What Information Is Most Important In Determining The Size Of An Orbital

3. Spacecraft Design Drivers, Infinite and Orbit

three.5 Orbital Mechanics

The nearly brilliant scientific minds of ancient times had radically unlike views of the motion and nature of the planets, sun, and stars. They accepted their view as a certainty – the nature of the universe equally they believed it. Some ancient scientists and astronomers did experiments that threw some of these behavior into question. It is important to the student to understand how our current understanding evolved, who were the scientists that led us to our electric current understanding, how they did it, and the difficulties they encountered when they presented their radical ideas. For the evolution of astronomy, nosotros should really start with the ancient Babylonians, Chaldeans, and Chinese, only we are concentrating on the evolution of orbital mechanics. From the get-go century A.D., the prevailing theory of the solar system was the Ptolemaic System (developed by Roman citizen Claudius Ptolemaeus) in which the Sunday and planets rotated around the Globe, although non centered on the Globe, using a complex system of epicycles. This system was very complicated but did match the observed movements fairly closely. The current sunday-centered (heliocentric) system was introduced by Copernicus, and that's where we'll kickoff.

Nicolaus Copernicus (1473-1543)

Nicolaus Copernicus was a Renaissance-era mathematician, astronomer, and Cosmic canon who formulated a model of the universe that placed the Sun rather than Earth at the center of the universe. Image from the town hall in Torun in 1580.

Roman Catholic cleric
Developed a heliocentric model of the solar system and published it in the book On the Revolution of the Celestial Spheres which he defended to Pope Paul III in 1543.
Martin Luther said, "This fool wishes u.s.a. to reverse the entire science of astronomy…sacred scriptures tell us that Joshua commanded the Sunday to stand still and non the Earth!"
The book was put on the forbidden list by the Church building in 1616 and not removed until 1835.
Copernican model of the solar arrangement was heliocentric with Globe rotating on its axis
- simpler than the Ptolemy model and better able to explicate the observed behavior of heavenly bodies (ascension and setting caused by rotation on-axis)
- seasons explained by annual revolution around the sun
Deficiencies in the Copernican model:
- kept planets in perfectly circular orbits
- could not prove Earth moved
- kept stars in a crystalline shell

Galileo Galilei (1564-1642)

Galileo has been called the "begetter of observational astronomy", the "father of modernistic physics", the "father of the scientific method", and the "father of modern science". Paradigm past Justus Sustermans.

Congenital the first telescope used for astronomy in 1609 – magnification was 20x
Dutchman, Zacharias Janssen, the showtime known telescope maker, fabricated a copy of an earlier Italian specimen of unknown origin in 1608
Made first detailed maps of the Moon (determined information technology was a solid earth with mountains & craters)
Discovered moons around Jupiter
In 1610 published Messenger of the Stars, which presented a formidable array of observational testify to disprove the Ptolemaic organisation
"Wandering stars" (planets) apparent retrograde and wandering motion was explained by Galileo

Tycho Brahe (1546-1601)

Tycho Brahe was a Danish nobleman, astronomer, and writer known for his authentic and comprehensive astronomical observations. Image by Edward Ender.

Danish nobleman who studied astronomy & built observatory castle
Lost his nose in a duel and had it reconstructed of gold, silver, & wax
Tenacious astronomical observer
Pushed instruments to limit (Brahe's quadrant) with observation precision of 4-five arc minutes
Observation of supernovas & comets led to the conclusion that solid crystalline spheres did non exist in infinite since things were not supposed to change in the superlunar realm
Did not accept Copernican organization
Non skilled mathematician
Died at the dinner table in the court of Rex Rudolf 2

Johannes Kepler (1571-1630)

Johannes Kepler was a German astronomer, mathematician, and astrologer. Epitome past an unknown artist in 1610.

German mathematician from a poor former noble family
Started working in 1600 for Tycho Brahe, who died in 1601
Challenged by Brahe to calculate the orbit of Mars, which Kepler did for over five years (1601-1606)
Kepler couldn't reconcile circular motion to Brahe's data – consistently had 8 minutes of arc discrepancy between theory and information
Finally fit the data when an ellipse was used with the Sun at ane focus
Kepler was so confident in Brahe's data and his fit, he made it into a law of motion

This became Kepler's Showtime Police of Planetary Motion:
- The orbits of the planets are ellipses with the Sunday at 1 focus
He noticed that the line between the Sun and Mars swept out equal areas in equal times, thus a planet must move faster every bit information technology approaches the Sun – this resulted in Kepler'south 2nd Police force:
- The line joining a planet to the Sun sweeps out equal areas in equal times
Kepler published these 2 laws in 1609 in the book Astronomia Nova … De Motu Stellae Mars (New Astronomy…On the Movement of Mars)
10 years afterward he added his third law, i.eastward., Kepler's Third Law:
- The foursquare of the period of a planet is proportional to the cube of its mean altitude from the Sun.

Sir Isaac Newton (1642-1727)

Sir Isaac Newton was an English mathematician, physicist, astronomer, theologian, and author. Image by Godfrey Kneller.

In 1665, while a pupil at Trinity College in Cambridge, the plague caused the university to close for 2 years
During these two years, which he spent at his home hamlet in the country, he formulated the three laws of motion, conceived the law of gravitation, developed the concept of differential calculus, and worked with optics (discovered spectrum)
His constabulary of gravitation resulted from his observation of a falling apple – nevertheless, due to an erroneous value for Earth's radius, he discounted his gravity work
At historic period 27 he was appointed to the Chair of Mathematics at Cambridge and at 30 was elected to the Regal Gild
In 1684 Robert Hooke boasted to Christopher Wren and Edmond Halley that he had worked out the laws of move governing the heavenly bodies. Wren was unimpressed with Hooke'southward solution and and so put upwardly a prize (40/-) for the correct answer. Halley took the problem to Newton who had solved the trouble in 1665.
At the urging of Halley, Newton recalculated his formulations and with the correct radius of the World, came upwards with the exact answer that matched observations.
Newton published his laws and discoveries in 1687 in the book Philosophiae Naturalis Principia Mathematica, better known today only equally Principia.
Newton was:
- elected to Parliament in 1689
- appointed Warden of the Mint in 1696
- elected president of the Regal Guild in 1703
- knighted in 1705
Newton's Laws of Motion:
- LAW i
  - Everybody continues in its state of rest, or of uniform motion in a right line unless information technology is compelled to change that state by forces impressed upon it.
- LAW 2
  - The alter of motility is proportional to the motive force impressed and is made in the direction of the right line in which that force is impressed.
- Constabulary 3
  - To every activeness at that place is always opposed an equal reaction; or, the mutual actions of ii bodies upon each other are e'er equal, and directed to contrary parts.
Newton'southward Law of Universal Gravitation
- The force of gravity between two bodies is directly proportional to the product of their two masses and inversely proportional to the square of the distance between them.

The Nuts

Kepler's three Laws of Planetary Motion, Newton's Police of Universal Gravitation, and Newton's three Laws of Move provide the edifice blocks upon which orbital mechanics are built. We will use these principles to develop the equations and parameters we need to describe orbital motion and to run into how satellites work and the different types of orbits that are available. We will later even have a cursory look at interplanetary trajectories, just let's start with the basics.

Newton'southward Laws of Motion

To show how nosotros get from these seven laws to what we know equally orbital mechanics, let's start with Newton's laws of motion. For convenience, nosotros have reworded Newton's Laws to the modern English form.

To prove how we get from these 7 laws to what we know every bit orbital mechanics, let's start with Newton's laws of motion. For convenience, we have reworded Newton's Laws to the modern English language form.

Paradigm past Trevor Sorensen with HSFL.

Showtime LAW

Every object will remain at rest or in uniform motion in a direct line unless compelled to alter its state past the action of an external strength. This is normally taken as the definition of inertia. The key point here is that if at that place is no net force acting on an object (if all the external forces cancel each other out), then the object will maintain a abiding velocity. If that velocity is zilch, then the object remains at balance. If an external force is practical, the velocity will change because of the force.

2nd Police force

The time charge per unit of alter of the momentum of a torso is equal in both magnitude and direction to the force imposed on information technology. This constabulary defines a force to be equal to a change in momentum (mass times velocity) per alter in time. The linear momentum and angular momentum are given by the following equations:

Linear Momentum: p = mv

Athwart Momentum:

where p is the angular momentum vector, m is the mass of the body, and 5 is the velocity vector. In the second equation, h is the angular momentum vector, I is the moment of inertia of the body, is the athwart velocity or rate, R is the position vector from the origin of the cartesian coordinate organization.

This is the famous equation that Strength = mass times acceleration (bold the mass stays constant).

Third Law

For every force, there is an equal and opposite reaction. This police is the basis for rocket propulsion, and releasing a filled balloon with the oral fissure open and seeing it fly effectually is an example of this law in work. Mathematically it looks similar this:

NEWTON'Due south Police force OF UNIVERSAL GRAVITATION

The forcefulness of gravity between two bodies is directly proportional to the product of their two masses and inversely proportional to the foursquare of the distance between them.

Newton'south Police of Universal Gravitation is expressed mathematically. Information technology includes a universal constant, Thousand, whose value was determined by Henry Cavendish in 1798.

previous effigy courtesy of Honeywell Applied science Solutions Inc. (HTSI)

Nosotros just showed the application of Newton's Law of Universal Gravitation for ii bodies, such as a satellite orbiting the Earth. Still, the law really takes into account all bodies with mass equally shown in the following figure. Consider a system in which there are n masses (m1, m2, m3,…mn). Applying Newton'due south Gravitational Law, each mass exerts an attractive force on all the other masses forth lines connecting the pair of masses. The body (mass) of interest is $m_i$ .

Applying Newton'south Gravitational Police, the force on mi by all northward masses is the vector sum of all gravitational forces acting on it:

$F_{OTHER}$ is the resultant forcefulness of all non-gravitational forces interim on the body:

$F_{OTHER}$ = $F_{drag} + F_{thrust} + F_{solar pressure} + F_{non-spherical perturbations}$ +…

$F_{TOTAL}$ = $F_{g} + F_{OTHER}$

Applying Newton'due south Second Law of Motility and bold $F_{OTHER}$ = 0

A closed-grade solution does not be for n>2.

Assume m1 is Earth and m2 is a satellite orbiting Earth, then it tin be shown that the acceleration of the satellite relative to Globe is:

2-body gravity. Perturbing effects due to other bodies

The perturbing effects of other bodies are shown in the tabular array. Effect of Other Bodies on Gravitational Acceleration for a satellite at an altitude of 370 km.

Alluring Body Acceleration in g's on Satellite

The Two-Body Problem

The equation shown for Newton's Law of Universal Gravitation for due north bodies is unsolvable in a closed-class solution. However, the equation is solvable for two bodies, and as seen from the tabular array the furnishings of the other planetary bodies are relatively small for a satellite orbiting the Earth, and then we will ignore the external furnishings and consider ii-body orbital mechanics. If you want precise solutions, and so the other bodies have to exist included and numerical methods used.

Satellite Flight Near to Earth is Approximated as a Two-Body Problem.

Assumptions

To treat the spacecraft motion as a two-body problem, we have to make the following iv important assumptions:

The mass of the satellite (m2) is negligible compared to that of the alluring principal torso (m1).
The coordinate arrangement chosen for a item problem is inertial. This supposition removes derivatives of the coordinate organization itself when differentiating vectors. Assume Earth-orbiting satellites apply the geocentric equatorial coordinate organization and interplanetary probes use the heliocentric system.
The bodies of the satellite and attracting primary trunk are spherically symmetrical with uniform density. This allows us to care for each as a betoken mass.
No other forces act on the system except for gravitational forces that act forth a line joining the centers of the two bodies. This also means the masses are constant (i.e., no modify in mass).

Equation of Motion

Assume two abiding masses interact by a force that is a office simply of the relative distance between the two masses and that is directed forth with the position m2 with respect to m1.

The geometry of Two Bodies in an inertial reference frame (coordinate system).

The differential equation of motion for the two-body trouble is:

where:

Center of Mass

Moment of a mass mi nigh a betoken P is the vector with magnitude $|r_{i}$ |• $m_i$ and direction $r_i$ where $r_i$ goes from $m_i$ to P. Moment of mass of the system nearly P for north masses is thus:

The Center of Mass (CM) is that point at which the total moment of mass = 0.

In a two-body arrangement nosotros can locate the CM (P'):

Now consider a coordinate system with the origin at m2 and x-centrality along the line between m1 and m2, the y-centrality in the plane formed by r1 and r2, and the z-axis completing the RH coordinate organization.

The centre of mass will be located on the line connecting the two masses. Its exact location is defined by:

Distance of CM from mass m2 along the line between m1 and m2 is:

Assuming origin is at m2 (i.eastward., $r_2$ = 0):

Where R = $r_12$ , the altitude between $m_1$ and $m_2$

Assuming origin is at m1 (i.e., r1 = 0):

Case Problem

What is the middle of mass of the Earth-Moon pair?

Given:

Canonical Units

Canonical units are dimensionless units used past astronomers to remove uncertainties. They are as well sometimes used in spacecraft orbital mechanics. Approved units are based on hypothetical circular reference orbits:

One obvious advantage of using Canonical Units is when you lot accept to calculate orbital mechanic equations by hand that the states the gravimetric parameter, which is Canonical Units is simply:

Constants of Movement

Because spacecraft operate in a conservative gravitational field, they conserve mechanical energy and athwart momentum. Using these two principles, we are able to hands determine and predict the motion of spacecraft.

Conservation of Mechanical Energy

Since the gravitational field is "bourgeois" an object moving under the influence of the gravitational field lone does not lose or proceeds full mechanical energy. Although mechanical energy remains constant, it exchanges one form, "kinetic energy" for another, "potential energy." The total mechanical energy (E) is often used in orbital mechanics with a constant mass, and then we usually apply a simplified term, the full mechanical energy per unit of measurement mass chosen the total specific mechanical energy:

Merely the total mechanical energy is the sum of the kinetic and potential energy, and so nosotros can express the specific mechanical energy in the course:

This equation is known equally the Vis-Viva Equation and is 1 of the near of import equations in orbital mechanics. The Vis-Viva Equation shows the full mechanical free energy per unit mass of the satellite converses. The specific potential energy is too equal to the gravitational potential function (5) per unit of measurement mass. One thing to note is that potential energy (PE) is nil at an altitude of infinity, and is increasingly negative between nix and the origin at r=0, i.e., PE<<0.

Conservation of Angular Momentum

Since the gravitational field is always directed radially towards the center of the large mass the athwart momentum of the object about the big mass does not change, the angular momentum per unit mass, called the specific athwart momentum (h), tin can be derived as:

h is ever perpendicular to the plane containing r and v. Since h is constant, r and v must remain in the same aeroplane, which is called the orbital plane. In the figure, we run across some of the terms we apply in describing the satellite motion and orientation in the orbital plane.

Definition of Terms for Satellite in Orbit.

Notation that satellites that desire to keep ane face up always pointing towards the Earth, use what is called an LVLH attitude hold, where LV is the Local Vertical and LH is the Local Horizontal as defined in the effigy. In orbit, the terms "up" and "downward" are generally in reference to the LV vector, which is the vector from the middle of the master mass (m1) to the satellite (m2).

In the figure the following angles are divers:

From which we get the following:

Instance Problem:

Trajectory Equation

From geometry, we know that the polar class of the equation for a conic department is:

Conic Sections and Respective Trajectories. Image by Microcosm Inc.

The family of curves called "conic sections" (i.e., circle, ellipse, parabola, hyperbola) stand for the simply possible paths for an orbiting object in the two-body problem. The circle and ellipse are closed-loop conics, while the parabola and hyperbola are open up conics. The focus of the conic orbit must be located at the heart of the cardinal body. The specific mechanical energy (ε) of a satellite (which is the sum of the kinetic and potential energies) does not change as the satellite moves along its conic orbit. However, there is an commutation of free energy between the two forms (P.E. and K.E.) which means that the satellite must deadening down as it gains altitude (i.due east., r increases) and speed up every bit r decreases so that ε remains constant. The orbital move takes place in a plane fixed in inertial infinite. The specific athwart momentum (h) of a satellite remains abiding. As r and v change along the orbit, the flight path angle () must change then as to keep h constant.

Elliptical Orbits

The virtually common type of conic section used in Earth-orbiting satellites is the ellipse. To understand these orbits, we must commencement be familiar with the geometry and characteristics of ellipses.

Apses of an Ellipse

Farthermost end-points of the major centrality of an ellipse are known as apses (singular apsis). The point closest to the primary focus is the periapsis (rp) and the point farthest from the chief focus is the apoapsis (ra). "Peri-" means near and "apo-" means far.

Tables: The Apses Names for Planetary and Other Bodies

The Geometry of the Ellipse

Equation of an ellipse in rectangular coordinates:

where a is the semi-major axis, and b is the semi-modest axis

Equation of an ellipse in polar coordinates (which was defined previously):

Eccentricity, e, defines the shape of the ellipse by comparing the ratio of the distance between the two foci and the length of the major axis:

except for parabola

Post-obit are another useful equations derived from the geometry of the ellipse that is useful in orbital mechanics.

Semi-Major Axis:

Distance Between Foci:

Eccentricity:

Apsides:

From the conic equation

At periapsis ( $r_p$ ), the true anomaly, 5=0 and the cos five =one.

At apoapsis ( $r_a$ ), the true anomaly, v=180deg and the cos v=-1.

From the structure of the ellipse, shown in Figure:

Semi-Latus Rectum:

Rewriting the trajectory equation, substituting for p:

Orbital Free energy

The equation for the specific mechanical free energy (Vis-Viva Equation) is of import for determining orbital characteristics, just there is another class that is as well very useful, and this is known every bit the Orbital Free energy Equation.

We start with the equation for angular momentum:

applying this equation at periapsis where the flight path angle =0,

Using the equation for specific mechanical energy () and noting energy is abiding everywhere on the orbit:

From this y'all can obtain the Orbital Energy Equation, which is true for all conics:

The Orbital Energy Equation shows that the specific mechanical free energy is inversely proportional to the orbit'due south semi-major axis, i.e., a only depends on ε, which depends only on r and v. The energy of a satellite forth the orbit determines which type of orbit it is in. For instance, firing a cannon horizontally from a high mountain:

In two-body motility, the shape of an orbit is adamant by the speed of the object.

Relationship of Speed to Trajectory Blazon.

Yet, if h = 0 regardless of ε then e = 1 (degenerate conic – point or line, but non parabola). All parabolas have e=1, but an orbit with due east=1 does non take to be a parabola – it could be a degenerate conic (point or line).

Catamenia of an Elliptical Orbit

Kepler'south 2d Constabulary of Planetary Motility states "Equal areas are swept out by the radius vector in equal time intervals."

It can too be shown that:

When we combine these equations nosotros obtain:

Since h is abiding this proves Kepler's 2nd Law. Integrating this equation through one bike (time period, P) noting that area of an ellipse is $A_{ellipse}$ = $\pi$ ab gives the equation for the total period of an orbit, which is the equation of Kepler'due south Tertiary Law of Planetary Motion, "The foursquare of the period of a planet is proportional to the cube of its mean distance from the Dominicus."

Therefore, the flow of an elliptical orbit depends just on the size of the semi-major axis, a.

Non-Elliptical Orbits

Although most Earth orbits are elliptical, some of them accept such a low eccentricity that they can be considered to commencement order as circular. When y'all want to escape from the Earth's gravitational field, then you must employ parabolic or elliptical trajectories, so it is important to examine the characteristics of these other trajectory types.

Round Orbits

Circular orbits are actually a special type of elliptical orbit, simply with an eccentricity e=0, which means that the periapsis and apoapsis radii are equal to the semi-major axis and the radius of the circular orbit.

Parabolic Trajectories

A parabolic trajectory is the lowest energy open-loop trajectory, in which a spacecraft tin can simply escape the gravitational field of the central body.

Thus, at any value of r, the specific KE ( $v_2$ /ii) will exist equal to the specific PE (-/r). The satellite will have merely plenty KE to follow the parabolic trajectory to ∞ where both KE and PE volition be zippo.

Escape Speed

The speed at which a probe can coast to an infinite altitude without falling dorsum is called the escape speed. Probe with escape speed in any direction travels on a parabolic trajectory. The escape speed is calculated using 2 points on a trajectory: (1) at a altitude r where escape speed is $v_{esc}$ , and (ii) at infinity (r=∞).

Escape Speed:

If the probe starts in a circular orbit, so the escape speed from a circular orbit is:

HyperbolicTrajectories

A parabolic trajectory produces the minimum v for escape from a central body in a gravitational field. If excess v is desired after an escape, then the trajectory must exist hyperbolic.

The angle betwixt asymptotes is δ, which is called the turning or handful angle. This is the modify in direction of v approach to v departure.

Turning Angle:

Hyperbolic Excess Speed

If ε > 0, then a is negative, east >1, and trajectory is hyperbolic.

If $v_{bo}$ > $v_{esc}$ where $v_{bo}$ is burnout speed and $v_{esc}$ is minimum escape speed, and then the probe approaches finite speed at infinity that is greater than nothing. This speed tin can be calculated given 5 and r since the specific mechanical energy is constant everywhere on the trajectory.

Using the energy equation you tin can summate the Hyperbolic Excess Speed:

The magnitude of the burn to reach the required excess speed (called the Oberth Maneuver) is:

The value of a can be establish from:

Coordinate Systems

Orbital mechanics uses many different coordinate systems to express the spacecraft trajectories in terms that are meaningful for the current mission or application. It depends on whether the spacecraft is in Earth orbit, a cis-lunar (Earth to Moon) trajectory, interplanetary trajectory, or orbiting some other gravitational torso. We will introduce the almost commonly used coordinate systems. To kickoff with we volition explain the central elements of a cartesian (orthogonal ) coordinate organization and how to specify such a coordinate system.

There are 5 steps in specifying a coordinate system:

ane. Pick the origin.

ii. Pick the fundamental plane.

iii. Pick a perpendicular to it.

4. Choice the principal direction.

5. Add the third centrality using the Right-Hand Rule (RHR).

Figure from "Fundamentals of Astrodynamics" by Bate, Mueller & White, Dover books.

Origin is the Dominicus.
The fundamental plane is the ecliptic (Earth's orbital plane).
Perpendicular is in the direction using RHR with the direction of Globe around the Sun.
The principal direction is the vernal equinox.
Y-axis according to RHR.

Note: Globe's rotation centrality precesses with a period of 26,000 years.

Geocentric-Equatorial Coordinate System (GECS)

Also known as Earth-Centered Inertial (ECI) or IJK Coordinate System

Origin is the heart of the Earth-mass (geocenter).
The primal plane is the equatorial plane.
Perpendicular is in the spin centrality in the northerly direction.
The principal direction is the vernal equinox.
Y-axis according to RHR.

Annotation: Axes practice non rotate with the Earth.

Origin is the center of the Earth-mass (geocenter).
The fundamental airplane is the equatorial plane.
Perpendicular is in the spin axis in the northerly management.
The principal management is in the fundamental aeroplane and points to the Greenwich meridian.
Y-axis according to RHR.

Note: GCS rotates with the Earth.

Origin is the geocenter or whatsoever point.
The fundamental plane is the celestial equatorial (Earth's equator extended to a celestial sphere).
The principal direction is to the vernal equinox.

Objects located by (RA) measured eastward from I in the equatorial plane and measured northward from the equatorial plane. This coordinate arrangement is used in astronomy.

Origin is the primary focus of the orbit.
The fundamental plane is the orbital plane of the satellite.
The principal direction (X-axis) points to periapsis.
Y-axis is rotated 90o in the management of orbital motility and lies in the orbital plane.
Z-centrality is along angular momentum vector (h) and completes the RH organisation.

Classical Orbital Elements

An orbit follows the laws of Newton and Kepler around its primal gravitational body (primary focus) and its motion is very predictable without major exterior disturbances (e.g., drag) or internal disturbances (e.g., thrust). It orbits in a basically inertial plane with respect to the center, and in that location are 6 parameters that define the orbit and the position of the satellite in the orbit. These parameters are called the Classical Orbital Elements (COEs). 5 of the COEs define the orbit, and the sixth defines the position of the satellite in the orbit at the given time. We will define the vi chief COEs and their alternatives.

1. Semi-Major Axis (a)

This defines the size of the orbit.

2. Eccentricity (east)

Defines the shape of the conic orbit,

3. Inclination (i) 0º ≤ i ≤180º

Defines the tilt of the orbital aeroplane with respect to the fundamental plane. i is equal to the angle between the orbital and equatorial planes, although mathematically, it is the angle between the spin axis (Z) and the angular momentum angle (h) of the orbital airplane.

four. Right Ascension of the Ascending Node (Ω)

Defines the swivel of the conic orbit wrt the cardinal airplane. Ω is the angle, in the fundamental plane, between the Ten-axis and the ascending node, which is the indicate where the satellite passes through the cardinal plane in a northerly direction and is measured counterclockwise (easterly) when viewed from to a higher place. UNDEFINED IN EQUATORIAL ORBITS! aka Longitude of the Ascending Node

5. Argument of Periapsis/Perigee (ω)

Defines the orientation of the orbit with the orbital plane. is the bending between the ascending node and periapsis in the orbital aeroplane and measured in the direction of satellite motility (0° ≤ ω ≤ 360°). UNDEFINED IN EQUATORIAL OR Circular ORBITS!

six. Time of Periapsis/Perigee Passage ( $t_o$ or T)

Defines the starting time of the orbit and is the time that the satellite was at periapsis/perigee. Standard practice replaces it with True Bibelot or epoch (νo) which defines the position of the satellite in the orbit at a given fourth dimension (epoch). (0° ≤ νo ≤ 360°) UNDEFINED FOR Circular ORBITS! Notation: ν, θ, and f are used to announce true anomaly. The first v COEs define the orbit, and the sixth COE defines where the satellite is in the orbit.

Alternate COEs

Nearly of the COEs have certain orbits for which they are undefined or at that place might be another parameter that provides the aforementioned information about a item characteristic of the orbit. The fix of parameters that are available to define the orbit when 1 of the original COEs is not possible is called the Alternating COEs.

1. Sem-Latus Rectum (p)

Alternate for semi-major axis.

2. Longitude of Perigee (Ii)

The angle between X-axis (i) and perigee measured eastward to ascending node than in the orbital plane to perigee i.e., II = Ω + ω. Replaces ω for equatorial orbits. UNDEFINED FOR Round ORBITS!

iii. Statement of Latitude at Epoch ( $u_o$ )

Angle in orbital plane between the ascending node and the radius vector to the satellite at time $t_o$ i.e., $u_o$ = ω + $v_o$ Replaces o for round orbits. UNDEFINED FOR EQUATORIAL ORBITS

4. Truthful Longitude at Epoch ( $l_o$ )

The angle between X-centrality and radius vector to the satellite at to measured eastward to ascending node and then in the orbital plane to satellite:

Orbital Inclination

Inclination tin range from 0 to 180 degrees, where a 90-degree inclination is called a polar orbit. 180 degrees of inclination is besides an equatorial orbit but the satellite orbital direction is the opposite of the 0-caste equatorial orbit. As 0 to 90-degree inclination looks symmetric to the ninety to 180-degree inclination, the inclination of orbits may be cleaved further down into the direction of the orbit: prograde (0 to 90) and retrograde (90 to 180). "The satellite's inclination depends on what the satellite was launched to monitor. Many of the satellites in NASA's Globe Observing Organization have a well-nigh polar orbit. In this highly inclined orbit, the satellite moves around the Earth from pole to pole, taking most 99 minutes to complete an orbit. During one-half of the orbit, the satellite views the daytime side of the Globe. At the pole, the satellite crosses over to the nighttime side of Earth" [NASA Earth Observatory].

Outcome of Launch Site on Inclination

Where the satellite is launched on the Earth'south surface and the direction of launch have straight effects on the inclination of the resulting orbit. The breadth of the launch site as well determines the minimum inclination possible without later propulsive maneuvers to modify the inclination. In the following figure, $L_o$ is the breadth of the launch site, $\beta_o$ is the launch azimuth, and i is the inclination.

Effigy from "Fundamentals of Astrodynamics" past Bate, Mueller & White, Dover books.

Since -90° $L_o$ 90° cos $L_o$ must always be positive. A posigrade orbit must thus exist easterly ( $\beta_o$ < 180°). Minimum orbital inclination achievable from launch site at $L_o$ is i = $L_o$ since for all i to be minimized, cos $L_o$ must be maximized, which implies $\beta_o$ = 90°. A satellite cannot be put directly into an equatorial orbit from a non-equatorial launch site.

Orbit Anomalies

Kepler invented "anomalies" (angles) originally used to define a planet'southward orbit motion effectually the Sun. The three anomalies are true anomaly, mean bibelot, and eccentric anomaly. We take already seen and used the true anomaly, and then we volition now look at the other two.

Mean Anomaly

Kepler defined a quantity related to truthful anomaly – the )hateful anomaly (Chiliad), which is the angle that the satellite would have moved since perigee if it were going at a abiding speed (mean motility, due north) in an imaginary circular orbit with the same period equally the actual elliptical orbit. Where:

Hateful anomaly is equal to truthful anomaly at perigee and apogee just for elliptical orbits, and at all times for circular orbits. For low eccentricity orbits, hateful anomaly provides a quick way of estimating the satellite's position.

Eccentric Anomaly

Kepler defined another angle related to a true anomaly called the eccentric anomaly (Eastward), which is found geometrically by circumscribing an elliptical orbit with a circle and relating Due east to M, using the true anomaly, o. Eccentric anomaly is equal to true and means anomaly at periapsis and apoapsis merely for elliptical orbits, and at all times for circular orbits.

NORAD Two-Line Elements (TLEs)

One of the virtually common methods of expressing Keplerian orbital elements is the NORAD two-line element (TLE) format. Originally adult past the North American Aerospace Defense force Command, TLEs are likewise used by NASA and Usa Space Command (USSPACECOM) too as many commercial and shareware satellite tracking programs. TLE element sets for many satellites are updated often and are available on websites similar CelesTrak:

http://celestrak.com/NORAD/elements/

Orbit Types

Orbits may exist classified by centric (the orbit center), altitude for geocentric orbits, inclination, directional, eccentricity, and synchronicity. To signify which planetary body the spacecraft orbits almost, diverse prefixes are concatenated to the discussion centric, like Jovicentric for Jupiter, but information technology is sufficient to say Jupiter orbit to imply a Jovicentric orbit. The most common centric orbit is geocentric or Earth orbit. As of April 2020, nosotros accept two,666 operational satellites currently orbiting Earth (cheque out this very cool open-source database of all operational satellites around Earth!) [UCS]. Earth (geocentric) orbits are usually classified by their altitude and shape, with some specialty orbits included. The most common orbit types are:

Low-World Orbit (LEO) – <2000 km altitude
- Dominicus-synchronous orbit (SSO) is a blazon of LEO
Medium Earth Orbit (MEO) – 2000 km<MEO<35,786km distance
- Besides known equally Intermediate Circular Orbit (ICO)
- Includes GPS satellites at 20,200 km altitude
Geosynchronous (GSO) or Geostationary Orbit (GEO) – 35,786 km altitude
Highly Elliptical Orbit (HEO)
- Sometimes HEO refers to High World Orbit beyond GEO

Examples of Typical Globe Orbits. Effigy courtesy of HTSI.

Low Globe Orbits

Depression World Orbit (LEO) is the simplest (cheapest) orbit to attain and is the nigh extensively used. Over 90% of artificial objects orbiting the World are in the LEO "corridor" – an surface area bounded on the depression-finish past atmospheric drag factors (at about 200 km altitude) and at the high-end by the lower van Allen radiation chugalug (at about 1,000 km to 2000 km altitude).

Sun-Synchronous Orbits

Image courtesy of Trevor Sorensen with HSFL.

A Lord's day-synchronous orbit (SSO) is a type of polar LEO that exploits the regression of the ascending node caused past the Globe'due south equatorial oblateness to "twist" the orbit at a rate of one revolution per year (0.9856 degrees per mean solar day). The result is that the orbit plane will always maintain the same angle (β) with respect to the Dominicus, and the satellite crosses the equator at the aforementioned local fourth dimension every orbit.

Since a satellite in a Sun-synchronous orbit always crosses the equator at the same time, it is common for their ascending (or descending) nodes to be measured in time rather than as an angle from the vernal equinox vector. The relationship can be seen in the following figure.

Annotation that the example in the figure is merely valid for the first 24-hour interval of spring when the vernal equinox happens to line up with the Earth-Sun line. For other times of the year, the time must exist corrected with the "Greenwich hour angle".

Definition of an SSO Using the Equator Crossing Time. Effigy courtesy HTSI.

SSO Dawn-Dusk Orbit. Effigy courtesy HTSI.

A dawn-sunset orbit is an SSO in which the ascending node occurs around local dawn (~06:00 h) and the descending node corresponds to local sunset (~18:00 h) on every orbit. If the equator crossings happen in the opposite order, the SSO is called a dusk-dawn orbit. These types of SSO straddle the terminator, the line betwixt night and day. The upshot is that the satellite is illuminated by the Sun almost continuously for most of the yr. This is advantageous for satellites with high power requirements.

Dawn-dusk orbits (or dusk-dawn orbits) still enter the Globe'southward shadow at certain times of the year because of the Lord's day's credible change in position relative to the Globe. These periods are chosen eclipse seasons.

For a dawn-dusk orbit, the most northern part of the orbit will exist in shadow when the Sunday appears "lowest" relative to the equator around the winter solstice. If the orbit is sunset-dawn, the eclipse season takes place around the summer solstice when the Dominicus appears "highest" relative to the equator.

Medium Earth Orbits

A Medium World Orbit (MEO), sometimes called an Intermediate Circular Orbit (ICO). These orbits lie betwixt the LEO (<2000 km) and GEO (35,786 km) altitudes, but they typically accept an distance of approximately ten,000 km and an orbital period of about half dozen hours. This orbit is popular for communications satellite constellations. MEO satellite constellations require fewer satellites than an LEO constellation to reach global coverage but at the expense of increased signal time delay (fifty-150 msecs circular trip delay equally opposed to simply twenty-40 msecs at LEO).

Global Positioning Organization (GPS)

The Global Positioning Arrangement (GPS) constellation is besides considered an MEO constellation just is actually in a 12-hour semi-synchronous orbit at a much higher distance – twenty,200 km than most of the MEO satellite constellations. 24 GPS satellites reside in half dozen orbital planes at 55o inclination, four satellites per aeroplane. There is petty atmospheric drag at 20,200 km, so the GPS orbits are quite stable. The master perturbation forces are the Earth'due south equatorial oblateness, solar radiations pressure, and third-body gravitational effects of the Dominicus and the Moon.

GPS Constellation – Courtesy of US Authorities.

GLONASS Navigation Constellation

GLONASS, or "Global Navigation Satellite Organization", is a Russian space-based satellite navigation system operating as office of a radio navigation satellite service. It provides an alternative to GPS and is the second navigational organization in operation with global coverage and of comparable precision. It provides better coverage at high latitudes than GPS because this area is of more importance to Russian federation. The GLONASS constellation consists of 24 satellites in three orbital planes (8 satellites per plane) at an altitude of xix,100 km and an orbital inclination of 64.viii deg.

https://en.wikipedia.org/wiki/GLONASS

Galileo Navigation Constellation

Galileo is a global navigation satellite system (GNSS) that went live in 2016, created past the European Marriage through the European Space Bureau (ESA), operated by the European GNSS Agency (GSA). One of the aims of Galileo is to provide an independent high-precision positioning system then European nations do not have to rely on the Us GPS, or the Russian GLONASS systems, which could exist disabled or degraded past their operators at any time. The Galileo GNSS consists nominally of 24 satellites in 3 orbital planes (eight satellites per airplane) at an altitude of 23,222 km and an orbital inclination of 56.0 deg.

https://en.wikipedia.org/wiki/Galileo_(satellite_navigation)

Constellation visibility from a indicate on the World

Geosynchronous Orbits

A Geosynchronous Orbit (GSO) is whatever Earth orbit, regardless of inclination or eccentricity, that has a menstruation of 24 hours. A Geostationary Orbit (GEO) is an important GSO with near-nada inclination and eccentricity, making this GSO both equatorial and circular. At an altitude of approximately 35,700 km, a satellite stationed in GEO goes about the equator at the aforementioned rate that the Earth is rotating. Therefore, it appears "stationary" as viewed from the Globe.

Although not strictly correct, both of these terms – geosynchronous and geostationary – are often used interchangeably to mean geostationary orbit. British science fiction author Arthur C. Clarke first proposed that this orbit could be used past communications satellites in a Wireless World commodity published in 1945. This is why GEO is sometimes called the Clarke Orbit.

Biography of Arthur C. Clarke

Highly Elliptical Orbits

A Highly Elliptical Orbit (HEO) is one that has a very high eccentricity, with an apogee altitude much greater than its perigee altitude. The nigh common types of HEO are the Molniya and Tundra orbits.

Molniya Orbit

This orbit was devised by Russian federation as a way of stationing its communications satellites to provide better coverage of the northern latitudes of the country (compared to the coverage that could be provided by equatorial geostationary orbits. Molniya orbits have a high eccentricity (eastward=0.75) and a stock-still apogee in the northern hemisphere, where the satellite remains for most eleven of the 12 hours of its orbital flow. Fixing the apogee is done by having the orbit at one of the critical inclinations of 63.4o or 116.6o. At these inclinations, the rate of alter in the argument of perigee due to the Earth's oblateness is zero. The time distribution of the orbit results from Kepler's Second Law or the conservation of athwart momentum. For a Molniya orbit, apogee = 39,354 km altitude and perigee = i,000 km.

Highly Elliptical Orbit (HEO) Figure courtesy HTSI.

Tundra Orbit

A Tundra orbit – with a period of 24 hours – is both highly elliptical and geosynchronous. Similar the Molniya orbit, the Tundra is tilted at ane of the critical inclinations so its apogee is stock-still in the northern hemisphere. Withal, the Tundra orbit is at a much higher altitude and has a menses of 24 hours instead of 12 hours. A telecommunications arrangement based on Tundra orbits would require only two satellites in two orbital planes whose ascending nodes are 180o apart. In contrast, a Molniya organisation needs three satellites in iii orbital planes with ascending nodes 120o apart. For a Tundra orbit, apogee = 53,622 km altitude and perigee = 17,951 km.

Recurrent and Sub-Recurrent Orbits

In a recurrent orbit, the footing track of the satellite repeats within 24 hours. A sub-recurrent orbit is ane in which the ground track repeats after a catamenia of time greater than 24 hours. Such orbits are necessary for Globe-observing satellites so that they can revisit ground targets in a regular and anticipated manner. They are accomplished by selecting a semimajor axis (altitude) with a flow that produces an integer number of revolutions per integer number of days. This distance is called the nominal semimajor axis. The number of days or the number of orbits between repetitions is chosen the repeat bike. For case, the sub-recurrent orbit of Radarsat has a repeat bike of 343 revolutions in 24 days. That is, if the satellite is over point X at a detail time, it volition once again be over bespeak X 24 days later.

Footing Tracks

The footing rail is the path that a satellite traces over the Earth's surface as it orbits. Information technology appears to shift westward during successive orbits due to the eastward rotation of the Earth. For an LEO satellite, the ground track has the appearance of a series of sine waves when displayed on a flat Mercator projection map. The highest North-South latitude circuit of the ground runway is equal to the inclination of the orbit. This website allows users to select a satellite from a list and encounter the Globe every bit viewed from that satellite: http://www.fourmilab.ch/earthview/satellite.html

Map in Mission Control, Houston Courtesy of NASA.

Basis Track Problem

The higher up plot shows the ground rails of a satellite in a circular posigrade orbit over a fourth dimension period. Assume the World rotates at 15°/60 minutes. Ignore J2 effects. Determine the post-obit characteristics of the orbit: inclination (i), period (P), semi-major axis (a), altitude (z), satellite'southward orbital speed ( $v_c$ ), angular momentum (h), and minimum speed necessary for it to escape from the World ( $v_{esc}$ ).

Orbital Maneuvers

Ofttimes a spacecraft needs to change its orbit/trajectory based on the requirements of its mission. Sometimes it might need to transfer into a dissimilar orbit for a diverseness of reasons, or it may need to change its delta-V in order to maintain its current altitude or position. This section looks at a variety of means to change orbits, starting with in-aeroplane orbit transfers.

In-Airplane Orbit Transfers

Apsidal Burns

Upshot of Non-Apsidal Burns in Plane

Posigrade Fire

A posigrade thrust at an arbitrary (non-apsidal) betoken in an elliptical orbit will rotate the line of apsides as well as increase the orbit size. This rotation will exist CW or CCW depending on whether burn occurs approaching apoapsis or periapsis. It moves the periapsis closer to the thrust indicate.

2. Retrograde Burn

A retrograde burn at a non-apsidal point in an elliptical orbit will rotate the line of apsides and decrease orbit size. Retrograde fire moves periapsis farther away from the thrust indicate with rotation of the line of apsides beingness CW or CCW depending on whether it was closer to apoapsis or periapsis.

Radial Burns

Inward

Moves S/C radially towards World, resulting in the negative, meaning the S/C is no longer in a circular orbit and is approaching periapsis, which will exist reached ninety deg ahead of burn down point. Apoapsis is raised by the same amount, so the menstruum stays the same.

ii. Outward

Circular orbit becomes elliptical with the positive, thus South/C is budgeted apoapsis, which occurs 90 deg ahead of thrust indicate. Orbit size and period remain constant.

Hohmann Transfers

In 1915 a German language engineer, Walter Hohmann, theorized a fuel-efficient style to transfer between orbits This method uses an elliptical transfer orbit tangent to both the initial and last orbits. Hohmann transfers are express to orbits in the aforementioned plane (coplanar), and circular orbits or those with their lines of apsides aligned (co-apsidal orbits). All ΔV burns during the transfer are tangent to the initial and final orbits, thus the velocity vector changes magnitude, but not management. Tangential burns (Flight Path Angle=0º) are the virtually of import aspects of Hohmann Transfers. Assume all ΔV burns are instantaneous.

From the free energy equation, ε =µ2a, when we add or subtract energy due to the ΔV burn down, we change the energy of the orbit and thus its size (semi-major axis, a). To move an south/c to a college orbit, nosotros have to increase a, so we must increase ε by increasing v. Conversely, to decrease a, nosotros must decrease ε.

Hohmann Transfer from Inner to Outer Orbit. Image by Microcosm Inc.

The geometry of Simple Aeroplane Change.

In a simple aeroplane change but the velocity vector'south management changes – its magnitude remains the same, i.eastward.,

$\mid{V_{initial}}\mid=\mid{V_{final}}\mid$

From the geometry in the figure higher up, the vectors form an isosceles triangle with the vertex angle being the plane change angle, $\theta$ .

Δ $V_{simple}=2v_{initial}sin\tfrac{\theta}{2}$

When θ = 60°, the isosceles triangle becomes equilateral, thus ΔV equals $v_{initial}$ (i.e., corporeality of energy to get into orbit in the first identify). Since ΔV increases with $v_{initial}$ , for elliptical orbits, the plane alter should exist washed equally close to apoapsis equally possible when v is minimum.

This combines a change in management with a change in magnitude (e.m., moving satellite from LEO with i = 28.five° to a GEO orbit with r = 42160 km and i = 0°. From geometry:

$\mid\Delta{V}_{simple}\mid+\mid\Delta{V}_{increase}>\mid\Delta{V}_{combined}\mid$

This means it is cheaper to do a combined ΔV burn than a uncomplicated plane change followed by a Hohmann transfer burn. Applying Law of Cosines:

$\Delta{V}_{combined=\sqrt{v^2_{initial+v^2_{final}-2v_{initial}v_{final}2cos\theta}$

It is cheaper to do airplane changes when $v_{initial}$ is to the lowest degree, i.e., about apoapsis.

Rendezvous

Special employ of orbit transfer is to enable ii spacecraft to rendezvous with one another. At that place are several different cases, depending on the initial orbits of the spacecraft.

Rendezvous Between Spacecraft in Coplanar Orbits

Coplanar Rendezvous using Hohmann Transfer.

For this blazon of rendezvous, the Interceptor and Target are in the same orbit. Applying ΔV in the direction to the Target will change the shape and size of the Interceptor'south orbit. Rendezvous is accomplished by the interceptor moving into a phasing orbit, which will return the interceptor to the same spot one orbit later in the time it takes the target to motility around to that same spot.

Case A – Target Ahead of Interceptor

Case B – Target Behind Interceptor

The angular distance the target must embrace to get to the rendezvous point is > 360°, thus the interceptor phasing orbit will be greater than the current orbit catamenia. To go into the phasing orbit, the interceptor must speed up, thus inbound a higher, slower orbit. This allows the target to catch up.

Lagrangian Libration Points

French scientist, Joseph Lagrange, discovered libration points in 1764 while attempting to solve the complex three-trunk trouble. A Lagrangian Libration Point (LLP) is an equilibrium point in the gravitational fields of ii major bodies where a small trunk will stay stationary with nil velocity and acceleration since the forces in the system balance each other out (rotating frame). The three co-axis points are chosen saddle points and are unstable The two orthogonal points are called trough points and are stable.

Sphere of Influence

All bodies in space exert a gravitational pull on other bodies that varies every bit the square of the distance betwixt the bodies based on Newton's Universal Law of Gravitation. Theoretically, a body's gravitational influence extends to infinity but practically, it's but constructive inside a certain volume of infinite called the Sphere of Influence (SOI). In the solar arrangement, the SOI depends on the planet'south mass and how close the planet is to the sun (the Sun'southward gravitational force overpowers the gravity of the closer planets).

The SOI of a planet is given by:

$R_{SOI}=r_{sp}(\tfrac{m_p}{m_s})^{0.4}$

$m_p$ = mass of planets

$m_s$ = mass of lord's day

$r_{sp}$ = distance betwixt planet and sunday (employ $a_p$ )