Hohmann Tansfer Trajectory

Hohmann Transfer Ascent

In Hohmann transfer orbit  , the satellite and the last stage of rocket first attains a low altitude circular parking orbit,just outside the densest part of atmosphere, usually this orbit is about 200 km or more, In most launches this altitude is taken to be 185km , shortly after parking orbit injection ,the vehicle is injected into an elliptic transfer trajectory , at perijee , this transfer ellipse is tangential to the parking orbit and at apogee tangential to the required satellite orbit. This transfer orbit is proposed by W.Hohmann , therefore is called "Hohmann transfer trajectory"

Direct Ascent Trajectory

Direct Ascent

In Direct Ascent the trajectory selected such that destination point lies in the required orbit,  the satellite which in most cases is attached to the final rocket stage, approaches this destination altitude , the final stage stage motor is ignited and the satellite is accelerated to the required orbital velocity.In some cases the rocket motor required for this acceleration is not the motor of last stage, but is an integral part of satellite itself.

Ascent Trajectories

Satellites are launched from the Earth's surface by means of multi-stage rockets.After ignition of the first stage rocket Engines,these Launch vehicles first ascent vertically, A few seconds after lift-off ,the rocket is tilted and it flies a curved trajectory until the satellite reaches the predetermined position called ascend trajectory.
since its a vast topic, we discuss only the types of ascent trajectories here.

Two basic types of ascent trajectories can be distinguished for satellite launch vehicles,
1.Direct Ascent ,
2.Hohmann Transfer Ascent.


1.Direct Ascent ,

In Direct Ascent the trajectory selected such that destination point lies in the required orbit,  the satellite which in most cases is attached to the final rocket stage, approaches this destination altitude , the final stage stage motor is ignited and the satellite is accelerated to the required orbital velocity.In some cases the rocket motor required for this acceleration is not the motor of last stage, but is an integral part of satellite itself.

2.Hohmann Transfer Ascent.


In Hohmann transfer orbit  , the satellite and the last stage of rocket first attains a low altitude circular parking orbit,just outside the densest part of atmosphere, usually this orbit is about 200 km or more, In most launches this altitude is taken to be 185km , shortly after parking orbit injection ,the vehicle is injected into an elliptic transfer trajectory , at perijee , this transfer ellipse is tangential to the parking orbit and at apogee tangential to the required satellite orbit. This transfer orbit is proposed by W.Hohmann , therefore is called "Hohmann transfer trajectory"

Rocket Principle of Giant Squid in Deep seas

Certain creatures found in nature employing this rocketry principle of ejecting mass stored in their bodies to create movement, best example shall be a Giant Squid in deep seas.

The Giant Squid is a large creature about 10 to 14 m long and is found in the deep oceans.A few have been caught off the coast or have been washed ashore the New Zealand coast. 

The Giant squid moves in the deep seas by taking the water in by expanding its antle. It holds the water and pressurizes it to about 40 kpa by contracting its mantle.The pressurized water and any other matter that it has gulped, such as,fish,eggs, etc., is squeezed out through a opening ,known as the funnel that can rotate in any direction.The gradual impulse from the squirting out of the water and waste pushes it by as much as 50 m for a single squirt.It is estimated to achieve a velocity of about 7 m/s in a single push of the water and waste.A schematic diag

Orbital and Sub Orbital Flight

A body in orbit is in a state of free fall i.e., the body freely moves under the influence of Gravitational field,A body when provided with an orbital velocity (Vo) tends to move along a straight line in the direction of the velocity, however ,the attractive force from the gravitational field of the planet tends to pull it towards the center. The body therefore falls towards of the planet. As the body falls,the orbit velocity Vo pushes it further away from the planet and the body always misses falling on the planet.As long as the velocity provided to a body is less than the escape velocity of gravitational attraction of the given planet will cause the body to fall towards it. when the body goes into orbit , the motion corresponds to a state of free fall.If the velocity provided is less than the orbital velocity , the body falls on the planet corresponding to a sub-orbital flight.
The motion of different planets in the solar system about the sun is also in a state of free fall.

Pseudo Force

Its a force that acts on all masses in a non inertial frame of reference, or Its a Fictitious force in a non-inertial frame of reference to correctly describe the motion of the body in the non-inertial frame of reference.
Example: passenger in any moving vehicle, The force given to the vehicle forces the objects or person inside the vehicle to move back until the interior gets balanced with the movement.

In a rotational frame of reference ,The pseudo force is a centrifugal force.

Example: when you travel faster in bike round and round around a circular place, their is a force which tries to pull you outwards.and also their is a force which balances the friction between the bike wheel and road.

Law of Harmonies

Law of Harmonies :
The ratio of the square of the orbital period of any two planets is equal to the ratio of the cube of their average distance from the sun

Law of Equal Areas

Law of equal Areas:
The imaginary line from center of the sun to the center of the planet sweeps out equal areas in equal interval of time.

Law of Ellipses

Law of Ellipses:
The Path of all the planets in our solar system about the sun are elliptical in shape with center of sun being located at the focus.

Kepler's Law

Johannes kepler, A mathematician and astronomer ,used the observations to formulate the following three basic laws of planetary motion
1.Law of Ellipses
2.Law of equal Areas
3.Law of Harmonies

Law of Ellipses:
The Path of all the planets in our solar system about the sun are elliptical in shape with center of sun being located at the focus.

Law of equal Areas:
The imaginary line from center of the sun to the center of the planet sweeps out equal areas in equal interval of time.

Law of Harmonies :
The ratio of the square of the orbital period of any two planets is equal to the ratio of the cube of their average distance from the sun

Types of orbit

In Rocket Propulsion, there are certain types of orbits which we use frequently for technical purposes, some of them listed below...

1.Geo-Synchronous orbit
2.Geo-Stationary orbit
3.Sun-Synchronous orbit
4.Molniya orbit
5.Retro Grade orbit
6.Polar orbit

Polar Orbit

The Inclination of an orbit is the orbital plane and the equatorial plane of the planet about which the body is orbiting.An orbit at an inclination of X degrees , An inclination of Zero denotes that the orbit is in the equatorial plane ,i.e the orbital plane and the equatorial plane coincide.An inclination of 90 degrees represents the orbit from pole to pole and is known as polar orbit.

Polar orbits are useful for the spacecrafts that carry out mapping or surveillance of the planets, This is because as the planet rotates ,the spacecraft has access to every point on the surface of the planet .In the case of polar orbit about the earth,when the orbital plane's path is as similar as earth's path in the solar orbit ,The Sun would maintain same inclination.

Retro Grade orbit

An Inclination of 180 degrees represents the orbiting body to move in the equatorial plane but in a direction opposite to the direction of rotation of the planet in the equatorial plane.It is known as Retro-Grade equatorial  orbit.

Molniya Orbit

A particular orbit known as Molniya orbit is highly elliptic with the half major and half minor axis being about 46,000 km and 6,800 km respectively.The inclination of the orbit is 63.4 degress . spacecrafts in Molniya orbit remain in the northern hemisphere for as much as 11 hours in an orbit  which is having a time period of 12 hours.

Sun Synchronous Orbit

The angle between the line joining the sun with the Earth and the orbital plane of the satellite remains constant,such an orbit is known as Sun-Synchronous orbit and provides constant illumination by the sun at a given point on the surface of the Earth as viewed by the space craft in polar orbit.

Geo Stationary Orbit

If the plane of the Geo Synchronous orbit is in the equatorial plane of the Earth and the body in that orbit rotates in the same direction as the Earth Rotates, then that orbiting body will always appear to be stationary to an observer standing on Earth's surface.It is then said to be Geo stationary orbit. this orbit is a particular type of Geo Synchronous orbit.

the concept of Geo Synchronous and Geo stationary not only applicable for Earth but also applicable for Every other planet in the galaxy and Natural satellites as long as they are rotating about an axis.

It is considered to be a circular orbit at a distance of about 35,786km from Earth surface.

This orbit is very much useful for
communication satellites,
broadcast satellites,
SBAS satellites (Satellite Based Augmentation System )
GNSS satellites (Global Navigation Satellite System) etc.,

Arthur Clarke , Author of several Science fiction books , had put forward this concept in 1945 , hence this referred to "Clarke's Orbit"

The first satellite to be placed in the Geo-Stationary orbit was Syncom-2 on July 26th 1963.

Angular Velocity

Angular velocity is the rate of change of angular displacement ,its a vector quantity, represented by the symbol omega , unit is rad/sec , for calculation simplifications below examples given in degrees/hour.

Earth rotates from east to west on its axis and completes one rotation each day, The angular velocity of earth  can be calculated as follows

consider earth rotates on its axis once in every 24 hour, the complete circle is 360 degrees i.e   radians , so per hour earth's angular velocity is

.

angular velocity in degrees/hour

Mercury -  0.255 (mercury spins about its axis with a period 58 days approximately so 360/24x58)
Venus - 0.061 (Venus spins about its axis with a period of 243 days approx. so 360/24x243)
Earth -  15.04 (Earth spins about its axis with a period of 24 hours i.e 1 day approx. so 360/24)
Mars -  14.62 (Mars spins about its axis 24.62 hours approx. so 360/24.62)
Jupiter - 36.29 (Jupiter spins about its axis 9.92 hours approx. so 360/9.92)
Saturn - 33.77 (Saturn spins about its axis 10.66 hours approx. so 360/10.66)
Uranus - 20.88 (Uranus spins about its axis 17.24 hours approx. so 360/17.24)
Neptune - 22.34 (Neptune spins about its axis 16.11 hours approx. so 360/16.11)
Pluto - 2.34 (Pluto spins about its axis 153.3 hours approx. so 360/153.3)

In same way angular velocity in rad per second can be calculated by 2pi/(24x3600)   i.e 60 min x 60 secso the angular velocity per sec is 7.273x10 ^ -5 rad/sec  for earth , for other planets just multiply 60x60 in denominator
.

Geo synchronous Orbit

Its the body or objects Orbit having an angular velocity equal to the angular velocity of the planet, around which orbit of the object or body takes place.

If a body is in circular orbit with the angular velocity equal to that of Earth and also rotates in the  same direction as that of Earth from east to west, its movement is synchronous with the rotation of earth and it is said to be Geo-Synchronous orbit.

for example :
Earth rotates from east to west on its axis and completes one rotation each day, The angular velocity of earth  can be calculated as follows

consider earth rotates on its axis once in every 24 hour, the complete circle is 360 degrees i.e   radians , so per hour earth's angular velocity is

.

so the object orbiting around earth orbit should move at an angular velocity of 15 degrees/hour to reach the Geo synchronous condition.

similarly for other planets, the Synchronous angular velocity can be calculated as follows,

angular velocity in degrees/hour

Mercury -  0.255 (mercury spins about its axis with a period 58 days approximately so 360/24x58)
Venus - 0.061 (Venus spins about its axis with a period of 243 days approx. so 360/24x243)
Earth -  15.04 (Earth spins about its axis with a period of 24 hours i.e 1 day approx. so 360/24)
Mars -  14.62 (Mars spins about its axis 24.62 hours approx. so 360/24.62)
Jupiter - 36.29 (Jupiter spins about its axis 9.92 hours approx. so 360/9.92)
Saturn - 33.77 (Saturn spins about its axis 10.66 hours approx. so 360/10.66)
Uranus - 20.88 (Uranus spins about its axis 17.24 hours approx. so 360/17.24)
Neptune - 22.34 (Neptune spins about its axis 16.11 hours approx. so 360/16.11)
Pluto - 2.34 (Pluto spins about its axis 153.3 hours approx. so 360/153.3)


In same way angular velocity in rad per second can be calculated by multiplying the denominator 24 by 3600 i.e     60 min x 60 sec so the angular velocity per sec is 7.273x10 ^ -5 rad/sec

Escape Velocity

It is the velocity required for a body to escape the attractive force of the planet or star.

Escape velocity of
Mercury - 4.300 km/s
Venus - 10.300 km/s
Earth - 11.190 Km/s
mars - 5.027 km/s
Jupiter - 59.500 km/s
Saturn - 35.600 km/s
Uranus - 21.200 km/s
Neptune - 23.600 km/s
Pluto - 1.200 km/s

sun - 617.700 km/s

Escape velocity of the planet or star can be calculated by the following formula

 

Ve - escape velocity
G - Gravitational constant -

M- Mass of the escaping body
r -  is the distance between the center of the body and center of the escaping body

for example :
Escape velocity of object on earth when it is in ground is about 11.19 km/s , but it will gets reduced to 3.76 km/s once the body reaches the height of 50,000 km from earth's surface