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Delta-v

Delta-v

In general physics, delta-v is simply the change in velocity. Depending on the situation delta-v can be referred to as a spatial vector (

\Delta \mathbf\,

) or scalar (

\Delta\,

). In both cases it is equal to the acceleration (vector or scalar) integrated over time: :

\Delta \mathbf = \mathbf_1 - \mathbf_0 = \int^_ \mathbf  \, dt

(vector version) :

\Delta = _1 - _0 = \int^_  \, dt

(scalar version) where:
-

\mathbf\,

\,

is initial velocity vector or scalar at time

t_0\,

,
-

\mathbf\,

\,

is target velocity vector or scalar at time

t_1\,

Astrodynamics

In astrodynamics delta-v is a scalar measure for the amount of "effort" needed to carry out an orbital maneuver, i.e., to change from one orbit to another. A delta-v is typically provided by the thrust of a rocket engine. The time-rate of delta-v is the magnitude of the acceleration, i.e., the thrust per kilogram total current mass, produced by the engines. The actual acceleration vector is found by adding the gravity vector to the vector representing the thrust per kilogram. Without gravity, delta-v is, in the case of thrust in the direction of the velocity, simply the change in speed. However, in a gravitational field, orbits which are not circular involve changes in speed without requiring any delta-v, while gravity drag can cause the change of speed to be less than delta-v. When applying delta-v in the direction of the velocity and against gravity the specific orbital energy gained per unit delta-v is equal to the instantaneous speed. For a burst of thrust during which both the acceleration produced by the thrust, and the gravity, are constant, the specific orbital energy gained per unit delta-v is the mean value of the speed before and the speed after the burst. The rocket equation shows that the required amount of propellant can dramatically increase, and that the possible payload can dramatically decrease, with increasing delta-v. Therefore in modern spacecraft propulsion systems considerable study is put into reducing the total delta-v needed for a given spaceflight, as well as designing spacecraft that are capable of producing a large delta-v. For examples of the first, see Hohmann transfer orbit, gravitational slingshot; also, a large thrust reduces gravity drag. For the second some possiblities are:
- staging
- large specific impulse
- since a large thrust can not be combined with a very large specific impulse, applying different kinds of engine in different parts of the spaceflight (the ones with large thrust for the launch from Earth).
- reducing the "dry mass" (mass without propellant) while keeping the capability of carrying much propellant, by using light, yet strong, materials; when other factors are the same, it is an advantage if the propellant has a high density, because the same mass requires smaller tanks. Delta-v is also required to keep satellites in orbit and is expended in orbital stationkeeping maneuvers.

Games

Delta-V is a trenchrunner game published by Bethesda Softworks in the 1980's

External link

- http://www.pma.caltech.edu/~chirata/deltav.html

Velocity

This article is about velocity in physics. For other meanings, see velocity (disambiguation). The velocity of an object is simply its speed in a particular direction. Note that both speed and direction are required to define a velocity.

Explanation

The velocity (v) is an physical quantity of the motion. A change in an object's velocity can therefore arise from either a change in its speed or in its direction. For example an aeroplane that is circling at a constant speed of 200km/h is changing its velocity because it is continously changing its direction. A aeroplane that is taking-off may go from zero to 200km/h in a straight line and so would also be changing its velocity. A change in velocity is called an acceleration. Objects are only accelerated if a force is applied to them. (The amount of acceleration depends the size of the force and the mass of the object being shifted, see Newton's Second Law of Motion.) In the case of the circling aeroplane, the pilot banks to use the force of lift from the wings to change direction. In another example the Space Shuttle orbits the earth at a constant speed but is constantly changing its velocity because of the circular orbit. In this case the force causing the acceleration is provided by the earth's gravity acting on the shuttle. The average speed v of an object moving a distance d during a time interval t is described by the formula: :

v = \frac

Acceleration is the rate of change of an object's velocity over time. The average acceleration of a of an object whose speed changes from v_i to v_f during a time interval t is given by: :

a = \frac

Where

v_i

= an object's initial velocity and

v_f

= the object's final velocity over a period of time t

Formal description

Velocity (symbol: v) is a vector measurement of the rate and direction of motion. The scalar absolute value (magnitude) of velocity is speed. Velocity can also be defined as rate of change of displacement or just as the rate of displacement, depending on how the term displacement is used. It is thus a vector quantity with dimension length/time. In the SI (metric) system it is measured in metre per second The instantaneous velocity vector v of an object that has position at time t is given by x(t) can be computed as the derivative :

v= = \lim_

The instantaneous acceleration vector a of an object that has position at time t is given by x(t) is :

\mathbf = \frac   = \frac

The equation for an object's velocity can be obtained mathematically by taking the integral of the equation for its acceleration beginning from some initial period time $t_0$ to some point in time later $t_n$ . The final velocity v_f of an object which starts with velocity v_i and then accelerates at constant acceleration a for a period of time t is: :

v_f = v_i + a t

The average velocity of an object undergoing constant acceleration is (v_i + v_f)/2. To find the displacement d of such an accelerating object during a time interval t, substitute this expression into the first formula to get: :

d = t \times \frac

When only the object's initial velocity is known, the expression :

d = v_i t + \frac

can be used. These basic equations for final velocity and displacement can be combined to form an equation that is independent of time, also known as Torricelli's Equation: :

v_f^2 = v_i^2 + 2 a d

The above equations are valid for both classical mechanics and special relativity. Where classical mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in classical mechanics, all observers agree on the value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. The kinetic energy (energy of motion) of a moving object is linear with both its mass and the square of its velocity: :

E_ = \begin \frac \end mv^2

The kinetic energy is a scalar quantity.

Polar coordinates

In polar coordinates, a two-dimensional velocity can be decomposed into a radial velocity, defined as the component of velocity away from or toward the origin, and transverse velocity, the component of velocity along a circle centred at the origin, and equal to the distance to the origin times the angular velocity. Angular momentum in scalar form is the distance to the origin times the transverse speed, or equivalently, the distance squared times the angular speed, with a plus or minus to distinguish clockwise and anti-clockwise direction. If forces are in the radial direction only, as in the case of a gravitational orbit, angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion.

Acceleration

In physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. It is thus a vector quantity with dimension length/time². In SI units, this is meter/second².

Explanation

To accelerate an object is to change its velocity over a period of time. In this strict scientific sense, acceleration can have positive and negative values – respectively called acceleration and deceleration (or retardation) in common speech – as well as change of direction. Acceleration is defined technically as "the rate of change of velocity of an object with respect to time" and is given by the equation :

\mathbf =

where :a is the acceleration vector :v is the velocity vector expressed in m/s :t is time expressed in seconds. This equation gives a the units of m/(s·s), or m/s² (read as "metres per second per second", or "metres per second squared"). An alternative equation is: :

\mathbf =

where :ā is the average acceleration (m/s²) :u is the initial velocity (m/s) :v is the final velocity (m/s) :t is the time interval (s) Transverse acceleration (perpendicular to velocity) causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion. For this centripetal acceleration we have :

\mathbf = - \frac \frac = - \omega^2 \mathbf

One common unit of acceleration is g, one g being the acceleration caused by the gravity of Earth at sea level at 45° latitude (Paris), or about 9.81 m/s². Jerk is the rate of change of an object's acceleration over time. In classical mechanics, acceleration

a \

is related to force

F \

and mass

m \

(assumed to be constant) by way of Newton's second law: :

F = m \cdot a

As a result of its invariance under the Galilean transformations, acceleration is an absolute quantity in classical mechanics.

Relation to relativity

After defining his theory of special relativity, Albert Einstein realized that forces felt by objects undergoing constant acceleration are indistinguishable from those in a gravitational field, and thus defined general relativity that also explained how gravity's effects could be limited by the speed of light. If you accelerate away from your friend, you could say (given your frame of reference) that it is your friend who is accelerating away from you, although only you feel any force. This is also the basis for the popular Twin paradox, which asks why only one twin ages when moving away from his sibling at near light-speed and then returning, since the aging twin can say that it is the other twin that was moving. General relativity solved the "why does only one object feel accelerated?" problem which had plagued philosophers and scientists since Newton's time (and caused Newton to endorse absolute space). In special relativity, only inertial frames of reference (non-accelerated frames) can be used and are equivalent; general relativity considers all frames, even accelerated ones, to be equivalent. With changing velocity, accelerated objects exist in warped space (as do those that reside in a gravitational field). Therefore, frames of reference must include a description of their local spacetime curvature to qualify as complete. Acceleration can be measured using an accelerometer.

References

-
-

External links and references

- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- [http://www.sandia.gov/LabNews/labs-accomplish/2005/pulp.html Experiments on Z produced a world-record peak velocity of 34 km/s] (that is about 76,000 mph)
- [http://www.sandia.gov/media/NewsRel/NR2001/flyer.htm Magnetic field shocklessly shoots pellets 20 times faster than rifle bullet] Category:Physical quantity Category:Classical mechanics ko:가속도 ja:加速度 simple:Acceleration th:ความเร่ง

Orbital velocity vector

orbital state vectors#velocity vector

Astrodynamics

Astrodynamics is the study of the motion of rockets, missiles, and space vehicles, as determined from Sir Isaac Newton's laws of motion and his law of universal gravitation. It is a specific and distinct branch of celestial mechanics, which focuses more broadly on Newtonian gravitation and includes the orbital motions of artificial and natural astronomical bodies such as planets, moons, and comets. Astrodynamics is principally concerned with spacecraft trajectories, from launch to atmospheric re-entry, including all orbital maneuvers, orbit plane changes, and interplanetary transfers. For a less technical treatment, see the article on space mathematics.

Laws of astrodynamics

The fundamental laws of astrodynamics are Newton's law of universal gravitation and Newton's laws of motion, while the fundamental mathematical tool is his differential calculus. Kepler's laws of planetary motion may be derived from these laws, when it is assumed that the orbiting body is subject only to the gravitational force of the central attractor. When an engine thrust or propulsive force is present, Newton's second law of motion applies, and Kepler's laws are temporarily invalidated. The formula for escape velocity is easily derived as follows. The specific energy (energy per unit mass) of any space vehicle is composed of two components, the specific potential energy and the specific kinetic energy. The specific potential energy associated with a planet of mass M is given by

- G M / r

while the specific kinetic energy of an object is given by

v^2/2

Since energy is conserved, the total specific orbital energy

v^2/2 - G M / r

does not depend on the distance,

r

, from the center of the central body to the space vehicle in question. Therefore, the object can reach infinite

r

only if this quantity is nonnegative, which implies

v\geq\sqrt

The escape velocity from the Earth's surface is about 11 km/s, but that is insufficient to send the body an infinite distance because of the gravitational pull of the Sun. To escape the solar system from the vicinity of the Earth requires around 42 km/s velocity, but there will be "part credit" for the Earth's orbital velocity for spacecraft launched from Earth, if their further acceleration (due to the propulsion system) carries them in the same direction as Earth travels in its orbit.

Formulae for ellipse

Orbits are ellipses, so, naturally, the formula for the distance of a body for a given angle corresponds to the formula for an ellipse in polar coordinates. The parameters of the ellipse are given by the orbital elements.

Historical approaches

Until the rise of space travel in the twentieth century, there was little distinction between astrodynamics and celestial mechanics. The fundamental techniques, such as those used to solve the Keplerian problem, are therefore the same in both fields. Furthermore, the history of the fields is essentially identical.

Kepler's equation

Kepler was the first to successfully model planetary orbits to a high degree of accuracy.

Derivation

To compute the position of a satellite at a given time (the Keplerian problem) is a difficult problem. The opposite problem—to compute the time-of-flight given the starting and ending positions—is simpler. We present a derivation for the time-of-flight equation here. Keplerian problem.]] The problem is to find the time

T

at which the satellite reaches point

S

, given that it is at periapsis

P

at time

t = 0

. We are given that the semimajor axis of the orbit is

a

, and the semiminor axis is

b

; the eccentricity is

e

, and the planet is at

Q

, at a distance of

ae

from the center

C

of the ellipse. The key construction that will allow us to analyse this situation is the auxiliary circle (shown in blue) circumscribed on the orbital ellipse. This circle is taller than the ellipse by a factor of

a/b

in the direction of the minor axis, so all area measures on the circle are magnified by a factor of

a/b

with respect to the analogous area measures on the ellipse. Any given point on the ellipse can be mapped to the corresponding point on the circle that is

a/b

further from the ellipse's major axis. If we do this mapping for the position

S

of the satellite at time

T

, we arrive at a point

R

on the circumscribed circle. Kepler defines the angle

PCR

to be the eccentric anomaly angle

E

. (Kepler's terminology often refers to angles as "anomalies.") This definition makes the time-of-flight equation easier to derive than it would be using the true anomaly angle

PQS

. To compute the time-of-flight from this construction, we note that Kepler's second law allows us to compute time-of-flight from the area swept out by the satellite, and so we will set about computing the area

PQS

swept out by the satellite. First, the area

PQR

is a magnified version of the area

PQS

: :

PQR = \frac PQS

Furthermore, area

PQS

is the area swept out by the satellite in time

T

. We know that, in one orbital period

\tau

, the satellite sweeps out the whole area

\pi a b

of the orbital ellipse.

PQS

is the

T / \tau

fraction of this area, and substituting, we arrive at this expression for

PQR

: :

PQR = \frac \pi a^2

Second, the area

PQR

is also formed by removing area

QCR

from

PCR

: :

PQR = PCR - QCR \;

Area

PCR

is a fraction of the circumscribed circle, whose total area is

\pi a^2

. The fraction is

E / 2 \pi

, thus: :

PCR = \fracE

Meanwhile, area

QCR

is a triangle whose base is the line segment

QC

of length

ae

, and whose height is

a \sin E

: :

QCR = \frac e \sin E

Combining all of the above: :

PQR = \frac \pi a^2
 = \fracE - \frac e \sin E

Dividing through by

a^2 / 2

: :

\fracT = E - e \sin E

To understand the significance of this formula, consider an analogous formula giving an angle

\theta

during circular motion with constant angular velocity

M

: :

MT = \theta \;

Setting

M = 2 \pi / \tau

and

\theta = E - e \sin E

gives us Kepler's equation. Kepler referred to

M

as the mean motion, and

E - e \sin E

as the mean anomaly. The term "mean" in this case refers to the fact that we have "averaged" the satellite's non-constant angular velocity over an entire period to make the satellite's motion amenable to analysis. All satellites traverse an angle of

2 \pi

per orbital period

\tau

, so the mean angular velocity is always

2 \pi / \tau

. Substituting

M

into the formula we derived above gives this: :

MT = E - e \sin E \;

This formula is commonly referred to as Kepler's equation.

Application

With Kepler's formula, finding the time-of-flight to reach an angle (true anomaly) of

\theta

from periapsis is broken into two steps: # Compute the eccentric anomaly

E

from true anomaly

\theta

# Compute the time-of-flight

T

from the eccentric anomaly

E

Finding the angle at a given time is harder. Kepler's equation is transcendental in

E

, meaning it cannot be solved for

E

analytically, and so numerical approaches must be used. In effect, one must guess a value of

E

and solve for time-of-flight; then adjust

E

as necessary to bring the computed time-of-flight closer to the desired value until the required precision is achieved. Usually, Newton's method is used to achieve relatively fast convergence. The main difficulty with this approach is that it can take prohibitively long to converge for the extreme elliptical orbits. For near-parabolic orbits, eccentricity

e

is nearly 1, and plugging

e = 1

into the formula for mean anomaly,

E - \sin E

, we find ourselves subtracting two nearly-equal values, and so accuracy suffers. For near-circular orbits, it is hard to find the periapsis in the first place (and truly circular orbits have no periapsis at all). Furthermore, the equation was derived on the assumption of an elliptical orbit, and so it does not hold for parabolic or hyperbolic orbits at all. These difficulties are what led to the development of the universal variable formulation, described below.

Perturbation theory

You can deal with perturbations just by summing the forces and integrating, but that is not always best. Historically, variation of parameters has been used which is easier to mathematically apply with when perturbations are small.

Modern techniques

Today, we do not use the same techniques that Kepler used, in general.

Conic orbits

For simple things like computing the delta-v for coplanar transfer ellipses, traditional approaches work pretty well. But time-of-flight is harder, especially for near-circular and hyperbolic orbits.

Transfer orbits

Transfer orbits get you from one orbit to another. Usually they require a burn at the start, a burn at the end, and sometimes a burn in the middle. The Hohmann transfer orbit typically requires the least delta-v, but any orbit that intersects both your origin and destination will work.

The patched conic approximation

The transfer orbit alone is not a good approximation for interplanetary trajectories because it neglects the planets' own gravity. Planetary gravity dominates the behaviour of the spacecraft in the vicinity of a planet, so it severely underestimates delta-v, and produces highly inaccurate prescriptions for burn timings. One relatively simple way to get a first-order approximation of delta-v is based on the patched conic approximation technique. The idea is to choose the one dominant gravitating body in each region of space through which the trajectory will pass, and to model only that body's effects in that region. For instance, on a trajectory from the Earth to Mars, one would begin by considering only the Earth's gravity until the trajectory reaches a distance where the Earth's gravity no longer dominates that of the Sun. The spacecraft would be given escape velocity to send it on its way to interplanetary space. Next, one would consider only the Sun's gravity until the trajectory reaches the neighbourhood of Mars. During this stage, the transfer orbit model is appropriate. Finally, only Mars' gravity is considered during the final portion of the trajectory where Mars' gravity dominates the spacecraft's behaviour. The spacecraft would approach mars on a hyperbolic orbit, and a final retrograde burn would slow the spacecraft enough to be captured by Mars. This simplification is sufficient to compute rough estimates of fuel requirements, and rough time-of-flight estimates, but it is not generally accurate enough to guide a spacecraft to its destination. For that, numerical methods are required.

The universal variable formulation

To address the shortcomings of the traditional approaches, the universal variable approach was developed. It works equally well on circular, elliptical, parabolic, and hyperbolic orbits; and also works well with perturbation theory. The differential equations converge nicely when integrated for any orbit.

Perturbations

The universal variable formulation works well with the variation of parameters technique, except now, instead of the six Keplerian orbital elements, we use a different set of orbital elements: namely, the satellite's initial position and velocity vectors

x_0

and

v_0

at a given epoch

t = 0

. In a two-body simulation, these elements are sufficient to compute the satellite's position and velocity at any time in the future, using the universal variable formulation. Conversely, at any moment in the satellite's orbit, we can measure its position and velocity, and then use the universal variable approach to determine what its initial position and velocity would have been at the epoch. In perfect two-body motion, these orbital elements would be invariant (just like the Keplerian elements would be). However, perturbations cause the orbital elements to change over time. Hence, we write the position element as

x_0(t)

and the velocity element as

v_0(t)

, indicating that they vary with time. The technique to compute the effect of perturbations becomes one of finding expressions, either exact or approximate, for the functions

x_0(t)

and

v_0(t)

Non-ideal orbits

The following are some effects which make real orbits differ from the simple models based on a spherical earth. Most of them can be handled on short timescales (perhaps less than a few thousand orbits) by perturbation theory because they are small relative to the corresponding two-body effects.
- Equatorial bulges cause precession of the node and the perigee
- Tesseral harmonics [http://mathworld.wolfram.com/TesseralHarmonic.html] of the gravity field introduce additional perturbations
- lunar and solar gravity perturbations alter the orbits
- Atmospheric drag reduces the semi-major axis unless make-up thrust is used Over very long timescales (perhaps millions of orbits), even small perturbations can dominate, and the behaviour can become chaotic. On the other hand, the various perturbations can be orchestrated by clever astrodynamicists to assist with orbit maintenance tasks, such as station-keeping, ground track maintenance or adjustment, or phasing of perigee to cover selected targets at low altitude Many of the options, procedures, and supporting theory are covered in standard works such as: 1. Bate, R.R., Mueller, D.D., White, J.E., Fundamentals of Astrodynamics, Dover Publications, New York, 1971. 2. Vallado, D. A., Fundamentals of Astrodynamics and Applications, 2nd Edition, McGraw-Hill, 2001 3. Battin, R.H., An Introduction to the Mathematics and Methods of Astrodynamics, New York, 1987. 4. Chobotov, V.A. (ed.), Orbital Mechanics, 3rd Edition, AIAA, Washington, DC, 2002. 5. Herrick, S. Astrodynamics, Van Nostrand Reinhold, London, 1971 (two volumes). 6. Kaplan, M.H., Modern Spacecraft Dynamics and Controls , Wiley, New York, 1976. 7. Logsdon, T., Orbital Mechanics, Wiley-Interscience, New York, 1997. 8. Prussing, J.E., and B.A. Conway, Orbital Mechanics, Oxford University Press, New York, 1993. 9. Sidi, M.J., Spacecraft Dynamics and Control, Cambridge University Press, New York, 1997. 10. Wiesel, W.E., Spaceflight Dynamics, McGraw-Hill, New York, 1996, 2nd edition. 11. Vinti, J.P., Orbital and Celestial Mechanics , AIAA, Reston, VA, 1998. or, on line: [http://www.psatellite.com/products/manuals/SCTUsersGuideBook1.pdf] and [http://www.psatellite.com/products/manuals/SCTUsersGuideBook2.pdf] The most elementary but very widely used reference is Bate, Mueller and White. It has several useful graphs off which one can read the rates of change of perigee and node due to earth oblateness, but there are typographical errors in a few equations. For example, in Eq. (9.7.5) the term in (3/2) J2 needs (r_e/r) squared and the term in J3 needs it cubed. The coefficient 315 in the J6 term, Eq.(9.7.6.) should be 245 (but the 315 in the J5 term is just fine). Battin's book may be too mathematical for many users.

Interplanetary superhighway and fuzzy orbits

It is now possible to use computers to search for routes using the nonlinearities in the gravity of the planets and moons of the solar system. For example, it is possible to plot an orbit from high earth orbit to Mars, passing close to one of the Earth's Trojan points. Collectively referred to as the Interplanetary Superhighway, these highly perturbative, even chaotic, orbital trajectories in principle need no fuel (in practice keeping to the trajectory requires some course corrections). The biggest problem with them is they are usually exceedingly slow, taking many years to arrive. In addition launch windows can be very far apart. They have, however, been employed on projects such as Genesis. This spacecraft visited Earth's lagrange L1 point and returned using very little propellant.

Reference

- Bate, Roger R., Mueller, Donald D., and White, Jerry E. (1971). Fundamentals of Astrodynamics. Dover Publications. ISBN 0-48-660061-0

External links

- [http://www.braeunig.us/space/orbmech.htm ORBITAL MECHANICS] (Rocket and Space Technology)
- [http://jat.sourceforge.net Java Astrodynamics Toolkit]
-

Thrust

Thrust is a reaction force described quantitatively by Newton's Second and Third Law. When a system expels or accelerates mass in one direction the accelerated mass will cause a proportional but opposite force on that system. Mathematically this means that the total force experienced by a system accelerating a mass m, is equal and opposite to the mass m times the acceleration a experienced by that mass: :F = −m·a

Examples

mass An aircraft generates forward thrust when the spinning propellers blow air, or eject expanding gases from a jet engine to the back of the aircraft. The forward thrust is proportional to the (mass of the air) multiplied by (average velocity of the airstream). Similarly, a ship generates forward thrust (or reverse thrust) when the propellers are turned to accelerate water backwards (or forwards). The resulting thrust pushes the ship in the equal and opposite direction to the sum of the momentum change in the water flowing through the propeller. A rocket (and all mass attached to it) is propelled forward by a thrust force equal to, and opposite of, the time-rate of momentum change experienced by the exhaust mass accelerating out from the combustion chamber through the rocket nozzle. This is the exhaust velocity with respect to the rocket, times the time-rate at which the mass is expelled. Of course, for a launch the thrust at lift-off should be more than the weight, and with a fair margin, because a "slow launch" would be very inefficient. Each of the three Space shuttle main engines can produce a thrust of 1.8 MN, and each of its two Solid Rocket Boosters 14.7 MN, together 34.8 MN. Compare with the mass at lift-off of 2,040,000 kg, hence a weight of 20.0 MN. The simplified Aid for EVA Rescue (SAFER) has 24 thrusters of 3.56 N each.

Gravity drag

In astrodynamics, gravity drag (or gravity losses) is inefficiency encountered by a spacecraft thrusting while moving against a gravitational field. If the gravitational acceleration vector is

g

and the thrust vector per unit mass (acceleration produced by the engine) is

a

, then the actual acceleration of the craft is

a-g

, while using delta-v at a time-rate of

a

; that is, the delta-v of the vehicle used is

|a|/|a-g|

times the actual increase in speed. In the case of a very large thrust during a very short time, a desired speed increase can be reached with little gravity drag, while for

a

only slightly more than

g

, the gravity drag is very large. For example, at the beginning of launch from Earth, a rocket does not have significant horizontal speed yet (which provides a centrifugal force away from the center of the Earth), so just staying aloft costs delta-v every second. Therefore a time consuming launch would be highly inefficient. When applying delta-v against gravity to increase specific orbital energy, it is advantageous to spend delta-v at as high speed as possible, rather than spending some, being decelerated by gravity, then spending some more, or spending it at less than full capacity. Gravity drag can be described as the extra delta-v needed because of not being able to spend all the needed delta-v instantaneously. This effect can be explained in two equivalent ways:
- The specific energy gained per unit delta-v is equal to the speed, so spend the delta-v when the rocket is going fast; in the case of being decelerated by gravity this means as soon as possible.
- It is wasteful to lift fuel unnecessarily: use it right away, and then the rocket does not have to lift it. These effects apply whenever climbing to an orbit with higher specific orbital energy, such as during launch to Low Earth orbit (LEO) or from LEO to an escape orbit.

Vector considerations

It is important to note that acceleration is a vector quantity, and the direction of the acceleration has a large impact on the overall efficiency. For instance, gravity drag would reduce a 2.6 G thrust directed upward to an acceleration of 1.6 G, for an efficiency of less than 62%. However, the same 2.6 G thrust could be directed at such an angle that it had a 1 G upward component, completely cancelled by gravity drag, and a horizontal component of 2.4 G, unaffected by gravity drag. Achieving 2.4 G acceleration with 2.6 G thrust gives an efficiency of over 92%. Note, however, that the objective of the craft is not only to maximize acceleration, or else it would direct its 2.6 G thrust downward, achieving 138% efficiency but never reaching orbit. Rather, the objective is to achieve the necessary specific orbital energy to sustain the desired orbit. On a planet with an atmosphere, the objective is further complicated by the need and to achieve the necessary altitude to escape the atmosphere, and to minimize the losses due to atmospheric drag during the launch itself.

Specific orbital energy

In astrodynamics the specific orbital energy

\epsilon\,\!

(or vis-viva energy) of an orbiting body traveling through space under standard assumptions is the sum of its potential energy (

\epsilon_p\,\!

) and kinetic energy (

\epsilon_k\,\!

) per unit mass. According to the orbital energy conservation equation (also referred to as vis-viva equation) it is the same at all points of the trajectory: :

\epsilon=\epsilon_k+\epsilon_p=-
=-\left(1-e^2\right)

where:
-

v\,\!

is the orbital speed of the orbiting body
-

r\,\!

is the orbital distance of the orbiting body
-

\mu\,\!

is the standard gravitational parameter
-

h\,\!

is the specific relative angular momentum of the orbiting body
-

e\,\!

is the orbit eccentricity It is expressed in J/kg = m²s^-2 or MJ/kg = km²s^-2.

Equation forms for different orbits

For an elliptical orbit specific orbital energy equation simplifies to: :

\epsilon = -\,\!

where:
-

\mu\,\!

is the standard gravitational parameter
-

a\,\!

is semi-major axis of the orbiting body For a parabolic orbit this equation simplifies to: :

\epsilon=0\,\!

For a hyperbolic trajectory this specific orbital energy equation takes form: :

\epsilon = \,\!

In this case the specific orbital energy is also referred to as characteristic energy (or

C_3\,\!

) and is equal to the excess specific energy compared to that for an escape orbit (parabolic orbit). It is related to the hyperbolic excess velocity

v_ \,\!

(the orbital velocity at infinity) by :

2\epsilon=2C_3=v_^2\,\!

It is relevant for interplanetary missions. Thus, if orbital position vector (

\mathbf\,\!

) and orbital velocity vector (

\mathbf\,\!

) are known at one position, and

\mu\,\!

is known, then the energy can be computed and from that, for any other position, the orbital speed.

Rate of change

For an elliptical orbit the rate of change of the specific orbital energy with respect to a change in the semi-major axis is: :

\frac\,\!

where:
-

\mu\,\!

is the standard gravitational parameter
-

a\,\!

is semi-major axis of the orbiting body In the case of circular orbits, this rate is one half of the gravity at the orbit. This corresponds to the fact that for such orbits the total energy is one half of the potential energy, because the kinetic energy is minus one half of the potential energy.

Additional energy

If the central body has radius R, then the additional energy of an elliptic orbit compared to being stationary at the surface is

\ -\frac+\frac = \frac

- For the Earth and

a

just little more than

R/2

this is

(2a-R)g

;

2a-R

is the height the ellipse extends above the surface, plus the periapsis distance (the distance the ellipse extends beyond the center of the Earth); the latter times

g

is the kinetic energy of the horizontal component of the velocity.

Examples

The International Space Station has an orbital period of 91.74 minutes, hence the semi-major axis is 6738 km [http://www.google.com/search?num=100&hl=en&lr=&newwindow=1&safe=off&q=%28%2891.74
- 60%2F2%2Fpi%29%5E2
- 398600%29%5E%281%2F3%29]. The energy is −29.6 MJ/kg [http://www.google.com/search?num=100&hl=en&lr=&newwindow=1&safe=off&q=398600%2F6738%2F2]: the potential energy is −59.2 MJ/kg, and the kinetic energy 29.6 MJ/kg. Compare with the potential energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 3.4 MJ/kg, the total extra energy is 33.0 MJ/kg. The average speed is 7.7 km/s, the net delta-v to reach this orbit is 8.1 km/s (the actual delta-v is typically 1.5–2 km/s more for atmospheric drag and gravity drag). The increase per meter would be 4.4 J/kg; this rate corresponds to one half of the local gravity of 8.8 m/s² [http://www.google.com/search?num=100&hl=en&lr=&newwindow=1&safe=off&q=398600e9%2F6738e3%5E2]. For an altitude of 100 km (radius is 6471 km) these figures are: The energy is −30.8 MJ/kg [http://www.google.com/search?num=100&hl=en&lr=&newwindow=1&safe=off&q=398600%2F%286371%2B100%29%2F2]: the potential energy is −61.6 MJ/kg, and the kinetic energy 30.8 MJ/kg. Compare with the potential energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 1.0 MJ/kg, the total extra energy is 31.8 MJ/kg. The increase per meter would be 4.8 J/kg; this rate corresponds to one half of the local gravity of 9.5 m/s². The speed is 7.8 km/s [http://www.google.com/search?num=100&hl=en&lr=&newwindow=1&safe=off&q=sqrt%28398600%2F%286371%2B100%29%29], the net delta-v to reach this orbit is 8.0 km/s [http://www.google.com/search?num=100&hl=en&lr=&newwindow=1&safe=off&q=sqrt%28398600%282%2F6371-1%2F%286371%2B100%29%29%29]. Taking into account the rotation of the Earth, the delta-v is up to 0.46 km/s less (starting at the equator and going east) or more (if going west).

Applying thrust

Assume:
- a is the acceleration due to thrust (the time-rate at which delta-v is spent)
- g is the gravitational field strength
- v is the velocity of the rocket Then the time-rate of change of the specific energy of the rocket is

\mathbf \cdot \mathbf

: an amount

\mathbf \cdot (\mathbf-\mathbf)

for the kinetic energy and an amount

\mathbf \cdot \mathbf

for the potential energy. The change of the specific energy of the rocket per unit change of delta-v is :

\frac

which is |v| times the cosine of the angle between v and a. Thus, when applying delta-v to increase specific orbital energy, this is done most efficiently if a is applied in the direction of v, and when |v| is large. If the angle between v and g is obtuse, for example in a launch and in a transfer to a higher orbit, this means applying the delta-v as early as possible and at full capacity. See also gravity drag. When passing by a celestial body it means applying thrust when nearest to the body. When gradually making an elliptic orbit larger, it means applying thrust each time when near the periapsis. When applying delta-v to decrease specific orbital energy, this is done most efficiently if a is applied in the direction opposite to that of v, and again when |v| is large. If the angle between v and g is acute, for example in a landing (on a celestial body without atmosphere) and in a transfer to a circular orbit around a celestial body when arriving from outside, this means applying the delta-v as late as possible. When passing by a planet it means applying thrust when nearest to the planet. When gradually making an elliptic orbit smaller, it means applying thrust each time when near the periapsis. If a is in the direction of v: :

\Delta \epsilon =  \int v\, d (\Delta v) = \int v\, a dt

Rocket equation

Tsiolkovsky's rocket equation, named after Konstantin Tsiolkovsky who independently derived it, considers the principle of a rocket: a device that can apply an acceleration to itself (a thrust) by expelling part of its mass with high speed in the opposite direction, due to the conservation of momentum. It says that for any maneuver or any journey involving a number of maneuvers: :

\Delta v = v_e \ln \frac

or equivalently :

m_1=m_0 e^

m_0=m_1 e^

where

m_0

is the initial total mass, and

m_1

the final total mass and

v_e

the velocity of the rocket exhaust with respect to the rocket (the specific impulse, or, if measured in time, that multiplied by gravity-on-Earth acceleration). :

1-\frac  =1-e^

is the mass fraction (the part of the initial total mass that is spent as reaction mass).

\Delta v

(delta v) is the integration over time of the magnitude of the acceleration produced by using the rocket engine (not the acceleration due to other sources such as gravity or drag). For the typical case of an acceleration in the direction of the velocity, this is the increase of the speed. In the case of an acceleration in opposite direction (deceleration) it is the decrease of the speed. Note that gravity or drag also change velocity, but they are not part of the quantity delta-v. Hence delta-v is not simply the change in speed or velocity. However, thrust is often applied in short bursts, and during these short periods the other sources of acceleration may be negligible, and the delta-v of one burst may be simply approximated by the speed change. The total delta-v can simply be found by addition, even though between bursts the magnitude and direction of the velocity changes due to gravity, e.g. in an elliptic orbit. Note that, as mentioned, at any time the magnitude of the acceleration contributes to the delta-v, hence always a non-negative value, regardless of whether the rocket is used for acceleration or deceleration. This again demonstrates that delta-v is not simply the change in speed or velocity: the latter may be zero if we first accelerate and than decelerate, but the delta-v accumulates. The equation is obtained by integrating the conservation of momentum equation :

mdv = v_e dm

for a simple rocket that emits mass at a constant velocity (

dm

is here the reaction mass; if it is the change of the rocket mass then there is a minus sign in the latter equation). Although an extreme simplification, the rocket equation captures the essentials of rocket flight physics in a single short equation. It happens that delta-v is one of the most important quantities in orbital mechanics, that quantifies how difficult it is to get from one trajectory to another. Clearly, to achieve a large delta-v, either

m_0

must be huge (growing exponentially as delta-v rises), or

m_1

must be tiny, or

v

must be very high, or some combination of all of these. In practice, this has been achieved by using very large rockets (increasing

m_0

), with multiple stages (decreasing

m_1

), and rockets with very high exhaust velocities. The Saturn V rockets used in the Apollo space program and the ion thrusters used in long-distance unmanned probes are good examples of this. The rocket equation shows a kind of "exponential decay" of mass, but not as a function of time, but as a function of delta-v produced. The delta-v that is the corresponding "half-life" is

v_e \ln 2 \approx 0.693 v_e

History

This equation was independently derived by Konstantin Tsiolkovsky towards the end of the 19th century. However, according to Dr. Mike Gruntman who teaches rocketry, a recently discovered pamphlet ("A Treatise on the Motion of Rockets" by William Moore) shows that the earliest known derivation of this kind of equation was in fact in Woolwich Arsenal in England in 1813, and was used for weapons research.

Stages

In the case of subsequently thrusting rocket stages, the equation applies for each stage, where for each stage the initial mass in the equation is the total mass of the rocket after discarding the previous stage, and the final mass in the equation is the total mass of the rocket just before discarding the stage concerned. For each stage the specific impulse may be different. For example, if 80% of the mass of a rocket is the fuel of the first stage, and 10% is the dry mass of the first stage, and 10 % is the remaining rocket, then :

\Delta v = v_e \ln 5=1.61 v_e

With three similar, subsequently smaller stages we have :

\Delta v = 3 v_e \ln 5=4.83 v_e

and the payload is 0.1% of the initial mass. A comparable SSTO rocket, also with a 0.1 % payload, could have a mass of 11% for fuel tanks and engines, and 88.9% for fuel. This would give :

\Delta v = v_e \ln(100/11.1)=2.20 v_e

If the engine of a new stage is ignited before the previous stage has been discarded and the simultaneously working engines have a different specific impulse (as is often the case with solid rocket boosters and a liquid-fuel stage), the situation is more complicated.

Energy

In the ideal case

m_1

is useful payload and

m_0-m_1

is reaction mass (this corresponds to empty tanks having no mass, etc.). The energy required can simply be computed as :

\frac(m_0-m_1)v_e^2

Seemingly this is just the kinetic energy of the reaction mass and not the kinetic energy required for the payload, but if e.g

v_e

=10 km/s and the speed of the rocket is 3 km/s, then the speed of the reaction mass changes only from 3 to 7 km/s; the energy thus "saved" corresponds to the increase of the specific kinetic energy (kinetic energy per kg) for the rocket. In general:

d\left(\fracv^2\right)=vdv=vv_edm/m=\frac\left(v_e^2-(v-v_e)^2+v^2\right)dm/m

Thus the specific energy gain of the rocket in any small time interval is the energy gain of the rocket including the remaining fuel, divided by its mass, where the energy gain is equal to the energy produced by the fuel minus the energy gain of the reaction mass. The larger the speed of the rocket, the smaller the energy gain of the reaction mass; if the rocket speed is more than half of the exhaust speed the reaction mass even loses energy on being expelled, to the benefit of the energy gain of the rocket; the larger the speed of the rocket, the larger the energy loss of the reaction mass. We have :

\Delta \epsilon =  \int v\, d (\Delta v)

where

\epsilon

is the specific energy of the rocket (potential plus kinetic energy) and

\Delta v

is a separate variable, not just the change in

v

. In the case of using the rocket for deceleration, i.e. expelling reaction mass in the direction of the velocity,

v

should be taken negative. The formula is for the ideal case again, with no energy lost on heat, etc. The latter causes a reduction of thrust, so it is a disadvantage even when the objective is to lose energy (deceleration). If the energy is produced by the mass itself, as in a chemical rocket, the fuel value has to be :

\fracv_e^2

, where for the fuel value also the mass of the oxidizer has to be taken into account. A typical value is

v_e=4.5 km/s

, corresponding to a fuel value of 10.1 MJ/kg. The actual fuel value is higher, but part of the energy is lost on heat that flows off as radiation. The required energy is :

E = \fracm_1\left(e^-1\right)v_e^2

Conclusions:
- for

\Delta v \ll v_e

we have

E\approx \fracm_1 v_e \Delta v

- for a given

\Delta v

, the minimum energy is needed if

v_e=0.6275 \Delta v

, requiring an energy of :

E = 0.772 m_1(\Delta v)^2

. :Starting from zero speed this is 54.4 % more than just the kinetic energy of the payload. Starting from a nonzero speed the required energy may be less than the increase in energy in the payload. This can be the case when the reaction mass has a lower speed after being expelled than before. For example, from a LEO of 300 km altitude to an escape orbit is an increase of 29.8 MJ/kg, which, using a specific impulse of 4.5 km/s, has a net cost of 20.6 MJ/kg (

\Delta v

= 3.20 km/s; the energies are per kg payload). This optimization does not take into account the masses of various kinds of engines. Also, for a given objective such as moving from one orbit to another, the required

\Delta v

may depend greatly on the rate at which the engine can produce

\Delta v

and maneuvers may even be impossible if that rate is too low. For example, a launch to LEO normally requires a

\Delta v

of ca. 9.5 km/s (mostly for the speed to be acquired), but if the engine could produce

\Delta v

at a rate of only slightly more than g, it would be a slow launch requiring altogether a very large

\Delta v

(think of hovering without making any progress in speed or altitude, it would cost a

\Delta v

of 9.8 m/s each second). If the possible rate is only

g

or less, the maneuver can not be carried out at all with this engine. The power is given by :

P= \frac m a v_e  = \fracF v_e

where

F

is the thrust and

a

the acceleration due to it. Thus the theoretically possible thrust per unit power is 2 divided by the specific impulse in m/s. The thrust efficiency is the actual thrust as percentage of this. If e.g. solar power is used this restricts

a

; in the case of a large

v_e

the possible acceleration is inversely proportional to it, hence the time to reach a required delta-v is proportional to

v_e

; with 100% efficiency:
- for

\Delta v \ll v_e

we have

t\approx \frac

Examples:
- power 1000 W, mass 100 kg,

\Delta v

= 5 km/s,

v_e

= 16 km/s, takes 1.5 months.
- power 1000 W, mass 100 kg,

\Delta v

= 5 km/s,

v_e

= 50 km/s, takes 5 months. Thus

v_e

should not be too large.

Examples

Assume an exhaust velocity of 4.5 km/s and a

\Delta v

of 9.7 km/s (Earth to LEO).
- SSTO rocket:

1-e^

= 0.884, therefore 88.4 % of the initial total mass has to be propellant. The remaining 11.6 % is for the engines, the tank, and the payload. In the case of a space shuttle, it would also include the orbiter.
- Two stage to orbit: suppose that the first stage should provide a

\Delta v

of 5.0 km/s;

1-e^

= 0.671, therefore 67.1% of the initial total mass has to be propellant. The remaining mass is 32.9 %. After deposing of the first stage, a mass remains equal to this 32.9 %, minus the mass of the tank and engines of the first stage. Assume that this is 8 % of the initial total mass, then 24.9 % remains. The second stage should provide a

\Delta v

of 4.7 km/s;

1-e^

= 0.648, therefore 64.8% of the remaining mass has to be propellant, which is 16.2 %, and 8.7 % remains for the tank and engines of the second stage, the payload, and in the case of a space shuttle, also the orbiter. Thus together 16.7 % is available for all engines, the tanks, the payload, and the possible orbiter.

Hohmann transfer orbit

In astronautics and aerospace engineering, the Hohmann transfer orbit is an orbital maneuver that moves a spacecraft from one orbit to another using a fairly low delta-v. It was named after Walter Hohmann, the German scientist who published it in 1925. (See also interplanetary travel.) A Hohmann transfer orbit will take a spacecraft from low Earth orbit (LEO) to geosynchronous orbit (GEO) in just over five hours (geostationary transfer orbit), from LEO to the Moon in about 5 days and from the Earth to Mars in about 260 days. However, Hohmann transfers are very slow for trips to more distant points, so when visiting the outer planets it is common to use a gravitational slingshot to increase speed in-flight. gravitational slingshot The Hohmann transfer orbit is one half of an elliptic orbit that touches both the orbit that one wishes to leave (labelled 1 on diagram) and the orbit that one wishes to reach (3 on diagram). The transfer (2 on diagram) is initiated by firing the spacecraft's engine in order to accelerate it so that it will follow the elliptical orbit. When the spacecraft has reached its destination orbit, it has slowed down to a speed not only lower than the speed in the original circular orbit, but even lower than required for the new circular orbit; the engine is fired again to accelerate it again, to that required speed. Since this transfer requires two powerful bursts, it can not be applied with a low-thrust engine. With that, the circular orbit can be gradually enlarged. Together this requires a delta-v that is up to 141% more (see also below), and, of course, the lower the thrust the longer it takes. When used to move a spacecraft from orbiting one planet to orbiting another, the situation becomes somewhat more complex. For example, consider a spacecraft travelling from the Earth to Mars. At the beginning of its journey, the spacecraft will already have a certain velocity associated with its orbit around Earth – this is velocity that will not need to be found when the spacecraft enters the transfer orbit (around the Sun). At the other end, the spacecraft will need a certain velocity to orbit Mars, which will actually be less than the velocity needed to continue orbiting the Sun in the transfer orbit, let alone attempting to orbit the Sun in an Mars-like orbit. Therefore, the spacecraft will have to decelerate and allow Mars' gravity to capture it. Therefore, relatively small amounts of thrust at either end of the trip are all that are needed to arrange the transfer. Note, however, that the alignment of the two planets in their orbits is crucial – the destination planet and the spacecraft must arrive at the same point in their respective orbits around the Sun at the same time, see launch window. Hohmann transfer orbits also work to bring a spacecraft from a higher orbit into a lower one – in this case, the spacecraft's engine is fired in the opposite direction to its current path, causing it to drop into the elliptical transfer orbit, and fired again in the lower orbit to brake it to the correct speed for that lower orbit. In certain situations where the semimajor axis of the final orbit is much greater then the semimajor axis of the initial orbit (usually by a factor of 12), it may be more advantageous to use a bi-elliptic transfer.

Calculation

Ignoring the delta-v needed to get to/from orbits around planets at either end of the journey, and just calculating the delta-v needed to get from one circular orbit to another coplanar circular orbit around a primary body, e.g. the Sun, the vis viva equation says, :

v^2 = \mu \left( \frac - \frac \right)

where:
-

v \,\!

is the speed of an orbiting body
-

\mu = GM\,\!

is the standard gravitational parameter of the primary body
-

r \,\!

is the distance of the orbiting body from the primary
-

a \,\!

is the semi-major axis of the body's orbit Therefore the Delta-v required for the Hohmann transfer can be computed as follows: :

\Delta v_P 
= \sqrt
  \left( \sqrt - 1 \right)

, Delta-v required at periapsis. :

\Delta v_A 
= \sqrt
  \left( 1 - \sqrt\,\! \right)

, Delta-v required at apoapsis. where:
-

r_1\,\!

is radius of lower orbit, and periapsis distance of Hohmann transfer orbit,
-

r_2\,\!

is radius of higher orbit, and apoapsis distance of Hohmann transfer orbit. Whether moving into a higher or lower orbit, by Kepler's third law, the time taken to transfer between the orbits is: :

t_H 
= \begin\frac12\end \sqrt
= \pi \sqrt

(one half of the orbital period for the whole ellipse) where:
-

a_H\,\!

is length of semi-major axis of the Hohmann transfer orbit.

Example; maximum delta-v

For the geostationary transfer orbit we have

r_2

= 42,164 km and e.g.

r_1

= 6,678 km (altitude 300 km). In the smaller circular orbit the speed is 7.73 km/s, in the larger one 3.07 km/s. In the elliptical orbit in between the speed varies from 10.15 km/s at the perigee to 1.61 km/s at the apogee. The delta-v's are 10.15 − 7.73 = 2.42 and 3.07 − 1.61 = 1.46 km/s, together 3.88 km/s. [http://www.google.com/search?num=100&hl=en&lr=&newwindow=1&safe=off&q=sqrt%28398600%2F6678%29
- sqrt%282%2F%286678%2F42164%2B1%29%29] Compare with the delta-v for an escape orbit: 10.93 − 7.73 = 3.20 km/s. Applying a delta-v at the LEO of only 0.78 km/s more would give the rocket the escape speed, while at the geostationary orbit a delta-v of 1.46 km/s is needed for reaching the sub-escape speed of this circular orbit. This illustrates that at large speeds the same delta-v provides more specific orbital energy, and, as explained in gravity drag, energy increase is maximized if one spends the delta-v as soon as possible, rather than spending some, being decelerated by gravity, and then spending some more (of course, the objective of a Hohmann transfer orbit is different). A Hohmann transfer orbit from a given circular orbit to a larger circular orbit, in the case of a single central body, costs the largest delta-v (53.6 % of the original orbital speed) if the radius of the target circle is 15.5 times as large as that of the original circle. For larger target circles the delta-v decreases again, and tends to

\sqrt-1

times the original orbital speed (41.4%). (The first burst tends to acceleration to the escape speed, the second tends to zero.)

Low-thrust transfer

It can be derived that going from one circular orbit to another by gradually changing the radius costs a delta-v of simply the absolute value of the difference between the two speeds. Thus for the geostationary transfer orbit 7.73 - 3.07 = 4.66 km/s, the same as, in the absence of gravity, the deceleration would cost. In fact, acceleration is applied to compensate half of the deceleration due to moving outward. Therefore the acceleration due to thrust is equal to the deceleration due to the combined effect of thrust and gravity. For escaping the Earth from LEO the required delta-v is 7.73 km/s, compare with the 3.20 km/s in the case of a single burst.

Interplanetary Superhighway

In 1997, a set of orbits known as the Interplanetary Superhighway was published, providing even lower-energy (though much slower) paths between different orbits than Hohmann transfer orbits.

External links

- http://www.pma.caltech.edu/~chirata/deltav.html
- [http://www.braeunig.us/space/orbmech.htm ORBITAL MECHANICS] (Rocket and Space Technology) Category:Astrodynamics Category:Spacecraft propulsion Category:Celestial mechanics

Gravitational slingshot

Image:StillJupiter.jpg Image:MovingJupiter.jpg

In orbital mechanics and aerospace engineering, a gravitational slingshot is the use of the motion of a planet to alter the path and speed of an interplanetary spacecraft. It is a commonly used maneuver for visiting the outer planets, which would otherwise be prohibitively expensive, if not impossible, to reach with current technologies. A slingshot maneuver around a planet changes a spacecraft's velocity relative to the Sun, even though it preserves the spacecraft's speed relative to the planet (as it must do, according to the law of conservation of energy). To a first approximation, from a large distance, the spacecraft appears to have bounced off the planet.

Explanation

Consider a spacecraft on a trajectory that will take it close to a planet, say Jupiter. As the spacecraft approaches the planet, Jupiter's gravity will pull on the spacecraft, speeding it up. After passing the planet, the gravity will continue pulling on the spacecraft, slowing it down. The net effect on the speed is zero, although the direction may have changed in the process. So where is the slingshot? The key is to remember that the planets are not standing still, they are moving in their orbits around the Sun. Thus while the speed of the spacecraft has remained the same as measured with reference to Jupiter, the initial and final speeds may be quite different as measured in the Sun's frame of reference. Depending on the direction of the outbound leg of the trajectory, the spacecraft can gain a significant fraction of the orbital speed of the planet. In the case of Jupiter, this is over 13 km/s. A slingshot may be simulated by rolling a steel ball past a magnet in one hand that is then moved away. Because both masses must not cross paths, the acceleration is oblique to the field and thus is similar to a sail vehicle tacking to work against the force.

Background

It is important to understand how spacecraft move from planet to planet. Taking Mars as an example destination, the simplest way to solve this problem is to use a Hohmann transfer orbit, an elliptical orbit with the Earth at perihelion and Mars at aphelion. If launched at the proper moment, the spacecraft will arrive at the aphelion just as Mars is passing by. These types of transfers are commonly used, e.g., for moving between orbits over the Earth, Earth-Moon and Earth-Mars transfers. A Hohmann transfer to the outer planets requires long times and considerable delta V (the velocity adjustments that consume rocket propellant). This is where the slingshot finds its most common applications. Instead of the Hohmann trajectory directly to, say Saturn, the spacecraft is instead sent in a path that is aimed only as far as, say, Jupiter, and the slingshot is then used to accelerate the spacecraft on towards Saturn. This way, the planet lends the spacecraft additional angular momentum, allowing it to reach Saturn using little or no fuel on top of that required to reach Jupiter. Also, during the spacecraft's close approach to Jupiter, the effectiveness of the rocket's propellant is magnified, such that small thrusts near Jupiter produce large changes in the spacecraft's eventual velocity. Such missions require careful timing, making the launch window a crucial part of the mission. A Hohmann transfer to Saturn would require a total of 15.7 km/s delta V (disregarding Earth's and Saturn's own gravity wells, and disregarding aerobraking) which is not within the capabilities of our current spacecraft boosters. A trip using multiple gravitational assists may take longer, but will use considerably less delta V, allowing a much larger spacecraft to be sent. Such a strategy was used on the Cassini probe, which passed by Venus twice, then Earth, and finally Jupiter on the way to Saturn. The 6.7-year transit is slightly longer than the six years needed for a Hohmann transfer, but cut the total amount of delta V needed to about 2 km/s, so that the large and heavy Cassini was able to reach Saturn even with the small boosters available. center center. The speed above is instantaneous distance in kilometers per second. The time and date is Orbiter UTC, which is from 1997-Oct-16 00:00:01 to 2008-Jul-07 00:00:00, notably there is one leap second during this period. Note also that the minimum velocity achieved during Saturnian orbit is more or less equal to Saturn's own orbital velocity, which is the ~5km/s velocity which Cassini matched to enter orbit.]] Another example is Ulysses, the ESA spacecraft which studied the polar regions of the Sun. All the planets orbit roughly in a plane aligned with the equator of the Sun; to move to an orbit passing over the poles of the Sun, the spacecraft would have to change its 30 km/s of the Earth's orbit to another trajectory at right angles to the plane of the Earth's orbit, a task impossible with current spacecraft propulsion systems. Instead the craft was sent towards Jupiter, aimed to arrive at a point in space just "in front" and "below" the planet. As it passed the planet, the probe 'fell' through Jupiter's gravity field, borrowing a minute amount of momentum from the planet; after it had passed Jupiter, the velocity change had bent the probe's trajectory up out of the plane of the planetary orbits, placing it in an orbit that passed over the poles of the Sun, rendering that region visible to the probe. This manoeuvre required only enough fuel to send Ulysses to a point near Jupiter, which is well within current technologies.

Powered slingshots

A well-established way to get more energy from a slingshot is to fire a rocket engine near the periapsis to increase the spacecraft's speed. Although a given rocket burn will provide the same change in velocity (delta-v) regardless of when it occurs, the resulting change in kinetic energy is proportional to the velocity at the time of the burn. Therefore, to maximize the effect of the burn, gaining the most kinetic energy for the propellant expended, the burn must occur at the moment when the velocity is at its maximum, which occurs at periapsis. From an energy conservation standpoint, the extra energy comes from the propellant being "left behind" in the planet's gravity well. If the ship travels at velocity

v

at the start of a burn that changes the velocity by

\Delta v

, then the change in specific orbital energy (SOE) is: :

v \Delta v + \frac

Once the space craft is far from the planet again, the SOE is entirely kinetic, since gravitational potential energy tends to zero. Therefore, the larger the

v

at the time of the burn, the greater the final kinetic energy, and the higher the final velocity. For example, a Hohmann transfer orbit from Earth to Jupiter brings a spacecraft into a hyperbolic flyby of Jupiter with a periapsis velocity of 60 km/s, and a final velocity (asymptotic residual velocity) of 5.6 km/s, which is 10.7 times slower. That means a burn that adds one joule of kinetic energy when far from Jupiter would add 10.7 joules at periapsis. Every 1 m/s gained at periapsis adds

\sqrt = 3.3

m/s to the spacecraft's final velocity. Thus, Jupiter's immense gravitational field has tripled the effectiveness of the space craft's propellant. See also specific energy change of rockets: :

\Delta \epsilon =  \int v\, d (\Delta v)

where

\epsilon

is the specific energy of the rocket (potential plus kinetic energy) and

\Delta v

is a separate variable, not just the change in

v

Limits to slingshot use

The main practical limit to the use of a slingshot is the size of the available masses in the mission. Another limit is caused by the atmosphere of the available planet. The closer the craft can get, the more boost it gets, because gravity falls with the square of distance. If a craft gets too far into the atmosphere, the energy lost to friction can exceed that gained from the planet. Interplanetary slingshots using the Sun itself are impossible because the Sun is at rest with respect to its own frame of reference and is therefore incapable of donating any angular momentum. However, thrusting when near the Sun has the same effect as the powered slingshot described above. This has the potential to magnify a spacecraft's thrusting power enormously, but is limited by the spacecraft's ability to resist the heat. An interstellar slingshot using the Sun is conceivable, involving for example an object coming from elsewhere in our galaxy and slingshotting around the Sun to boost its galactic travel. The energy and angular momentum would then come from the Sun's orbit around the Milky Way. The time scales involved for such an operation are considerably beyond current human capabilities, however. There's also another, theoretical limit based on general relativity. If a spacecraft gets close to the Schwarzschild radius of a black hole (the ultimate gravity well), space becomes so curved that slingshot orbits require more energy to escape than the energy that could be added by the black hole's motion. Also, a spinning mass produces frame-dragging. A spinning black hole actually is surrounded by a region of space, called the ergosphere, within which standing still (with respect to the black hole's spin) is impossible, as space itself slips in the same direction as the black hole's spin at the speed of light. Suffice it to say that there is a subtle relativistic effect (the Lense-Thirring effect) which can transfer angular momentum between any spinning mass and a passing object.
See also

- Delta-v budget
- Pioneer 11
- Voyager 1
- Voyager 2
- MESSENGER
- Michael Minovitch
External links

- [http://www.dur.ac.uk/bob.johnson/SL/ Slingshot effect]
- [http://www.esa.int/esaCP/SEMXLE0P4HD_index_0.html Animation of Cassini Huygens gravitational sling shot]
- [http://www.mathpages.com/home/kmath114.htm Gravitational Slingshot Theory]
- [http://saturn.jpl.nasa.gov/mission/gravity-assist-primer.cfm A Quick Gravity Assist Primer] Category:Spacecraft propulsion Category:Celestial mechanics ja:スイングバイ

Staging (rocketry)

Description

A multistage (or multi-stage) rocket is, like any rocket, propelled by the recoil pressure of the burning gases it emits as it burns fuel. What characterizes it as "multi-stage" is that it successively jettisons one or more stages as they become empty. It is effectively one or more rockets (stages) stacked on top of each other in order to reduce the total amount of mass which needs to be accelerated to the final speed/height. Generally each stage consists of one or more motors, plus fuel and oxidiser tanks for a liquid rocket or the casing for a solid rocket. In rocketry, this concept is known as staging. The first stage is at the bottom and is usually the largest, the second stage above it and is usually the next largest, etc. In the typical case (linear staging), when the first stage's motor(s) fire, the entire rocket is propelled upwards. When the first stage's motor(s) run out of fuel, the first stage is detached from the rest of the rocket (usually with some kind of small explosive charge) and falls away. This leaves a smaller rocket, with the second stage on the bottom, which then fires. This process is repeated until the final stage's motor burns out. explosive

Advantages

The main reason for multi-stage rockets and boosters is that once the fuel is burnt, the space and structure which contained it and the motors themselves (in the case of liquid-fuelled rockets) are useless and only add weight to the vehicle which slows down its future acceleration. By dropping the boosters and/or stages which are no longer useful, the rocket lightens itself. The thrust of the future stages is able to provide more acceleration than if the earlier stages or boosters were still attached, or than a single, large rocket would be capable of. When a stage drops off, the rest of the rocket is still travelling near to the speed that the whole assembly reached at burn-out time. This means that it needs less total fuel to reach a given velocity and/or altitude. A further advantage is that each stage can use a different type of rocket motor, with each stage/motor tuned for the conditions in which it will operate. Thus the lower stage booster can use a motor suited for use at atmospheric pressure, while the upper stages can use motors suited to near vacuum conditions. Lower stages tend to require more structure than upper as they need to bear their own weight plus that of the stages above them, optimizing the structure of each stage decreases the weight of the total vehicle and provides further advantage.

Disadvantages

On the downside, staging requires the vehicle to loft motors which are not being used until later, as well as making the entire rocket more complex and harder to build. Nevertheless the savings are so great that every rocket currently used to deliver a payload into orbit uses staging. In more recent times the usefulness of the technique has come into question. Since the costs of space launches appear to mostly composed of the operational costs of the people involved (as opposed to fuel or equipment), reducing these costs appears to be the best way to lower the overall launch cost. Since staging is expensive in terms of manpower, a new movement has concentrated on single stage to orbit designs that do not have separate stages.

Development

single stage to orbit single stage to orbit single stage to orbit single stage to orbit The earliest experiments with multistage rockets were made Austrian Conrad Haas, the arsenal master of the town of Sibiu, Transylvania (now in Romania) in 1551. This concept was developed independently by at least four individuals:
- the Polish Kazimierz Siemienowicz
- the Russian Konstantin Tsiolkovsky
- the American Robert Goddard
- the ethnically German, Transylvanian-born Hermann Oberth.

Orbital stationkeeping

In astrodynamics orbital stationkeeping is a term used to describe a particular set of orbital maneuvers used to keep a spacecraft in assigned orbit, either low earth orbit (LEO), or geostationary orbit (GEO). It is especially important for satellite communications systems since maintaining proper satellite position over long periods of time is crucial for the operation of those systems. Stationkeeping maneuvers require inclusion of a particular delta-v in the mission's delta-v budget. Due to its usually low requirements for propulsive impulses the stationkeeping is usually performed using attitude control system's thrusters. Stationkeeping is particularly hard for spin-stabilized satellites.

Stationkeeping in LEO

Stationkeeping is necessary for some objects such as the International Space Station or the Mir or Salyut stations. The International Space Station has an operational altitude above Earth between 330 and 410 km. Due to atmospheric drag, the space station is constantly losing orbital energy. In order to compensate for this loss, which would eventually lead to a reentry of the station (time depends on solar activity), it is being reboosted to a higher orbit from time to time. The chosen orbital altitude is a trade-off between the delta-v needed to reboost the station and the delta-v needed to send payloads and people to the station. The upper limitation of orbit altitude is due to the constraints imposed by the Soyuz spacecraft.

Stationkeeping in GEO

Due to luni-solar perturbations and the ellipticity of the Earth equator, an object placed in a GEO without any stationkeeping would not stay there. It would start building up inclination at an initial rate of about 0.85 degrees per year. After 26.5 years the object would have an inclination of 15 degrees, decreasing back to zero after another 26.5 years. Therefore, a lot of energy has to be devoted to maneuvers that compensate this tendency. This part of the GEO stationkeeping is called North-South control. The ellipticity of the Earth equator is causing an East-West drift if the satellite is not placed in one of the stable (75 degrees longitude east, 105 degrees longitude west) or unstable (15 degrees longitude west, 165 degrees longitude east) equilibrium points. Nevertheless, this part of GEO stationkeeping, called East-West control requires significantly less amount of fuel than North-South control. Therefore, in some cases aging satellites are only East-West controlled. This would still guarantee that the satellite is always visible to a steerable antenna. Taking into consideration the relatively long periods of operation of modern GEO satellites (up to 15 years) the delta-v expended over such a period can be substantial (about 46 m/s per year). It is therefore crucial for GEO satellites to have the most fuel-efficient propulsion system. Some modern satellites are therefore employing a high specific impulse system like plasma or ion thrusters.

Delta-v budget

Delta-v budget (or velocity change budget) is a term used in astrodynamics and aerospace industry for velocity change (or delta-v) requirements for the various propulsive tasks and orbital maneuvers over phases of the space mission. Sample delta-v budget will enumerate various classes of manoeuvres, delta-v per manoeuvre, number of manoeuvres required over the time of the mission. In the absence of an atmosphere and landings where the ground is hit with some speed, the delta-v is the same for changes in orbit the other way around: gaining and losing speed cost an equal effort.

Launch/landing budget

- Launch to LEO — this not only requires an increase of velocity from 0 to 7.8 km/s, but also typically 1.5–2 km/s for atmospheric drag and gravity drag
- Re-entry from LEO

Stationkeeping budget

Interplanetary budget

External links

- [http://www.pma.caltech.edu/~chirata/deltav.html Delta V pages at Caltech] Category:Astrodynamics Category:Spacecraft propulsion

Orbital maneuver

An orbital maneuver is a change from one orbit to another, accomplished by applying thrust. In deep space it is called deep-space maneuver (DSM).

Impulsive maneuvers

An impulsive maneuver approximates a finite thrust maneuver by adding an instantaneous velocity change to an ephemeris record while maintaining the position. During the planning phase of most space or rocket missions, designers will first calculate orbital changes using impulsive maneuvers. This greatly reduces the complexity of finding the correct orbital transitions. The instantaneous changes in velocity are referred to as delta-v (

\Delta\mathbf\,

), the total delta-v for all maneuvers required in the mission is called a delta-v budget. With a good approximation of the delta-v budget designers can estimate the fuel to payload requirements of the spacecraft. Using these approximations are most useful when finite thrusts are to be executed in short bursts. Finite maneuvers like these are possible with high thrust-to-weight propulsion systems, e.g. chemical rockets. However, even for long burns, impulsive maneuver approximations remain very accurate outside the Earth's atmosphere.

Non-impulsive maneuvers

Applying a low thrust over longer periods of time is referred to as non-impulsive maneuvers (even though any thrust can be said to produce an amount of impulse). They are less efficient as energy can be lost due to gravity drag. However those maneuvers can be the only option when efficient but low thrust-to-weight propulsion systems are used (e.g. ion engines). They are not possible for a launch.

Finite Burn Trajectories

For a few space missions high fidelity models of the trajectories are required to meet the mission goals. Calculating a finite burn requires a detailed model of the spacecraft and its thrusters. The most important of details include: mass, center of mass, moment of inertia, thruster positions, thrust vectors, thrust curves, specific impulse, thrust centroid offsets and fuel consumption.

Orbital stationkeeping

Stationkeeping in LEO

Stationkeeping in GEO

Specific impulse

The specific impulse (commonly abbreviated I_sp) of a propulsion system is the impulse (change in momentum) per unit of propellant. Depending on whether the amount of propellant is expressed in mass or in weight (by convention weight on the Earth) the dimension of specific impulse is that of speed or time, respectively, differing by a factor of g, the gravitational acceleration at the surface of the Earth.

General considerations

Essentially, the higher the specific impulse, the less propellant is needed to gain a given amount of momentum. In this regard a propulsion method is more fuel-efficient if the specific impulse is higher. This should not in any way be confused with energy-efficiency, which can even decrease as specific impulse increases, since many propulsion systems that give high specific impulse require high energy to do so. In addition it is important that thrust and specific impulse not be confused with one another. The specific impulse is a measure of the thrust per unit of propellant that is expelled, while thrust is a measure of the momentary or peak force supplied by a particular engine. In fact, propulsion systems with very high specific impulses (such as ion thrusters: 3,000 seconds) are power limited to producing low thrusts, due to the relatively high weight of power generators. When calculating specific impulse, only propellant that is carried with the vehicle before use is counted. For a chemical rocket the propellant mass therefore would include both fuel and oxidizer; for air-breathing engines only the mass of the fuel is counted, not the mass of air passing through the engine.

Examples

Specific impulse of various propulsion technologies

An example of a specific impulse measured in time is 459 seconds, or, equivalently, an effective exhaust velocity of 4500 m/s, for the Space Shuttle Main Engines when operating in vacuum. An air-breathing engine typically has a much larger specific impulse than a rocket: a jet engine may have a specific impulse of 2000-3000 seconds or more at sea level. In some ways, comparing specific impulse seems unfair in the case of jet engines and rockets. However in rocket or jet powered aircraft, specific impulse is approximately proportional to range, and rockets do indeed perform much worse than jets at sea level. The highest specific impulse for a chemical propellant ever test-fired in a rocket engine was lithium, fluorine, and hydrogen (a tripropellant): 542 seconds (5320 m/s). However, the combination is impractical, see rocket fuel. Nuclear thermal rocket engines differ from conventional rocket engines in that thrust is created strictly through thermodynamic phenomena, with no chemical reaction. The nuclear rocket typically operates by passing hydrogen gas over a superheated nuclear core. [http://www.lascruces.com/~mrpbar/rocket.html Testing in the 1960s] yielded specific impulses of about 850 seconds (8340 m/s), about twice that of the Space Shuttle engines. A variety of other non-rocket propulsion methods, such as ion thrusters, give much higher specific impulse but with much lower thrust; for example the Hall effect thruster on the Smart 1 satellite has a specific impulse of 1640 s (16100 m/s) but a maximum thrust of only 68 millinewtons. The hypothetical Variable specific impulse magnetoplasma rocket(VASIMR) propulsion should yield a minimum of 10,000-300,000 m/s but will probably require a great deal of heavy machinery to confine even relatively diffuse plasmas, so they will be unusable for very-high-thrust applications such as launch from planetary surfaces.

Specific impulse in seconds

For all vehicles specific impulse (impulse per unit weight-on-Earth of propellant) in seconds can be defined by the following equation: :

\mathrm=I_ \cdot \frac   \cdot g_ \,

where: Thrust is the thrust obtained from the engine. I_sp is the specific impulse measured in seconds.

\frac

is the mass flow rate, which is minus the time-rate of change of the vehicle's mass, since fuel is being expelled. g₀ is the acceleration at the Earth's surface. This I_sp in seconds value is somewhat physically meaningful - it is the number of seconds a unit weight of fuel would last if the engine would apply a unit force (if an engine could be scaled proportionately). As such it is a value that can be used to compare engines; much like 'miles per gallon' is for cars. The advantage that this formulation has is that it may be used for rockets, where all the reaction mass is carried onboard, as well as aeroplanes, where most of the reaction mass is taken from the atmosphere. In addition, it gives a result that is independent of units used (provided the unit of time used is the second).

Rocketry - specific impulse in seconds

In rocketry, where the only reaction mass is the propellent, an equivalent way of calculating the specific impulse in seconds is also frequently used. In this sense, specific impulse is defined as the change in momentum per unit weight-on-Earth of the propellent: :

I_=\frac

where I_sp is the specific impulse measured in seconds

v_

is the average exhaust speed along the axis of the engine g₀ is the acceleration at the Earth's surface It may seem odd that the acceleration or weight at the Earth's surface is in the definition, while the rocket may be far from the Earth. However, accelerations are often measured in terms of g₀; for example, astronauts should not be subjected to an acceleration more than a few times this value. Additionally, in Imperial units the relationship between force and mass is defined to involve the acceleration due to gravity. Thus pounds (force) and pounds (mass), both used in rocketry, when divided, must be additionally multiplied by g₀ to get the acceleration in more usual units. The official Imperial unit of mass the slug, which is not popular for obvious reasons, was introduced to make Imperial units more like the SI units and avoid this multiplication. This, the common use of pounds for both force and mass, is in fact the chief reason g₀ enters so often into rocketry definitions, and is likely the reason two definitions of specific impulse are in common use. When expressed in units of seconds, the specific impulse can be interpreted in the following ways:
- the impulse divided by the sea-level weight of a unit mass of propellant
- the time one kilogram of propellant lasts if a force equal to the weight of one kilogram is produced, for example for a hypothetical hovering over the Earth (imagine the fuel to be supplied from outside, so that the mass on which the thrust is applied does not reduce by spending fuel)
- the time one pound mass of propellant lasts if a force of one pound is produced, for example for a hypothetical hovering vehicle over the Earth (imagine the fuel to be supplied from outside, so that the mass on which the thrust is applied does not reduce by spending fuel)
- alternatively, for engines that can not produce a large thrust: approximately the time one kilogram of propellant lasts if an acceleration of 0.01 g of a mass of one 100 kilogram is produced
- 100 times the time an acceleration g can be produced (i.e. a thrust equal to the weight on Earth of the current mass) with a propellant mass of 1 % of the current total mass (100 times the time it takes in this case to reduce the total mass by 1 %)
- the time an acceleration g can be produced with a propellant mass of 63.2 % of the