1. Space Mechanics

2. Introduction to Space Mechanics
It concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment. (from Wikipedia)
Space (Outer Space)
Earth’s atmosphere
Solar system
Celestial mechanics (motions of celestial objects)
Orbital mechanics, also called ‘astrodynamics’ (motion of artificial satellites/space probes/ballistic missiles and their orbits/trajectories – Keplerian orbits and real orbits)
Various missions and analysis
Earth mission, Interplanetary mission, Lunar mission
Limit to our scope

3. A thin layer of gases extends over 100 kms above Earth’s surface.
Earth’s Atmosphere
Temperature variation
A thin layer
of gases
extends over
100 kms
above Earth’s
surface.
Alt.
Temp.

4. Introduction to Celestial Space
Celestial Sphere

5. Introduction to Celestial Space
Ecliptic

6. Introduction to Celestial Space
Obliquity of the Ecliptic

7. Introduction to Celestial Space
Right Ascension and Declination

8. Introduction to Celestial Space
Right Ascension and Declination

9. Introduction to Celestial Space
Vernal Equinox and Autumnal Equinox

10. Introduction to Celestial Space
Equinoxes and Solstices
Figure:
The Sun and the seasons
Point
Usual Date
Right Ascension
Declination
Vernal Equinox
March 20
0 hours 0°
Summer Solstice
June 21
6 hours
23.5° N
Autumnal Equinox
September 23
12 hours
Winter Solstice
December 22
18 hours
23.5° S

11. Introduction to Celestial Space
Solar day (24 hrs), Sidereal day (23 hrs 56 mins 4 secs)

12. Introduction to Celestial Space
Important units followed for distance in space
Astronomical Unit (AU)
An astronomical unit is a unit of length equal to 149,597,870,700 metres (92,955, miles) or approximately the mean Earth–Sun distance.
Length of the semi-major axis of the Earth's elliptical orbit around the Sun. The average distance is about 150 million kilometres (149.6 × 10^6 km).
Light Year
The distance which the light travels in one year.
(9.46 × 10^12 km)

13. Coordinate Systems in Space
Coordinate Systems (for space applications)

14. Coordinate Systems in Space
Coordinate Systems for space missions (…contd)
Also called local horizontal
coordinate system

15. Coordinate Systems in Space
Coordinate Systems for space missions (…contd)
Ecliptic (or heliocentric) coordinate system

16. Coordinate Systems in Space
Summary of Coordinate Systems (for space mission)

17. Copernicus (1473 – 1543) Sun-centered Solar system
Orbital Mechanics
Copernicus (1473 – 1543) Sun-centered Solar system

18. Overview of Orbital Mechanics
Newton’s Law of Universal Gravitation
Kepler’s Laws of Planetary Motion
Orbital Elements (parameters) for Keplerian Orbits
Two-boby problem and Orbit (Trajectory) Equation
Orbital Maneuvers
Orbit Perturbation Analysis
Tycho Brahe Johannes Kepler Sir Issac Newton (1546 – 1601) (1571 – 1630) (1642 – 1727)

19. Kepler Laws of Planetary Motion
Kepler’s Laws of Planetary Motion:
First Law: The orbit of each planet
is an ellipse, with the Sun at one focus.
Second Law: The line joining the planet
to the Sun sweeps out equal areas in
equal times.
Third Law: The square of the period of
a planet is proportional to the cube of its ,
mean distance from the Sun.

20. Classical Orbital Elements (for Keplerian Orbits)
These orbital elements will mathematically describe an orbit in space.
These parameters are enough to define the location of a body in space moving in any Keplerian orbit.
Semi-major axis (a)
Eccentricity (e)
Inclination (i)
Argument of Periapsis (w)
Right ascension of ascending node (W)
True anomaly (q) or Time of Periapsis passage

21. Semi-major axis and Eccentricity
Semi-major axis (a) is one-half of the major axis and is a satellite’s mean distance from its primary (for eg., Earth).
Semi-major axis describes the size of an orbit.
Eccentricity (e) is the distance between the foci divided by the length of the major axis and it describes the shape of an orbit.
Orbit
Semi-major axis
Eccentricity
Circle
= radius
Ellipse
> 0
0 < e < 1
Parabola
infinity 1
Hyperbola
< 0
> 1

22. Inclination
Inclination (i) is the angle between a satellite’s orbital plane and the equator of its primary. It gives the tilt of an orbital plane.
(1) 0° inclination indicates an orbit about primary’s equator in the
same direction as the primary’s rotation.
(2) 90° inclination indicates polar
orbit (for surveillance of earth).
(3) 0° < i < 90° inclination indicates
prograde orbit (direct orbits).
(4) > 90° to 180° inclination
indicates retrograde orbit .

23. Inclination
Inclination (i) is also defined as the angle between the angular momentum (h) vector and the unit vector in the Z-direction.

24. Nodes of an orbit
Nodes (n) are the points where an orbit crosses a plane, such as a satellite crossing the Earth’s equatorial plane.
If the satellite crossing the Earth’s equatorial plane going from south to north, the node is ascending node.
If the satellite crossing the Earth’s equatorial plane going from north to south, the node is descending node.

25. Right ascension of ascending node
The right ascension (longitude) of ascending node (W) is the node’s celestial longitude.
It is the angle from the vernal equinox direction to the ascending node measured in the direction of primary’s rotation.

26. Argument of Periapsis
Periapsis is a point in an orbit (where the major-axis line intersecting) closest to the primary. (For sun-centered orbit, the point is perihelion and for earth-centered orbit, it is perigee)
The argument of periapsis (w) is the angular distance between the ascending node and the point of periapsis measured in the direction of satellite’s motion.

27. True anomaly Orbit Equation
True anomaly (q), also called polar angle of an orbit (ellipse), is the angle measured in the direction of satellite’s motion from the direction of perigee to the position vector.
Orbit Equation

28. Conic Sections

29. Conic Sections – Keplerian Orbits

30. Orbits – Basic Classifications
Centric Inclination Eccentricity
Heliocentric Inclined Conic sections
Lunar Non-inclined
Jovicentric
Neptunocentric Low Earth Orbit
Uranocentric Geosynchronous Orbit
Geocentric Medium Earth Orbit
Highly Elliptical Orbit
Source:

31. Geocentric Orbits

32. Two-body Problem Bodies are spherically symmetric.
Figure: Relative Motion of two bodies two-body equation
of motion
Bodies are spherically symmetric.
No external nor internal forces other than gravitational force.

33. Earth-Satellite System
Constants of Satellite motion
Conservation of total energy (or mechanical energy)
Conservation of angular momentum
Note: At apogee and perigee,

34. Keplerian Orbit – Properties

35. Keplerian Orbit – Velocities

36. Kepler’s Equation and Time of Flight
Eccentric anomaly (E) and Mean anomaly (M)
Kepler’s Equation
Mean Motion (n)

37. Some Useful Relations

38. Introduction to Orbital Maneuvers
An Orbital maneuver is the use of propulsion systems to change the orbit of a spacecraft.
Hohmann Transfer Bi-elliptical Transfer

39. Reference(s)
Wiley J.Larson & James R.Wertz, Space Mission Analysis and Design, Microcosm Press & Kluwer Academic Publishers, 1999.
Roger R. Bate, Donald D. Mueller & Jerry E. White, Fundamentals of Astrodynamics, Dover Publications, 1971.
First Year Higher Secondary Text Book of Physics, Vol 1.
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