Space Mechanics

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

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.

Introduction to Celestial Space Celestial Sphere

Introduction to Celestial Space Ecliptic

Introduction to Celestial Space Obliquity of the Ecliptic

Introduction to Celestial Space Right Ascension and Declination

Introduction to Celestial Space Right Ascension and Declination

Introduction to Celestial Space Vernal Equinox and Autumnal Equinox

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

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

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 (92,955,807.273 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)

Coordinate Systems in Space Coordinate Systems (for space applications)

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

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

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

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

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)

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.

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

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

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 .

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

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.

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.

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.

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

Conic Sections

Conic Sections – Keplerian Orbits

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: http://en.wikipedia.org/wiki/List_of_orbits

Geocentric Orbits

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.

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

Keplerian Orbit – Properties

Keplerian Orbit – Velocities

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

Some Useful Relations

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

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. E-mail: Feel free to ask your queries: senthil.avionics@gmail.com