Celestial Mechanics
Celestial Mechanics
Orbital Maneuvers
Earth, Moon, Mars, and Beyond
Rodolfo G. Ngilangil Jr. MBA, BSAE
Impulsive Maneuvers
Impulsive maneuvers are those in which brief firings of onboard rocket motors change the magnitude and direction of the velocity vector instantaneously. The position of the spacecraft is considered to be fixed during the maneuvers. This is true for high-thrust rockets with burn times short compared with the coasting of the spacecraft.
Orbital Maneuvers - 2
Hohmann Transfer - Definition
The Hohmann transfer is an elliptical orbit tangent to both circles at its apse line. The periapse and apoapse of the transfer ellipse are the radii of the inner and outer circles.
The Hohmann transfer is the most energy efficient twoimpulse maneuver for transferring between two coplanar circular orbits sharing a common focus. [Conditional]
Orbital Maneuvers - 3
Hohmann Transfer - Seatwork Challenge
: Orbits 1 and 2 are two circular orbits with altitudes of 300 km and 35,786 km, respectively. An elliptical orbit is tangent to both circles at its apse line. Determine the velocities of a 2,0000kg spacecraft at point A and B on both elliptical and circular orbits.
Orbital Maneuvers - 4
Hohmann Transfer - 2 v1 = vp = va =
µ rp v2 =
µ ra 2ra rp (ra + rp )
µ
µ 2rp ra (ra + rp )
∆v = ∆v A + ∆vB = (vp − v1 ) + (v2 − va )
ε1 = − ε3 = − ra > rp
µ
2rp
ε2 = −
=−
µ
2ra
= =
µ µ µ 2rp 2ra − + − rp (ra + rp ) rp ra ra (ra + rp ) µ⎛ ⎜
µ
µ
2a
µ ra + rp
ε1 < ε3 < ε 2
⎞ 2rp ⎞ µ⎛ 2ra ⎟ ⎜1 − −1⎟ + rp ⎜ (ra + rp ) ⎟ ra ⎜ (ra + rp ) ⎟ ⎠ ⎠ ⎝ ⎝ µ ⎡ 2ra / rp ⎛ rp ⎞ rp ⎤ ⎜1 − ⎟ + = −1⎥ ⎢ rp ⎢ (1 + ra / rp ) ⎜ ra ⎟ ra ⎥ ⎝ ⎠ ⎣ ⎦
Orbital Maneuvers - 5
Hohmann Transfer - 3
∆v = ∆v = ∆v = ⎤ 2ra / rp ⎛ rp ⎞ r ⎜1 − ⎟ + p − 1⎥ ⋅⎢ rp ⎢ (1 + ra / rp ) ⎜ ra ⎟ ra ⎝ ⎠ ⎥ ⎣ ⎦
µ ⎡
r 2ra / rp ⎛ rp ⎞ ∆v ⎜1 − ⎟ + p − 1 = ⎜ r ⎟ v1 ra (1 + ra / rp ) ⎝ a ⎠ 2r ⎛ 1 ⎞ 1 −1 ⎜1 − ⎟ + r (1 + r ) ⎝ r ⎠
r=
Orbital Maneuvers
Earth, Moon, Mars, and Beyond
Rodolfo G. Ngilangil Jr. MBA, BSAE
Impulsive Maneuvers
Impulsive maneuvers are those in which brief firings of onboard rocket motors change the magnitude and direction of the velocity vector instantaneously. The position of the spacecraft is considered to be fixed during the maneuvers. This is true for high-thrust rockets with burn times short compared with the coasting of the spacecraft.
Orbital Maneuvers - 2
Hohmann Transfer - Definition
The Hohmann transfer is an elliptical orbit tangent to both circles at its apse line. The periapse and apoapse of the transfer ellipse are the radii of the inner and outer circles.
The Hohmann transfer is the most energy efficient twoimpulse maneuver for transferring between two coplanar circular orbits sharing a common focus. [Conditional]
Orbital Maneuvers - 3
Hohmann Transfer - Seatwork Challenge
: Orbits 1 and 2 are two circular orbits with altitudes of 300 km and 35,786 km, respectively. An elliptical orbit is tangent to both circles at its apse line. Determine the velocities of a 2,0000kg spacecraft at point A and B on both elliptical and circular orbits.
Orbital Maneuvers - 4
Hohmann Transfer - 2 v1 = vp = va =
µ rp v2 =
µ ra 2ra rp (ra + rp )
µ
µ 2rp ra (ra + rp )
∆v = ∆v A + ∆vB = (vp − v1 ) + (v2 − va )
ε1 = − ε3 = − ra > rp
µ
2rp
ε2 = −
=−
µ
2ra
= =
µ µ µ 2rp 2ra − + − rp (ra + rp ) rp ra ra (ra + rp ) µ⎛ ⎜
µ
µ
2a
µ ra + rp
ε1 < ε3 < ε 2
⎞ 2rp ⎞ µ⎛ 2ra ⎟ ⎜1 − −1⎟ + rp ⎜ (ra + rp ) ⎟ ra ⎜ (ra + rp ) ⎟ ⎠ ⎠ ⎝ ⎝ µ ⎡ 2ra / rp ⎛ rp ⎞ rp ⎤ ⎜1 − ⎟ + = −1⎥ ⎢ rp ⎢ (1 + ra / rp ) ⎜ ra ⎟ ra ⎥ ⎝ ⎠ ⎣ ⎦
Orbital Maneuvers - 5
Hohmann Transfer - 3
∆v = ∆v = ∆v = ⎤ 2ra / rp ⎛ rp ⎞ r ⎜1 − ⎟ + p − 1⎥ ⋅⎢ rp ⎢ (1 + ra / rp ) ⎜ ra ⎟ ra ⎝ ⎠ ⎥ ⎣ ⎦
µ ⎡
r 2ra / rp ⎛ rp ⎞ ∆v ⎜1 − ⎟ + p − 1 = ⎜ r ⎟ v1 ra (1 + ra / rp ) ⎝ a ⎠ 2r ⎛ 1 ⎞ 1 −1 ⎜1 − ⎟ + r (1 + r ) ⎝ r ⎠
r=