Orbital Mechanics Flashcards
What are some (4) milestones in the field of planetary motion and rocket science?
- Moon and planets move in elliptical orbits, Tycho Brahe
- Kepler’s 3 laws of planetary motion
- Newton’s laws
- Tsiolkovsky’s rocket equation
What are some (12) important variables of orbital geometry?
- R, spacecraft’s position vector, measured from Earth’s center
- V, spacecraft’s velocity vector
- F and F’, primary and vacant focus of the ellipse
- Rp, radius of the perigee (closest approach)
- Ra, radius of the apogee (farthest approach)
- 2a, major axis
- 2b, minor axis
- 2c, distance between the foci
- a, semimajor axis
- b, semiminor axis
- v/lowercase nu, true anomaly
- phi, flight-path angle
What is the general formula of an ellipse?
(x^2)/(a^2) + (y^2)/(b^2) = 1
What is a conic section (1) and what forms can it take (2)?
- It is a curve formed by passing a plane through a right circular cone
- The angular rotation of the plane determines whether the conic section becomes a:
- Circle
- Ellipse
- Parabola
- Hyperbola
What is the eccentricity of a circle (1), ellipse (2), parabola (3) and hyperbola (4)?
- e = 0
- e^2 = 1 - (b^2/a^2)
- e = 1
- e^2 = 1 + (b^2/a^2)
What are Kepler’s 3 laws of planetary motion?
- The Law of Ellipses
- The Equal Areas Law
- The Harmonic Law
How is Kepler’s 3rd law defined?
The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis of the orbits
How is Kepler’s 2nd law defined?
A line joining a planet and the sun sweeps out equal areas during equal intervals of time
How is Kepler’s 1st law defined?
The orbit of every planet is an ellipse with the Sun at one of the foci
How can the gravitational constant of Earth be derived from Newton’s Law of Universal Gravitation and Newton’s Second Law of Motion?
By setting them equal with m2 being the mass of an object (F = m2*a) and m1 being the mass of Earth, a becomes g = 9.81 m/s^2
How is the energy of two orbiting bodies defined?
Etotal = Ek + Ep (+deltaE(gains+losses))
- Ek is the kinetic energy
- Ep is the potential energy
What kind of field is gravity (1) and how does it affect the total energy of two orbiting bodies (2)?
- A conservative field, meaning energy is conserved
- The total energy remains constant
How is it called if the specific orbital energy is <0?
Bound orbit
What is a property of the specific orbital energy throughout its trajectory?
It is the same at every point of the trajectory
What is the specific orbital energy in an elliptic (1), a parabolic (2) and a hyperbolic (3) orbit equal to?
- The negative of the additional energy required to accelerate a mass of one kg to escape velocity (parabolic orbit)
- 0
- The excess energy compared to that of a parabolic orbit (here also referred to as characteristic energy)
What are the six quantities, called classical orbital elements, to desribe an orbit?
- Orbital size: Semimajor axis, a
- Orbital shape: Eccentricity, e
- Orientation of the orbital plane in space: Inclination, i
- Orientation of the orbital plane in space: Longitude of Ascending Node, capital omega –> Right Ascension of the Ascending Node (RAAN)
- Orientation of the orbit with the plane: Argument of Periapsis/Perigee, lowercase omega
- Object’s location in the orbit: True anomaly, v/lowercase nu
What does Direct/Prograde (1) and Indirect/Retrograde (2) mean?
An orbit moving:
1. In the direction of Earth’s rotation
2. Against the direction of Earth’s rotation
What is the Right Ascension of the Ascending Node (RAAN)?
An angle along the equator between the principal direction, I, and the point where the orbital plane crosses the equator from south to north, measured eastward
What is the argument of Perigee?
An angle between the ascending node and perigee, measured in direction of the spacecraft’s motion
What is the true anomaly?
An angle between perigee and the spacecraft’s position vector, measured in direction of the spacecraft’s motion
What are some (3) types of coordinate systems?
- Rectangular coordinates
- Polar coordinates
- Cylinder coordinates
What are some (4) origin points of coordinate systems and how are they called?
- Sun, heliocentric
- Earth, geocentric
- Center of gravity, barycentric
- Observation point, topocentric
What are some (2) position of the x-y-plane?
- Ecliptical (x-y-plane parallel to the ecliptic)
- Equatorial (x-y-plane parallel to Earth’s equatorial plane)
What is the first point of Aries?
Location in the sky where the Sun crosses the celestial equator from south to north at the March equinox
Why does the first point of Aries change over time?
Due to the precession of Earth’s axis
What are 3 types of Earth orbits defined by inclination?
- Equatorial orbit (Satellite covers equator)
- Polar orbit (Satellite covers all parts of Earth as planet rotates)
- Inclined orbit (Satellite covers range of latitudes in N/S hemisphere)
What are 2 types of Earth orbits defined by shape?
- Circular orbit
- Elliptical orbit
How high is Low-Earth orbit (LEO)?
160 to 2000 km
How high is Medium-Earth orbit (MEO)?
2000 to <36000 km
How high is High-Earth orbit (HEO)?
Higher than 35800 km
What is a geostationary orbit (GEO) (1) and how high is it (2)?
- An orbit where the orbital period of the satellite equals the rotational period of Earth
- 35800 km
What is a Sun Synchronous orbit (SSO) (1) and how high is it (2)?
- A near polar orbit, which precesses slowly over Earth, crossing the equator at the same local solar time each day, thus always maintaining the same relationship with the Sun.
- 600 to 800 km
What is a Geotransfer orbit (GTO) (1) and how high is it (2)?
- An orbit which is used to “transfer” objects from a LEO to a geostationary orbit
- 200 to 800 km in perigee, 36000 km in apogee
What is a Molniya orbit (1), how high is it (2), what is its inclination (3) and what is its orbital period (4)?
- An orbit with close perigee and high apogee which passes over mainly North America and Siberia in its apogees (8 hours to pass) and the South Atlantic and Pacific in its perigees (4 hours to pass), thus making it suitable for surveillance or communications in North America and Russia
- 400 km in perigee, 40000 km in apogee
- 63.4°
- 12h
What is a geosynchronous orbit?
A general form of the geostationary orbit, which doesn’t necessarily need to be circular or tracking the Earth’s equator
How high is the orbit of the ISS (1) and what is its orbital period (2)?
- 408 km
- 90 to 93 minutes
What is a ground track?
The path traced by an imaginary line between satellite and Earth’s center
Where do orbits drift when their orbital period is not equal to Earth’s rotation?
Westward
How does the inclination of a circular orbit affect the ground track?
The degree of inclination represents the total latitude reachable by the ground track
What is the Instantaneous Access Area?
The total area a given instrument could potentially see at a given moment
What is the Footprint Area?
Area a given instrument sees at a given moment
What is a Tundra orbit?
Similar to Molniya orbit, but its orbital period is 24h and it’s designed to not cover 2 places but 1
What are advantages (1) and disadvantages (2) of highly elliptical orbits (HEO), such as Molniya and Tundra?
- Stays in the northern hemisphere a long time, throws no shadow in apogee
- Crosses van Allen belts, 3 (Molniya) / 2 (Tundra) satellites required for coverage
How is Tsiolkovsky’s rocket equation defined (1) and what does it describe (2)?
- deltav = ce * ln(m0/mB)
- deltav: change of velocity
- ce: effective exhaust velocity of the combustion gases
- m0: start mass of the launcher (including propellant)
- mB: mass of the launcher at burn-out - It describes the motion of vehicles that follow the basic principle of a rocket (device that can apply acceleration to itself by expelling part of its mass with high velocity)
What is an orbit change (1) and what program had refining this process as one of its objectives (2)?
- Transferring an object from one orbit to another
- Gemini program, 1960’s
What is essential for the energy savings of a Hohmann transfer?
The velocity changes to transfer from one orbit to another are tangential to initial and final orbit
How are the 2 steps of a Hohmann transfer?
- First burn takes a spacecraft out of its initial, circular orbit and puts it into an elliptical transfer orbit
- Second burn takes it out of its transfer orbit and puts it into the final, circular orbit
What changes in a velocity vector of a spacecraft in a plane change of orbit and what remains the same?
Magnitude stays the same, the direction changes
What is the relationship between initial and final velocities in the orbits during a plane change?
deltaVsimple = 2Vinitial * sin(lowercase theta/2)
- deltaVsimple: velocity change for a simple plane change
- Vinitial = Vfinal: velocities in initial and final orbit
- lowercase theta: plane-change angle
Where must the velocity change take place in order to change the inclination of an orbit?
At the ascending or descending node, because the orbit then pivots around this node, changing the inclination
What can a plane change also be used for (1) and why is that useful (2)?
- To change the right ascension of the ascending node (RAAN)
- If we want a remote-sensing satellite to pass over a certain point on Earth at a certain time
Why should a plane change in an elliptical orbit be performed at its apogee?
Because deltaVsimple increases as the initial velocity increases, thus making it easier to perform a plane change when the initial velocity is low at the apogee
What is a “rendezvous” (1) and when was it needed in the past (2)?
- One spacecraft arriving at the same place at the same time as another one
- During the Apollo missions, where the astronauts returning from moon needed to rendezvous with the command module in lunar orbit / Space Shuttles and the ISS
What is a co-planar rendezvous?
A meeting of two objects with orbits in the same plane
What needs to be calculated in order to perform a co-planar rendevous?
- Angular velocity
- Time of flight
- Lead angle of the target
- Final “head start” angle
- Wait time
What does outbound and inbound refer to for a co-planar rendezvous?
- Outbound: Rocket leaving smaller orbit for bigger orbit, e.g. Earth to Mars
- Inbound: Rocket leaving bigger orbit for smaller orbit, e.g. Earth to Venus
What is a co-orbital rendezvous (1) and how is it performed (2)?
- A meeting of two objects in the same orbit, with one ahead of the other
- One object enters a phasing orbit which is slower or faster than the initial orbit to catch the other object
What are the two possibilities to perform a co-orbital rendezvous?
- Slow Down to Speed Up: To catch another spacecraft ahead, interceptor slows down to enter smaller phasing orbit with shorter period (lower energy)
- Speed Up to Slow Down: To catch another spacecraft behind, interceptor speeds up to enter bigger phasing orbit with longer period (more energy)
What needs to be calculated in order to perform a co-orbital rendezvous?
- Angular velocity of the target
- Travel angle of the target
- TOF of the target = Period of phasing orbit
- Semimajor axis of the phasing orbit
- Necessary delta v’s
What is a bi-elliptic transfer (1) and how is it performed (2)?
- Transfer encompassing two ellipses, eventually leading to the desired orbit
- -First burn boosts the spacecraft into first elliptical orbit with apoapsis at some point rb
- Second burn sends spacecraft into second elliptical orbit with periapsis at the radius of the final desired orbit
- Third burn injects spacecraft into desired orbit
How does a bi-elliptic transfer compare to a Hohmann transfer?
It requires one more boost and generally a longer travel time, but can have a lower total delta v when the ratio of final to initial semimajor axis is 11.94 or greater
How does spiraling work?
Gradual spacecraft boost leading to ever greater orbits, eventually leading to desired orbit
What is a gravity assist (1), what can it be used for (2) and when was it first used (3)?
- Use of the relative movement and gravity of an astronomical orbit
- Alters speed and path of a spacecraft, can save fuel and reduce expenses
- First used in 1959 by Soviet probe Luna 3
What are orbit pumping (1) and orbit cranking (2)?
- Gravity assist that changes the magnitude of a spacecraft’s velocity
- Gravity assist that changes the direction of travel
What are Type 1 (1) and Type 2 (2) interplanetary trajectories?
- If the interplanetary trajectory carries the spacecraft less than 180° around the sun
- If the interplanetary trajectory carries the spacecraft more than 180° around the sun
What is an impulsive manoeuvre?
Mathematical model of a manoeuvre as an instantaneous change in the spacecraft’s velocity, burn time tends to zero (not possible in physical world, but model describes effect of manoeuvre on orbit pretty well)
What is an atmospheric entry (1) and what two main types does it consist of (2)?
- Movement of an object from outer space into and through the atmosphere of a planet, dwarf planet or natural satellite
- Uncontrolled entry (astronomical objects like space debris)
- Controlled entry or re-entry (entry, descent and landing / EDL of a spacecraft)
Where does atmospheric entry occur for Earth (1), Mars (2) and Venus (3)?
- 100km (Karman line)
- 80km
- 250km
What is an attached (1) and detached (2) shock wave during re-entry?
- Streamlined spacecrafts, shock wave may attach to the tip and transfer a lot of heat (localized heating)
- Blunt spacecrafts, shock wave detaches and curves in front of the vehicle, leaving boundary of air between shock wave and vehicle surface
How does the Ballistic Coefficient differ between streamlined and blunt vehicles?
BC of streamlined vehicle > BC of blunt vehicle
What is a re-entry corridor?
The 3-dimensional corridor a vehicle must pass to not skip out or burn up
What requirements need to be balanced for a re-entry system?
- Vehicle design (size and shape, thermal-protection systems)
- Trajectory design (re-entry velocity and re-entry flight path angle)
What is aerobraking?
Orbiting spacecraft brushes against the top of a planetary atmosphere, slows down spacecraft and lowers its altitude (saves cost and mass)
What are the first and second cosmic velocities?
- 7.9 km/s
- 11.2 km/s
