Download Orbital Maneuvers and Hohmann Transfer for Communication Satellites and more Study notes Engineering in PDF only on Docsity!
COLLEGE OF ENGINEERING by: Dr. Naser Al-Falahy ELECTRICAL ENGINEERING
ORBIT MANOUVERS
Orbital plane change (inclination) It is an orbital maneuver aimed at changing the inclination of an orbiting body's orbit. This maneuver is also known as an orbital plane change as the plane of the orbit is tipped. This maneuver requires a change in the orbital velocity vector (delta v) at the orbital nodes (i.e. the point where the initial and desired orbits intersect, the line of orbital nodes is defined by the intersection of the two orbital planes).
In general, inclination changes can take a very large amount of delta v to perform, and most mission planners try to avoid them whenever possible to conserve fuel. This is typically achieved by launching a spacecraft directly into the desired inclination, or as close to it as possible so as to minimize any inclination change required over the duration of the spacecraft life.
When both orbits are circular (i.e. e = 0) and have the same radius the Delta-v (Δ v i) required for an inclination change (Δ v i) can be calculated using:
where:
v is the orbital velocity and has the same units as Δ v i (Δi ) inclination change required.
Example
Calculate the velocity change required to transfer a satellite from a circular 600 km orbit with an inclination of 28 degrees to an orbit of equal size with an inclination of 20 degrees.
SOLUTION,
r = (6,378.14 + 600) × 1,000 = 6,978,140 m , ϑ = 28 - 20 = 8 degrees
Vi = SQRT[ GM / r ] Vi = SQRT[ 3.986005×1014 / 6,978,140 ] Vi = 7,558 m/s
Δ v i = 2 × Vi × sin(ϑ/2) Δ v i = 2 × 7,558 × sin(8/2) Δ v i = 1,054 m/s
WEEK 4
COLLEGE OF ENGINEERING by: Dr. Naser Al-Falahy ELECTRICAL ENGINEERING
Orbital altitude change In orbital mechanics, the Hohmann transfer orbit is an elliptical orbit used to transfer between two circular orbits of different altitudes, in the same plane.
The orbital maneuver to perform the Hohmann transfer uses two engine impulses, one to move a spacecraft onto the transfer orbit and a second to move off it. This maneuver was named after Walter Hohmann, the German scientist who published a description of it in his book.
The diagram shows a Hohmann transfer orbit to bring a spacecraft from a lower circular orbit into a higher one. It is one half of an elliptic orbit that touches both the lower circular orbit that one wishes to leave (labeled 1 on diagram) and the higher circular orbit that one wishes to reach ( 3 on diagram). The transfer ( 2 on diagram) is initiated by firing the spacecraft's engine in order to accelerate it so that it will follow the elliptical orbit; this adds energy to the spacecraft's orbit. When the spacecraft has reached its destination orbit, its orbital speed (and hence its orbital energy) must be increased again in order to change the elliptic orbit to the larger circular one.
The total energy of the body is the sum of its kinetic energy and potential energy, and this total energy also equals half the potential at the average distance, (the semi-major axis):
Solving this equation for velocity results in the vis-viva equation:
where:
v is the speed of an orbiting body μ = GM is the standard gravitational parameter of the primary body, assuming M+m is not significantly bigger than M. r is the distance of the orbiting body from the primary focus a is the semi-major axis of the body's orbit. = (r 1 + r 2 )/
Hence, the energy of the transfer orbit is greater than the energy of the inner orbit (a = r1), and smaller than the energy of the outer orbit (a = r2). The velocities of the transfer orbit at perigee and apogee are given, from the conservation of energy equation, as
COLLEGE OF ENGINEERING by: Dr. Naser Al-Falahy ELECTRICAL ENGINEERING
The following diagram illustrates the geometry of the coplanar Hohmann transfer:
Example
A spacecraft is in a circular parking orbit with an altitude of 200 km. Calculate the velocity change required to perform a Hohmann transfer to a circular orbit at geosynchronous altitude.
Sol:
Given: rA = (6,378.14 + 200) × 1,000 = 6,578,140 m
rB = 42,164,170 m a = (rA + rB) / 2 a = (6,578,140 + 42,164,170) / 2 a = 24,371,155 m
V1 = SQRT[ GM / rA ] = SQRT[ 3.986005×10^14 / 6,578,140 ] V1 = 7,784 m/s
V 2 = SQRT[ GM / rB ] = SQRT[ 3.986005×10^14 / 42,164,170 ] V2 = 3,075 m/s
Vp = SQRT[ GM × (2 / rA - 1 / atx)] Vp = SQRT[ 3.986005×1014 × (2 / 6,578,140 - 1 / 24,371,155)]
COLLEGE OF ENGINEERING by: Dr. Naser Al-Falahy ELECTRICAL ENGINEERING
Vp = 10,239 m/s
Vap = SQRT[ GM × (2 / rB - 1 / atx)] Vap = SQRT[ 3.986005×1014 × (2 / 42,164,170 - 1 / 24,371,155)] Vap = 1,597 m/s
V 1 = Vp – V 1 = 10,239 - 7, V 1 = 2,455 m/s
V 2 = V 2 – Vap = 3,075 - 1, V 2 = 1,478 m/s VT = V 1 + V 2 = 2,455 + 1,478 , VT = 3,933 m/s
Ex- For the geostationary transfer orbit we have = 42,1 64 km and e.g. = 6,678 km
(altitude 300 km).
In the smaller circular orbit the speed is 7.73 km/s, in the larger one 3.07 km/s. In the
elliptical orbit in between the speed varies from 10.15 km/s at the perigee to 1.61 km/s at the apogee.
The delta-v's are 10.15 − 7.73 = 2.42 and 3.07 − 1.61 = 1.46 km/s, together 3.88 km/s.
Students need to validate the above exercise!
Exercise
A communication satellite was carried by the Space Shuttle into low earth orbit (LEO) at an altitude of 322 km and is to be transferred to a geostationary orbit (GEO) at 35, 860 km using a Hohmann transfer. Determine the characteristics of the transfer ellipse and the total Δv required.
From the inclination change equation, we see that if the angular change is equal to 60 degrees, the required change in velocity is equal to the current velocity.
