s). The M-file should return an empty set (use open brackets []) for characteristics that do not exist, such as period for a parabolic or hyperbolic trajectory. Test your M-file by solving Problem 2.1.
2.25 Write an M-file that will calculate a satellite’s orbital “state” for a particular loca-tion in an Earth orbit. The desired outputs are radial posiloca-tion r (in km), velocity magnitude v (in km/s), and flight-path angle γ (in deg). The M-file inputs are semimajor axis a (in km), eccentricity e, and true anomalyθ (in deg). Test your M-file by solving Problem 2.4.
2.26 The second stage of a launch vehicle is approaching its main-engine cutoff (MECO). Suppose the vehicle’s guidance system has the following simplified equa-tions for orbital radius and velocity at MECO as a function of flight-path angle
r= 6, 878 + 12γ km, v= 7 613−1 5γ km/s
where flight-path angleγ is in radians. Plot perigee altitude, semimajor axis, and eccentricity as a function of MECO flight-path angle for the range−10 ≤ γ ≤ 10 . Using these plots, determine the MECO flight-path angle that results in the max-imum-energy elliptical orbit with a 200 km altitude perigee.
Mission Applications
2.27 GeoEye-1 is an Earth-observation satellite that provides high-resolution images for Google. The orbital period and eccentricity of GeoEye-1 are 98.33 min and 0.001027, respectively. Determine the perigee and apogee altitudes of GeoEye-1.
2.28 The Chandra X-ray Observatory (CXO) used a sequence of two transfer orbits to increase orbital energy (additional subsequent transfer orbits were used to eventually achieve the highly elliptical operational CXO orbit presented in Example 2.3). Figure P2.28 shows that the two transfer orbits are tangent at the perigee altitude of 300 km above the surface of the Earth. Determine the eccentricity and orbital period of each transfer orbit.
2.29 A US reconnaissance satellite is in an elliptical orbit with a period of 717.8 min.
Ground-tracking stations determine that its perigee altitude is 2,052 km. What is the apogee altitude of this satellite?
2.30 The Apollo 17 command and service module (CSM) orbited the moon in a 116 km altitude circular orbit while two astronauts landed on the lunar surface.
Determine the orbital velocity and period of the CSM.
2.31 The Meridian 4 is a Russian communication satellite that was launched in May 2011 on a Soyuz-2 rocket. The operational (target) orbit of the Meridian 4 satellite is an elliptical orbit with perigee and apogee altitudes of 998 and 39,724 km, respectively. The Meridian 4 satellite reached its target by following an elliptical transfer orbit that is tangent to the target orbit at apogee (Figure P2.31; not to scale). The perigee altitude of the transfer orbit is 203 km. Determine:
a) The perigee velocity on the transfer orbit
b) The transit time from perigee to apogee on the transfer orbit c) The apogee velocity on the transfer orbit
d) The apogee velocity on the Meridian 4 target orbit.
Transfer orbit 1:
Apogee altitude = 13,200 km
Perigee altitude = 300 km Transfer orbit 2:
Apogee altitude = 72,000 km
Figure P2.28
2.32 Lunar Orbiter 1 (1966) was the first US spacecraft to orbit the moon. It was initially inserted into a lunar orbit with angular momentum h = 3,509.8 km2/s and specific energyξ = –0.886641 km2/s2. Determine the following:
a) Periapsis (“perilune”) and apoapsis (“apolune”) altitudes b) Velocity at perilune and apolune
c) Radial and transverse velocity components at true anomalyθ = 220 . d) Orbital period (in min).
2.33 The Apollo lunar module (LM) used its ascent propulsion system (APS) to depart the moon’s surface. After over 7 min of powered flight, the APS engine was shut down and the LM was at an altitude of 18 km above the moon with velocity v= 1.687 km/s and flight-path angleγ = 0.4 . Determine the periapsis (“perilune”) and apoapsis (“apolune”) altitudes of the LM’s orbit after engine cutoff.
Problems 2.34–2.36 involve the Mars Reconnaissance Orbiter (MRO) space-craft which approached the target planet in March 2006 and subsequently per-formed a propulsive maneuver to slow down and enter a closed orbit about Mars.
2.34 The MRO spacecraft approached Mars on a hyperbolic trajectory with eccentric-ity e = 1.7804 and asymptotic approach speed v−∞= 2.9572 km/s. Determine the altitude, velocity, and flight-path angle of the MRO spacecraft at its closest approach to Mars.
2.35 The MRO spacecraft fired its rocket engines at periapsis of the hyperbolic approach to slow the spacecraft’s velocity to 4.5573 km/s for insertion into a closed orbit about Mars. Using the MRO hyperbolic trajectory information in Problem 2.34, determine the orbital period and eccentricity of the MRO space-craft after the orbit-insertion burn (the rocket burn did not change the periapsis radius– it is the same as the periapsis radius of the hyperbolic approach trajectory as determined in Problem 2.34).
2.36 The MRO used atmospheric drag at each periapsis pass (“aerobraking”) to slow down and reduce the orbital energy. After the aerobraking phase (and a small pro-pulsive maneuver), the operational orbit for the MRO spacecraft has periapsis and apoapsis altitudes of 250 and 316 km above the surface of Mars, respectively.
Determine the following:
vp
va
Meridian 4 target orbit:
998 km perigee altitude 39,724 km apogee altitude Transfer orbit:
203 km perigee altitude 39,724 km apogee altitude
Figure P2.31
a) Semimajor axis, a b) Eccentricity, e
c) Orbital period (in min)
d) True anomaly,θ, when the altitude is 300 km e) Flight-path angle,γ, when the altitude is 300 km.
Problems 2.37 and 2.38 involve the Lunar Atmosphere and Dust Environment Explorer (LADEE) spacecraft, which was launched in September 2013 (see Exam-ple 2.6).
2.37 The LADEE spacecraft was launched into a highly elliptical orbit about the Earth by a Minotaur V booster with perigee and apogee altitudes of 200 and 278,000 km, respectively (see Figure 2.14 and Example 2.6). Determine the altitude, velocity, and flight-path angle of the LADEE spacecraft at true anomalyθ = 300 . 2.38 After a coasting translunar trajectory, the LADEE spacecraft was inserted into an
elliptical orbit about the moon by performing a series of retrorocket propulsive burns. The orbital period of the LADEE spacecraft was 4 h and its orbital eccen-tricity was e = 0.2761 (Figure P2.38; not to scale). Determine the periapsis and apoapsis altitudes (or“perilune” and “apolune” altitudes) of the LADEE spacecraft in its lunar orbit.
Problems 2.39 and 2.40 involve the Stardust capsule which returned to Earth in January 2006 on a hyperbolic approach trajectory after sampling particles from the comet Wild-2.
2.39 When the Stardust capsule arrived at the “edge” of the Earth’s atmosphere (the so-called“entry interface” altitude of 122 km), it had inertial velocity v = 12.9 km/s and flight-path angleγ = –8.21 . Determine the following:
a) Specific energy,ξ b) Semimajor axis, a c) Eccentricity, e
d) True anomaly,θ, at entry interface e) Arrival hyperbolic excess speed, v−∞
f) Arrival asymptotic true anomaly,θ−∞.
moon Period = 4 hr
e= 0.2761
Figure P2.38
2.40 Using the entry-interface state of the Stardust capsule from Problem 2.39, deter-mine the velocity, flight-path angle, and true anomaly of the capsule when it was at radius r = 384,400 km (roughly the distance from the Earth to the moon).
2.41 The Pegasus launch vehicle reaches its second-stage burnout at an altitude of 192 km, inertial velocity v = 5.49 km/s, and flight-path angleγ = 25.8 . The launch vehicle then coasts in this orbit until it ignites its third stage when flight-path angle decreases to 2.2 . Determine:
a) Semimajor axis of the Pegasus orbit after second-stage burnout.
b) Eccentricity of the Pegasus orbit after second-stage burnout.
c) The altitude and velocity of the Pegasus launch vehicle when the third stage is ignited.
Problems 2.42 and 2.43 involve the Juno spacecraft which departed Earth in early August 2011 and arrived at Jupiter in early July 2016.
2.42 The Juno spacecraft approached Jupiter on a hyperbolic trajectory with eccentric-ity e = 1.0172 and semimajor axis a =–4.384(106) km. Determine the asymptotic approach speed v−∞ and radial distance from Jupiter at its closest approach.
2.43 The Juno spacecraft fired a retrorocket at its periapsis (“perijove”) position to slow down and establish a highly elliptical orbit about Jupiter with semimajor axis a = 4,092,211 km and eccentricity e = 0.981574. Determine the orbital period and apoapsis (“apojove”) radius of Juno’s orbit.