Lecture

Coordinate systems used to describe the postion and motion of an object in space

Vernal equinox is at 21th of March, which is also the ascending node (between equatorial plane and ecliptic)

The direction in which the sun is seen from earth at vernal quinox

Earths precession causes the position of the equinoxes to move “westward”. -> point of aries changes

• x points to Aries

• z normal to ecliptic

• y right hand system

• I to Aries

• K normal to equatorial plane

• J right hand system

• P shows to periapsis of orbit around earth

• W normal to orbit

• Q right hand system

• RS in orbit plane whilre r is radial and W is normal to plane

• NT in orbit plane while T is in velocity direction, W is normal to plane

• solar day is 24h rotating 361°

• sidereal day is 23h 56m 4s for 360°

Satellites path on Earth surface.

Orbit projection to nadir

The inclination is shown on the maximum latitude for the groundtrack

Earths rotation makes the GT drift westwards

Highly elliptical orbits

• spy satellites

• apogee over locations of interest for longer time period

Geostationary orbits. Orbit period same as rotation period of earth.

Height is 35700km

Radius is 42000km

Geosynchronos Orbit

• Orbit Period is earth rotation period

• not necessarly e=i=0

coverage is limited by inclination, therefore reachable latitude

The width of the ground track what a satellite would see at a certain altitude

• Access Area: total Area a instrument could potentially see

• Footprint Area: FOV, Area a instrument sees at a given moment

Constellation with t satellites flying on p orbits uniformly distributed with inclination i and a relative spacing between neighboring orbits

• f: Phase (True Anomaly) between two nearest satellites on two adjacent orbits. I.e. if a satellite in the nth orbit is just at the ascending node (n = 0), then in the neighboring orbit n+1 a satellite has the True Anomaly of

The angle between the sun vector and the orbit plane

• Distance determination via time-of-flight method (1m - 20m accuracy)

• Radial velocity via Doppler shift (0,1 - mm/s accuracy)

• tangential burn: along track burn, changes a and energy

• transverse burn: in orbit plane but normal to flight path

• normal burn: normal to flight path and normal to orbit plane -> inclination change

• apogee/perigee for e,a theta=0=180

• Nodes for i omega+theta=0=180

• Orthogonal to nodes ,RAAN omega+theta = 90 =270

• for apogee at perigee

• for perigee at apogee

Delta-v will be lowest when orbital velocity v is smallest at time of kickburn. Therefore kickburns are often done in apoapsis

A unique 2 impulse transfer that takes a S/C from Point 1 to Point 2 on a destination orbit.

For uniqueness one needs a constraint:

• lowest delta v

• shortest time

• ….

What is the orbit path that will take me in the shortest time from a given point P1 on an exit orbit to a given point P2 on a target orbit? And: Which kick-burns do I have to set at P1 und P2?

A Lambert transfer between two circular orbits with the lowest delta-v demand

To reduce target error. The less the ratio of radii is the smaller the error.

But this takes more time

Also mid course corrections can be done but uses more delta v

Hohmann transfer

• Bi-parabolic transfer:

  Accelerate to escape velocity, then return the satellite from "infinity" to the desired orbit

• Bi-elliptic transfer:

  Accelerate to near escape velocity, then return the satellite to the desired orbit.

Supersynchronous geostationary transfer orbit:

• bi-elliptic transfer from LEO to GEO with inclination change

• used because launch side latitude must be corrected for GEOs. Cheaper when further away from earth

1. SSTO

2. ITO

3. Circulazation burn for GEO

• Fuel efficiency high specific impulse

• Solar power utilization

• More payload possible (less space needed for fuel)

• Precision

  
  

The Hill equations describe the relative motion of two nearby orbiting objects in space

1. Launch

2. Phasing

3. Homing

4. Closing

5. Final approach

beta = Latitude

phi = Launch azimuth (Angle between north and launch trajectory)

S/C orbit radius smaller than target -> increased speed lets the S/C “catch up” to target.

Each orbit the S/C closes in on the target by an angle delta theta

Hohmann transfer from S1 20km to S2 3,5km behind ISS. Phase decreases further in angle

Radial Boost Transfer Maneuver, alike throw in z axis -> generates elliptic trajectory.

Moves the S/C from S2 3,5km to S3 250m behind the ISS and is now in the Approach Ellipsiod just outside the keep out sphere

Forced Motion Straight Line Approach from S3 to S4 (20m). Entering the Approach cone inside KOS.

With a translation force x but also in z direction to prevent rising of the orbit

Now linear equations of motions are used

Kinematics = Motion of a rigid body WITHOUT external force -> Nutation

Dynamics = Motion of a rigid body WITH external force -> Nutation + Precession

The main axis with the largest (oblate) and smallest (prolate) moment of inertia is dynamically stable in the short term. The mean moment of inertia is not stable

In the long term (dissipation, nutation decreases) only the oblate rotation is stable -> "flat spin rotation“

Circular motion of the rotation axis around the angular momentum vector

Nutation damper via ball in tube, energy dissipates due to friction

The circular motion of the angular momentum vector, because of an external force (gravitation) which induces a Torque

Sun and Moon gravitation act on mass buldge on equator.

• Atmospheric Drag and Torque

• External Torques due to Eddy currents

• Gravity gradient torque

The parts of an extended body that are closer to the Earth experience a slightly larger gravitational force and a slightly smaller centrifugal force, and the parts on the outside experience exactly the opposite.

COP and COM are often offsetted, therefore inducing a moment around the COM

Stable pitch oscillation around “y” axis if

because of Torque.

If the oscillation is damped the S/C orbits around the Earth with constant orientation to the center of the Earth.

a: Semimajor axis

e: Eccentricity

i: Inclination

Ω: Right Ascension of the Ascending Node (RAAN)

small omega: Argument of Perigee

theta: true anomaly

• Geoid: Gravitational anomalies -> potato shape

• 2nd approxmiation earth is a sphere + oblate rotational ellipsoid + potato

The oblatness induces a torque on the satellite which leads to a rotation of the orbital plane.

• RAAN_dot < 0 for i < 90°

• RAAN_dot > 0 for i > 90°

Argument of Perigee rotates because of perturbations.

• omega_dot > 0 for i < 63,42°

• omega_dot < 0 for i > 63,42°

Sun-Synchronous Orbits:

RAAN rotates exactly 360° per year, but RAAN needs to be >0 so i > 90° needed

• Type I:

• Type II:

2,2 +- 0,2

semi-major-axix and eccentricity

Different parts of earth surface induce different gravitational pull periodically

Orbits of bodies which have periods of a simple integer ratio

3/2 1/2 etc

repeat ground track satellites retrace their ground track pattern after a number of k revolutoins and l days

-> resonance with earths rotation

• Triaxiality: satellites ground track oszillates east- and westwards due to the “potato” shaped earth and therefore gravity perturbations, where the satellite is accelerated and decellerated due to gravity “spots” on earth

• solar pressure -> eccentric

North-South movement of satellite due to the influence of the sun and moon, which induce a out of orbit-plane force

With a symmetric RAAN adjustment or inclination adjustment (but more expensive)

m << other two masses

Point where gravitational forces of 2 large bodys on a smaller body balances out with the centrifugal force.

Points are stable or semi-stable

Subset of Lagrange points on 1 line

• horizontal Lyapunov orbit

• vertical Lyapunov orbit

• Halo orbit

• Lissajous orbit

• Quasi-halo orbit

• Heteroclinic: joining to different equilibrium points in phase space

• Homoclinic: joining a saddle point to itself

Yes, with increasing distance from the points a S/C gains speed. The velocity plus the rotation of the system leads to a coriolis force which transports the S/C back to a path around the Lagrange points

Sphere of Influence, a sphere where the gravity of a planet exceeds the one from the sun

• outer planet: in direction of v_planet

• inner planet: opposite to v_planet

v_planet of departure planet not target

• steering loss: Losses due to the vectoring of engine to stay on path. Diverting thrust away from optimum

• drag losses: self explanatory

• gravitational losses: rocket has to start vertically to gain height. Ideally it would be better to start tangential to earth surface to build up centrifugal force

• Perform course changes at smallest velocities possible, that is in the early ascent phase (e.g. roll of the Shuttle into orbit inclination)

• Minimize gravitational losses (Turn horizontally as early as possible). This increases drag and steering losses. So a turn too early it not optimal.

Use gravitational pull g*sin(gamma) for reducing flight path angle gamme, and turn into the final more or less horizontal orbit.

Has to be iniciated via pitch maneuver 3-5°

Due to drag, a flight trajecory is optimal when it is initially steeper and later more horizontal than an exclusive gravity turn. This is achieved by a so-called constant pitch-rate maneuver (gamma_dot=const)

9 km/s

Reentry with such a large lift that lift plus centrifugal force outweigh gravity and carry the S/C back out into space.

With the lift, the S/C is adjusted exactly so that it cancels the gravitational force together with the centrifugal force.

a = 0.4 + 0.3x2^n

• Mercury: n = –infinity

• Venus: n = 0

• Earth: n = 1

• Mars: n = 2

• Planetoids: n = 3

• Jupiter: n = 4