Orbital Mechanics Flashcards
What is a sidereal day?
The time it takes for the earth to perform one full rotation on its axis. This is different to a solar day which considers the earth’s motion around the sun also.
What is a Satellite’s ballistic ratio?
ratio of its mass to the product of its area with its drag coefficient
What is apsidal rotation?
The earth’s bulge gives rise to uneven forces acting on the mass in orbit. These forces result in rotation over time of the line of apsides. i.e. the orbit rotates within the orbital plane. dω/dt.
What are Kepler’s three laws?
- Planets move in a plane and their orbits are described by ellipses/
- The radius vector from the Sun to the planet sweeps out equal areas in equal times.
- Orbital period = 2pi (a3/GM)^0.5
In an elliptical orbit around the Sun, what are the following terms describing:
1. Aphelion
2. Perihelion
3. Semi-major axis
4. Semi-minor axis
5. Foci
6. Semi-distance between Foci
- Furthest point from the Sun in a heliocentric orbit
- Nearest point to the Sun in a heliocentric orbit
- half the distance between the periapsis and the apoapsis (i.e. half the length of the line of apsides)
- Distance to the orbit in the direction normal to the line of apsides at a
- Two two focus points in the orbit. M is expected to be at one of them while the other is empty for the 2-body problem
- Distance between the foci divided by 2, c. e=c/a.
What are the four main assumptions of the ideal Keplerian orbit in a 2-body system?
- M»m; One of the foci is at the centre of M.
- Both M and m are spherical and homogeneous
- Motion occurs in free space and no influence from other bodies.
- Only the force of gravity acts on m from M
What are the 5 Keplerian elements?
- a - semi major axis
- e - eccentricity
- i - inclination angle
- Ω - Right angle of ascension node (RAAN)
- ω - Argument of perigee
What are the four trajectory solutions to motion under gravity in the ideal 2-body problem? How are they derived
Four conic sections:
1. Circle
2. Ellipse
3. Parabola
4. Hyperbola
Taking different cuts from a circular-based cone (see google)
In orbital geometry, what is the local horizontal?
the line that runs perpendicular to the distance vector connecting m to M
In an elliptical orbit of a satellite around a planet, what are the following terms describing:
1. True anomaly
2. Flight path angle
3. Radius of apoapsis
4. Radius of periapsis
- Angle between the distance vector and the periapsis
- Angle between the local horizontal and the velocity vector
- AP prefix is the largest distance from M to the ellipse
- Peri is the smallest. Apsis is the general case which becomes ‘helion’ and ‘gee’ for Sun and Earth.
Explain the main features of a perifocal coordinate system
- Centre of M is the origin
- Location of the periapsis defines the direction of the x-axis
What are the three mission segments
Space segment: payload and bus management
Launch segment: appropriate launch vehicle to get spacecraft to its orbit
Ground segment: mission control
What does SMAD mean?
Space mission analysis and design
To overcome the seemingly impossible rocket mass requirements, what technique is used?
Staging allows for waste mass to be deposited to make launch more fuel efficient
What are the four main assumptions of the two-body problem?
- M»_space; m
- M and m are spherical and homogeneous
- the only bodies present are M and m
- no other forces act on M and m
What permutations to the rwo-body problem make it more like a real orbit model?
- geopotential field
- atmospheric drag
- lunisolar attraction
- solar radiation pressure
What two main effects from the j2 harmonic allow for SSOs, Molniya and Tundra orbits?
Precession of the line of nodes (dΩ/dt)
Rotation of the line of apsides (dω/dt)
What two main effects from the j2 harmonic allow for SSOs, Molniya and Tundra orbits?
Precession of the line of nodes (dΩ/dt)
Rotation of the line of apsides (dω/dt)
What two forces must be balanced in order for a body to maintain its orbit
Centripetal and gravitational force
GMm/r2 = mv2/r
What three conditions are necessary for an orbit to be geostationary (GEO)?
- Orbital period must be equal to sidereal day
- Eccentricity must be zero (i.e. circular orbit)
- The inclination angle must be 0
if latter two conditions not met orbit is Geosynchronous not geostationary
Orbit perturbations cause changes to e and i such as from lunisolar attraction and solar radiation pressure
How is SSO achieved?
What is the associated equation?
achievd by setting the rate of change of nodal precession equal to earth’s rotation around the sun in degrees per day which turns out to be 0.986 degrees per day.
What are the primary components of a spacecraft?
The payload (mission-specific equipment) and the bus (housekeeping systems like power, thermal control, and communication).
Why is systems engineering critical in space mission design?
It ensures interdisciplinary coordination, robustness, and that the mission is delivered on time and within budget.
What is the role of trade-off analysis in mission design?
To balance mission requirements, constraints, and design choices iteratively until an optimal solution is achieved.
What is Specific Impulse in rocketry?
A measure of rocket efficiency, defined as thrust per unit weight flow of propellant, with higher I_sp indicating more efficient engines.
Why are launch sites near the equator advantageous?
The Earth’s rotational velocity provides additional boost, reducing propellant requirements for equatorial and geostationary orbits.
What factors determine the choice of a launch site?
Payload weight and size, target orbit (inclination and altitude), infrastructure, risk, cost, and geopolitical considerations.
What is the significance of the launch azimuth angle?
It determines the inclination of the orbit, with lower inclinations requiring launches closer to the equator.
What is the vis viva equation?
v 2 = μ ( 2/r - 1/a), relating orbital velocity to distance from the focal body, r, and semi-major axis a.
What is a Hohmann Transfer Orbit?
A two-impulse elliptical transfer orbit that connects two circular orbits in the same plane with minimal energy.
How do plane change maneuvers impact mission design?
They require significant
ΔV, making them energy-intensive, especially in low Earth orbit.
What causes orbital perturbations?
Non-ideal effects like Earth’s oblateness, atmospheric drag, solar radiation pressure, and third-body effects (Moon and Sun).
What is the purpose of sun-synchronous orbits (SSOs)?
To maintain a constant solar illumination angle, ideal for imaging and environmental monitoring satellites
What is the LTAN in SSOs?
Local Time of Ascending Node, defining the time of day when the satellite crosses the equator heading north
What are the characteristics of geostationary orbits (GEO)?
Altitude of ~35,786 km, zero inclination, orbital period of 24 hours, appearing stationary to ground observers.
How are Molniya orbits used?
For high-latitude coverage, with highly elliptical orbits and an inclination of ~63.4° to minimize apsidal rotation.
How does the Earth’s rotation assist rocket launches?
Provides a velocity boost of up to ~0.46 km/s at the equator, reducing fuel requirements.
What does the diagram of a Hohmann transfer illustrate?
Two burns: one to transfer from a lower circular orbit to an elliptical orbit, and another to circularize at the target orbit.
What is the Geocentric Equatorial Coordinate System (GECS)?
A 3D inertial reference frame centered at Earth’s center, with axes aligned with the Earth’s equator and vernal equinox.
What is the perifocal coordinate system used for
Describing an orbit in the plane of motion, centered at the primary body with the periapsis as the reference direction.
Describing an orbit in the plane of motion, centered at the primary body with the periapsis as the reference direction.
Drag reduces semi-major axis
a and causes energy loss, strongest at perigee but evident at apogee due to orbit shape change.
What are sun-synchronous orbits’ advantages for radar missions?
Constant illumination conditions optimize radar imaging performance, especially in terminator-aligned orbits.
How do constellations like GPS ensure global coverage?
By positioning satellites in multiple orbital planes with sufficient spacing to maintain visibility from any location on Earth.
Air density falls off with height but there are two factors influence where these curves start and end what are they?
- Air density is lower at night than in the day
- Air density is higher during periods of large solar activity
how does velocity of a spacecraft change as it experiences atmospheric drag?
Its velocity will increase as the semi-major axis decreases
-the Hamiltonian will decrease as the fall in potential energy is greater than the increase in kinetic
how is the flatness of earth calculated?
(D_E - D_p ) / D_E where D_E is the distance from centre along the equator to the edge and p for pole
What causes the line of nodes to precess?
The uneven forces experienced by the satellite from the equator and the pole drives the angular momentum vector to precess
What is the equation for the force experienced due to solar radiation pressure on a satellite?
F=KAP
K: reflectivity coefficient between 0 and 2
A: Projected area of the satellite
P: momentum flux from the sun
Why is lunisolar attraction onlly considered at GEO? What are the effects on the inclination angle?
Because at lower altitudes, the earth’s mass distribution dominates.
Periodic:
13.66 day cycle with 0.0035 degree shift (half a moon orbit)
182.65 day cycle with 0.023 degree shift (half a year)
- Saros cycle leading to inclination drifts of about 0.75 to 0.95 degrees per year.
How can we modify the keplerian elements to account for perturbations to the orbit over time?
Add a time dependence to the mean motion, RAAN and argument of perigee:
n(to) + dn/dt . (t-to)
what are the three advantages of Geostationary orbits?
- no tracking required for ground station antennas
- large coverage of earth’s surface at 42.6%
- can interconnect earth stations with large distances between them
what are the three disadvantages of Geostationary orbits?
- polar regions get no coverage
- large propagation delay because of altitude
- high path loss
For LEO orbits, above what inclination angle do SSO orbits require?
> 96 degrees (retrograde orbits)
What is a retrograde orbit?
Inclination angle > 90 degrees i.e. travels against the motion of the earth’s rotation
what condition must be met to obtain a Molniya orbit?
the rate of change of line of apsides must be set equal to 0 by solving the 1-5cos^2i term giving an inclination angle of 63.4 degrees
what is the advantage and disadvantage of being in SSO above the terminator (6am - 6pm)?
- always solar flux on the satellite so godo for high-power payloads like radar
- not good for imaging
what are the best LTAN for imaging?
- 3am and 9am
- 10 - 11 am best for mid latitudes
What is a typical eccentricity of a Molniya orbit?
0.6 - 0.75
What is the orbital period of a Molniya satellite?
Exactly half a sidereal day
Asides from latitudinal coverage, why else is a Molniya orbit useful?
Sweeps the exact same geographical spot every other orbit
When transferring between different orbits using a Hohmann transfer, are the velocity increases of the two burns positive or negative?
When moving to a bigger orbit they are both positive, when moving to a smaller orbit, they are both negative
Describe the 3 steps for a launch vehicle to lift a satellite into geostationary orbit
Depending on launch location it either:
1. ascends to a parking orbit, it then injects its payload into a GTO once it arrives at the perigee
2. ascends to the altitude and reaches the desired injection velocity to achieve GTO.
- The payload then enters GTO and once it reaches the apogee of the GTO it then performs a burn to join the GEO and make any inclination changes needed.
Suppose we have a rocket of mass ‘M’, which increases its velocity by an amount ‘dV’ after burning an amount of fuel ‘dM’, and ejecting the exhaust at a relative velocity ‘ve’, then by conservation: MdV= vedM. What equation does this lead to if integrating from t_init to t_final
- integrate
- deltaV = V_e . ln (m_final) - ln (m_init) = V_e . ln [(m+M) / M]
What is the equation for the exhaust velocity?
V_e = g . I_sp
What launch site is ideal for launching geostationary satellites and why?
Kourou because its positioned close to the equator so less propellent is needed to achieve injection velocity and to make inclination changes to the orbit
if a satellite needs to change inclination once it has achieved orbit, what equation is used to determine the velocity change required for the corresponding change to inclination? When is the best time to perform this manouvre
deltaV = 2V sin (deltai / 2)
When the satellite is slowest, i.e. at apogee
how long are launch windows and why do they exist?
- About 10 minutes
- Because orbits usually require some fixed relationship with celestial bodies such as the sun
what is typical altitude of a SSO?
600 - 800 km
