sp 110 orbits. orbits basic requirements orbital motion in the traditional sense requires two...

SP 110SP 110

OrbitsOrbits

OrbitsOrbits

Basic requirementsBasic requirements

Orbital motion in the traditional sense requires two Orbital motion in the traditional sense requires two componentscomponents

1. Attractive force1. Attractive force

2. Relative motion away from the common axis2. Relative motion away from the common axis

Energy is easiest to use for the description of orbitsEnergy is easiest to use for the description of orbits

1. Attractive force – potential energy1. Attractive force – potential energy

2. Relative motion – kinetic energy2. Relative motion – kinetic energy

OrbitsOrbits

Gravity is the attractive force for objects in orbit in Gravity is the attractive force for objects in orbit in space, although other attractive forces allow space, although other attractive forces allow orbits such as electrically charged particles with orbits such as electrically charged particles with relative motion that can orbit. These include relative motion that can orbit. These include subatomic particles, atoms and molecules, and subatomic particles, atoms and molecules, and larger particleslarger particles

Gravitational force – use gravitational potential Gravitational force – use gravitational potential energyenergy

EEpp = -GMm/r = -GMm/r G = gravitational constant, M and m are the G = gravitational constant, M and m are the two masses, and r is the separationtwo masses, and r is the separation

Relative motion – use kinetic energyRelative motion – use kinetic energyEEkk = 1/2mV = 1/2mV22 m = one of the masses, V is the relative m = one of the masses, V is the relative

motionmotion

OrbitsOrbits

The sum of the kinetic and potential energies is also The sum of the kinetic and potential energies is also importantimportant

EEtotaltotal = E = Epp + E + Ekk = -GMm/r + ½ mV = -GMm/r + ½ mV22

If EIf Etotaltotal is negative, the negative component is negative, the negative component (gravitational potential an attractive force) is (gravitational potential an attractive force) is greater than the kinetic energy (relative motion), greater than the kinetic energy (relative motion), and the two objects can be in orbitand the two objects can be in orbit

If EIf Etotaltotal is positive, the kinetic energy (relative velocity) is positive, the kinetic energy (relative velocity) component is larger than the potential energy component is larger than the potential energy component (attractive force), and the two objects component (attractive force), and the two objects are not bound and there is no repeated orbitare not bound and there is no repeated orbit

OrbitsOrbits

Two restrictions on these orbits needed Two restrictions on these orbits needed to keep the discussion simpleto keep the discussion simple

1. Two-body orbits only1. Two-body orbits only

2. The off-axis relative motion must be 2. The off-axis relative motion must be sufficient to keep the two bodies from sufficient to keep the two bodies from collidingcolliding

Orbits - KeplerOrbits - Kepler

Johannes Kepler (1570-1630) first described planetary Johannes Kepler (1570-1630) first described planetary orbits with three lawsorbits with three laws

1. The planets orbit the Sun in elliptical orbits with the Sun at one focus

2. The planets sweep out equal areas in these ellipses in equal times (conservation of angular motion)

3. The period of orbit squared equals the semimajor axis (average separation) cubed

P2 = a3 P = period in Earth-years, a = semimajor axis in Astronomical Units


Implications of the three Keplerain laws

1. The planets orbit the Sun in elliptical orbits with the Sun at one focus

Each body orbits about the center of mass, but each body follows an elliptical path around the other, with the other at one focus

2. The planets sweep out equal areas in equal times Angular momentum is conserved Total orbital energy is constant In the elliptical orbit, decreasing separation corresponds to

decreasing potential energy (less negative), and therefore faster orbital speeds (increasing kinetic energy to maintain the constant), and vice-versa for greater separation

3. The period of orbit squared equals the semimajor axis cubed Orbital velocity decreases with increasing separation


Keplerain orbitsKeplerain orbits PP22 = a = a33

Although the assumption Although the assumption here has been that there here has been that there are only two bodies are only two bodies involved in orbits, involved in orbits, orbital motion described orbital motion described as “Keplerian” can as “Keplerian” can involve many bodies in involve many bodies in orbit, including the orbit, including the planets solar systemplanets solar system

Kepler’s PKepler’s P22=a=a33 law is limited law is limited to objects in orbit to objects in orbit around the Sun (planets, around the Sun (planets, asteroids, comets, etc)asteroids, comets, etc)

Orbits - NewtonOrbits - Newton

Issac Newton (1643-1727) Issac Newton (1643-1727) developed the basic laws of developed the basic laws of motion and the theory of motion and the theory of gravity, and the relationship gravity, and the relationship between mass and forcebetween mass and force

Newton’s equation relating period Newton’s equation relating period to semimajor axis is to semimajor axis is generalized for any two-body generalized for any two-body orbit (satellites around Earth orbit (satellites around Earth or the Moon, stars around the or the Moon, stars around the galaxy centers, etc.) galaxy centers, etc.)

PP22 = a = a33 [4 [4ππ22/G(M+m)]/G(M+m)]

Orbits - NewtonOrbits - Newton

Newton’s period-semimajor axis equation can be used for Newton’s period-semimajor axis equation can be used for any gravitational orbit that approximates a two-body any gravitational orbit that approximates a two-body orbitorbit

Example:Example: Find the semimajor axis of a geosynchronous Find the semimajor axis of a geosynchronous orbitorbit

PP22 = a = a33 [4 [4ππ22/G(M+m)]/G(M+m)]

a = [pa = [p22G(M+m)/4G(M+m)/4ππ22]]1/31/3 p = 24 hours (86,400 sec), M p = 24 hours (86,400 sec), MEarthEarth

= 5.97x10= 5.97x102424 kg, G=6.67x10 kg, G=6.67x10-11-11NmNm22/kg/kg22, m is mass of , m is mass of satellite and is insignificant (compared to Earth)satellite and is insignificant (compared to Earth)

a = 3.54x10a = 3.54x1077m = 35,400 km (22,500 mi)m = 35,400 km (22,500 mi)

Orbit Definitions

Orbits - DefinitionsOrbits - Definitions

Basic ellipse

Longest axis = major axis

a = semi major axis = 1/2 major axis

Smallest axis = minor axis (perpendicular to major axis)

b = semi minor axis = 1/2 minor axis


EccentricityEccentricity – a measure of – a measure of the flatness of the orbitthe flatness of the orbit

e = e = [1- (b/a)[1- (b/a)22]]1/21/2

e varies from zero to 1 for a e varies from zero to 1 for a bound orbit and greater bound orbit and greater than 1 for an unbound orbit than 1 for an unbound orbit (e = 1 for a parabolic orbit)(e = 1 for a parabolic orbit)

e = 0 for a circular orbite = 0 for a circular orbit

e = 0.999999999 = flat e = 0.999999999 = flat (highly eccentric) orbit(highly eccentric) orbit


Inclination angleInclination angle – the angle between orbit plane – the angle between orbit plane and reference plane, or between the orbit and reference plane, or between the orbit plane and the equatorplane and the equator

i = 0 to 360i = 0 to 360oo (0 to 2 (0 to 2ππ))


InclinationInclination - the angular difference - the angular difference between the orbital plane and a between the orbital plane and a reference planereference plane

Nodes Nodes - the two intersecting - the two intersecting points between an orbit and a points between an orbit and a reference plane reference plane

Descending nodeDescending node - point of - point of intersection of object in orbit intersection of object in orbit that is descending through the that is descending through the reference planereference plane

Ascending nodeAscending node - point of - point of intersection of object in orbit intersection of object in orbit that is ascending through the that is ascending through the reference planereference plane

Line of nodesLine of nodes - the line - the line intersecting the two nodes, intersecting the two nodes, (also the intersection of the (also the intersection of the orbit plane and the reference orbit plane and the reference plane)plane)


PeriapsisPeriapsis (or (or periastronperiastron) ) is the closest approach is the closest approach distance for a 2-body distance for a 2-body orbit orbit

Perigee Perigee - closest - closest point in the orbit point in the orbit between the Earth and between the Earth and the object in orbitthe object in orbit

PerihelionPerihelion - closest - closest point in orbit of point in orbit of planets to the Sunplanets to the Sun


ApoapsisApoapsis (or (or apoastronapoastron) ) is the farthest point in is the farthest point in orbit between two bodies orbit between two bodies in a 2-body orbit in a 2-body orbit

ApogeeApogee - farthest point - farthest point in orbit between an in orbit between an object and the Earth object and the Earth

AphelionAphelion - farthest - farthest point in orbit between point in orbit between an object and the Sunan object and the Sun


EccentricityEccentricity – a measure of – a measure of the flatness of the orbitthe flatness of the orbit

e = e = [1- (b/a)[1- (b/a)22]]1/21/2

If a = b, the orbit is circular, If a = b, the orbit is circular, thus (b/a)thus (b/a)22 = 1, and = 1, and therefore e = therefore e = [1- (1)[1- (1)22]]1/2 1/2 = 0= 0

Orbit and Position Reference

Orbit ReferenceOrbit Reference

The most common type of orbit position reference is The most common type of orbit position reference is the geocentric equatorial reference coordinate the geocentric equatorial reference coordinate system that employs the primary reference plane system that employs the primary reference plane as the Earth’s equatoras the Earth’s equator

This is a Cartesian reference with the X and Y axis on This is a Cartesian reference with the X and Y axis on the equatorial plane, and the polar axis along the Z the equatorial plane, and the polar axis along the Z axisaxis

The X-axis is the primary axis pointing to the vernal The X-axis is the primary axis pointing to the vernal equinox – the position of the Sun at the definition equinox – the position of the Sun at the definition of spring (apparent passage through the plane of of spring (apparent passage through the plane of the ecliptic)the ecliptic)


To identify the position of a satellite in orbit, the To identify the position of a satellite in orbit, the geocentric equatorial coordinates are used as the geocentric equatorial coordinates are used as the positional reference system positional reference system

The orbit and orbit plane is used to identify the The orbit and orbit plane is used to identify the satellite position using a set of lengths and angles satellite position using a set of lengths and angles known as the orbital elementsknown as the orbital elements

Can also be done in Cartesian coordinatesCan also be done in Cartesian coordinates

A second coordinate system is used to identify an A second coordinate system is used to identify an observer’s position with respect to the geocentric observer’s position with respect to the geocentric equatorial referenceequatorial reference

Provides relative angles to satellite from the Provides relative angles to satellite from the observer’s positionobserver’s position


The seven orbital elements are: a = semimajor axis

i = inclination angle

e = eccentricity

Ω = Right Ascension (or longitude) of the ascending node - angle between X-axis (vernal equinox) and the ascending node

ω = Argument (or longitude) of the perigee - angle between the ascending node and the perigee

ν = True anomaly - angle between perigee and position of orbiting object

T = Time since perigee passage

Orbital Energy

Orbital EnergyOrbital Energy

Changing an orbit requires energyChanging an orbit requires energy

An orbit has a fixed semimajor axis based on the sum of the An orbit has a fixed semimajor axis based on the sum of the kinetic and potential energieskinetic and potential energies

Changing the semimajor axis changes the average potential Changing the semimajor axis changes the average potential energy and the constantenergy and the constant

Therefore, it takes energy (thrust) to boost to a higher or lower Therefore, it takes energy (thrust) to boost to a higher or lower orbitorbit

Higher orbit requires added energy (forward thrust = Higher orbit requires added energy (forward thrust = increased speed)increased speed)

Lower orbit requires lower energy (retrograde thrust = Lower orbit requires lower energy (retrograde thrust = decreased speed)decreased speed)

Orbital EnergyOrbital Energy

Orbit Circular EllipticalEscape /Parabolic

Hyperbolic

Condition Bound Bound Neither Unbound

Total Energy (Ek + Ep)

Negative (Ek < Ep)

Negative (Ek < Ep)

Zero (Ek = Ep)

Positive (Ek > Ep)

Eccentricity e = 0 0≤ e <1 e = 1 e > 1

Based on the total energy, there are three types of orbits - bound, unbound , and escape (neither bound nor unbound)

•Bound Orbits - Elliptical orbits - potential greater than kinetic energy

•Unbound Orbits - Hyperbolic orbits - potential less than kinetic

•Escape orbits - Parabolic orbits - potential equal to kinetic

Orbit Types

Orbit TypesOrbit Types

The various types of orbits can describe the orbital energy and the orbital shape (eccentricity mostly), or by reference orbit orientation, orbital period or planetary surface coverage for the orbiting satellites

Communication, remote sensing, and surveillance all require specific orientation throughout the satellite operation. A number of the types and uses for these orbits are given below.

1. Bound (elliptical) orbitA bound orbit, which is also an elliptical orbit, has relative kinetic energy less than the combined gravitational potential energy

2. Unbound (hyperbolic) orbitA hyperbolic orbit is unbound, meaning that the relative kinetic energy is greater than the combined gravitational energy

3. Escape (parabolic) orbitA parabolic orbit is neither bound nor unbound since the relative kinetic energy is exactly equal to the combined gravitational potential energy. The parabolic orbit conditions are the same as escape velocity.


4. Prograde orbit A prograde orbit has an inclination angle less than 900 which follows the same direction as the Earth's or the orbited planet’s rotation

5. Retrograde orbitA retrograde orbit has an inclination greater than 900 which travels in reverse direction to Earth's rotation

6. Polar orbitA polar orbit has an inclination of 900 which allows world-wide coverage over a period of hours to days depending on altitude. This is an orbit commonly used for meteorological and surveillance satellites


7. Geosynchronous orbit A geosynchronous orbit has an orbital period equal to the Earth's rotation period of 24 hours (23h56m4.09s). The semimajor axis of this orbit is 42,164 km. The inclination for this orbit is often but not required to be 0o

8. Geostationary orbitA geostationary orbit is also geosynchronous, but has an equatorial orbit (i = 00), with an eccentricity of zero. This provides a fixed communications platform with respect to the Earth, and is used for communications, remote sensing (whether satellites, for example.), and surveillance.


9. Sun-synchronous orbitA sun-synchronous satellite orbit maintains a constant orientation between the Sun and Earth which is useful for several applications. The most common are the remote sensing applications satellites (Landsat, for example) and astronomical observation satellites (IRAS, for example)

These applications require either full back-illumination, or complete shadowing from the Sun. To accomplish this, a nearly polar orbit is used. If the orbit were polar, the orientation of the orbit would be fixed with respect to the Sun, but not the Earth

This would show a 0.98o per day change in orientation as the Earth does an make the Sun-satellite orientation seasonal, as the Earth is. Hence, the spacecraft needs a -0.98o per day retrograde motion in the orbital plane to counter the Earth's orbital motion around the Sun.

To do this, an orbital retrograde rotation can be made by using the oblateness of the Earth to place a torque on the orbit and produce a precession of the nodes (rotation of the orbital plane)

A range of altitudes and corresponding inclinations are available for this type of orbit. Landsat satellites use a = 709 km and I = 98.2o


10. Molniya orbit Russia has traditionally used communication satellite orbits that are highly-inclined with high eccentricity for their high-latitude ground stations with the extended orbit (apogee) over the higher latitudes

The Molniya's 12 hour orbit period also allows for communications between the Asian and North American continents, since it is one-half of the sidereal day (43,082 sec)

The inclination for this orbit is 63.4o, with a semimajor axis of 26,562 km, and with varying apogee and perigee values that satisfy the desired period and semimajor axis

11. Tundra orbitA tundra orbit is an eccentric, high-inclination (63o) orbit similar to the Molniya orbit but with a period twice as long (one sidereal day)

Like the Molniya orbit, this tundra orbit is used for communications at latitudes far from the equator


12. Parking orbitA parking orbit is a temporary orbit commonly used for spacecraft checkout operations before departure from Earth

13. Graveyard orbitA graveyard orbit is a permanent, higher-than-normal orbit used to remove defective or aging spacecraft from the busy geostationary region


14. Walking orbit A walking orbit gets its name from the rotation or precessional motion of the orbit due to the asymmetrical shape of the planet. An example of this is the precession of the sun-synchronous orbit due to the Earth's oblate shape (larger equatorial diameter than polar diameter because of its rapid rotation).

15. Halo orbit A halo orbit is found not around a celestial object but around either the Lagrange L1 or L2 stability regions (objects within or close to the L1 and L2 regions are not in a stable orbit)

Orbit Transfers

Transfer OrbitsTransfer Orbits

To get from one planet to another, a spacecraft must follow an elliptical, heliocentric orbit for most of its flight path

As the spacecraft departs the planet of origin, the target planet must be in correct position so that spacecraft arrival is coincident with the target planet

This simple requirement allows planet-to-planet flights only at specific times, and similarly, any return flights only at times of correct alignment of the two planets


Basic orbit transfers

1. Hohmann Transfer - the most energy efficient coplanar orbit transfer has an arrival point 180o from the departure point

2. Spiral (low thrust) - this is a long-period spiral orbit from launch to arrival that is best suited for high efficiency, low thrust propulsion systems (ion engines are a good example)

3. Direct (high thrust) - the direct, short-period transfer requires greater thrust to accelerate the spacecraft quickly to desired orbit at the expense of greater transfer energy and a larger propulsion system

4. Gravity assist augmented propulsion - the gravitational attraction of a spacecraft in a close encounter with a large planet can be used to boost the kinetic energy of the spacecraft with reference to the Sun

This technique can also be used to slow spacecraft to get to a closer heliocentric orbit


Traditional orbit transfer definitions

Type I (direct) transfer - transfer angle is less than the Hohmann transfer ellipse (<180o)

This is also known as a direct transfer orbit

Hohmann transfer - transfer angle of 180o

Type II transfer - greater than 180o and less than 360o (spiral)

Type III transfer - greater than 360o and less than 540o

(spiral)

Type IV transfer - greater than 540o and less than 720o

(spiral)


A sketch of the orbital paths of the three common types of transfer orbits


Hohmann transfer for interplanetary missions

The simple expression of a Hohmann transfer can be calculated in Earth years since the transfer orbit (trajectory cruise) occurs within the Sun's gravitational influence

The ellipse representing the transfer will have the perihelion at the inner planet orbit and the aphelion at the larger orbit planet

This technique also is used for Earth-orbit satellite transfers from a low-Earth orbit to higher orbits, an example being a boost from low-Earth parking orbit to a more distant geostationary orbit

Note that the transfer ellipse touches both the larger and smaller circular orbits. That means that the semimajor axis of the transfer orbit is one-half of the two orbits added together


Calculations for the Hohmann transfer to the planets

atransfer = ½ (aA + aB)

A period calculation uses the same third law of Kepler, but only one-half since the second half represents a return path to the departure planet

ptransfer = ½ (atransfer3/2)

Start with the semimajor axis values for the planets

aEarth = 1.00 aVenus = 0.72 AU aMars = 1.52 AUaJupiter = 5.20AU aSaturn = 9.56 AU aUranus = 19.22 AU aNeptune = 30.11 AU aPluto = 39.55 AU


Earth-Venus pHohmann = ½ [(aHohmann )3/2 ] = ½ [(aA + aB)/2]3/2

= ½ [(1 AU + 0.72 AU)/2]3/2 = 0.40 yr

Earth-Mars PHohmann = ½ [(1 AU + 1.52 AU)/2]3/2 = 0.72 yr

Earth-Jupiter PHohmann = ½ [(1 AU + 5.20 AU)/2]3/2 = 2.73 yr

Earth-Saturn PHohmann = ½ [(1 AU + 9.56 AU)/2]3/2 = 6.07 yr

Earth-Uranus PHohmann = ½ [(1 AU + 19.22 AU)/2]3/2 = 16.1 yr

Earth-Neptune PHohmann = ½ [(1 AU + 30.11 AU)/2]3/2 = 30.7 yr

Earth-Pluto PHohmann = ½ [(1 AU + 39.55 AU)/2]3/2 = 45.7 yr

Gravity Assist

Gravity AssistGravity Assist

Gravity assist

Augmenting spacecraft propulsion with a gravity-assisted boost is a technique that exchanges momentum between an orbiting planet and a spacecraft during a close flyby of a moving planet

The gravity assist propulsion boost can be used to either increase or decrease the spacecraft velocity relative to the Sun which can provide significant ΔV changes that may not be possible with conventional boosters

Missions to Mercury, for example, used or are using gravity assist to augment propulsion to overcome the extremely high gravitational pull from the Sun to reach the planet


Missions beyond Jupiter are also not possible without gravity assist since boosters are not available that are capable of taking interplanetary spacecraft beyond Jupiter, depending on the spacecraft mass

Gravity assist was first used on the Mariner 10 mission to Mercury, and has since been used for all missions to the Giant Planets, with the exception of the Pioneer 10 which targeted only Jupiter Although the mission targeted Jupiter in a close

approach trajectory, the resulting boost was simply a part of the observation objectives and not an intentional boost to reach a more distant planet


Even the Galileo spacecraft that was launched to Jupiter required a gravity assist because of its huge mass

A close flyby of the Earth and Venus were used in a sequence labeled VEEGA (Venus-Earth-Earth Gravity Assist) since the launcher and upper-stage booster were insufficient to get the spacecraft to Jupiter

The Voyager II spacecraft was boosted to Jupiter, then used gravity assists to get to Saturn, Uranus, and Neptune, then outside of the solar system


In graphic form, with three of the four Jovian planets' gravity added, the solar system gravitational potential looks like this, with colored arrows showing the energy to/from orbits


A plot of the escape velocity throughout the solar system can be A plot of the escape velocity throughout the solar system can be made by taking the square root of the absolute value of the made by taking the square root of the absolute value of the potential energy, as shown belowpotential energy, as shown below


Voyager II spacecraft was boosted to Jupiter, with gravity assists to get to Saturn, Uranus, and Neptune, then outside the solar system which is depicted below

Synodic Period and Interplanetary Launch

Opportunities

Launch OpportunitiesLaunch Opportunities

The period that the position of a planet is repeated in the sky is called the synodic period

The calculation of the synodic period is simple but has two forms. One is for orbits inside the Earth's orbit, the other for outside.

To calculate the synodic period of Venus, for example, and the launch opportunity cycle between Earth and Venus, use:

1/Psynodic = 1/PEarth - 1/PVenus

or Psynodic = 1/(1/PEarth - 1/PVenus ) = 1/(1 - 1/0.615) yr

= -1.60 yr = 584 days


Similar calculations produce the synodic periods for the other planets with respect to Earth. The same calculations can be made for relative position periods of any two planets

The practical application of the synodic period is that it provides the frequency of launch opportunities from one planet to another

Mercury 116 days Venus 584 days (19.2 mo) Mars 780 days (25.6 mo) Jupiter 399 days (13.1 mo) Saturn 378 days Uranus 370 days Neptune 367 days Pluto 367 days Moon 30 days


Launch opportunities from Earth to the planets are dictated primarily by the synodic period and the phase position between the two planets

Since the Earth is in a different orbital plane than all other planets, the alignment of the intersection of the planes (line of nodes) and the phase between the planets is an important consideration in launch timing

In addition, none of the planets including Earth are in circular orbits, making the optimum departure and arrival points a compromise between three primary variables. Those are:

1. Orbit phase between launch and arrival planets2. Line of nodes of the two planetary orbit planes3. Position in orbit of the two planets


Finding the minimum time or a minimum in transfer energy of an actual transfer is a compromise from the ideal orbits - circular and coplanar – using actual orbits

The actual planetary orbit variables generate a range of solutions that are plotted for ease in planning and interpretation

The plots are called pork chop plots because of their shape and have two general best solutions corresponding to the Type I and Type II transfers

From missions to Mars, the Type I would be optimal for manned missions because of their shortest transfer period

The Type II solution would be optimal for cargo or instrumentation missions since the energy is lower than the Type I solutions


On the right is a graph of the Type I and Type II flight limitations for the 2005 opportunity to Mars

The minimum energy is shown in the closed concentric shapes, with the lowest for Type II missions centered near 12 km2/s2 that corresponds to a flight duration of approximately 370 days


A plot of the future 2.1 year flight opportunities for Mars is shown on the right for Type II (cargo) missions.

Flight energy (C3 is total energy in km2/s2) is listed in the top series and time of flight shown on the bottom

Mission to Mars

Mission to MarsMission to Mars

A quick and dirty estimate of the flight time to Mars and back using 2-dimensional orbits begins with a simple Hohmann transfer calculation

Transfer orbit semimajor axis is simply the average of the Earth and Mars orbits, or (1+ 1.52)/2 AU = 1.26 AU

The corresponding transfer period is 1.263/2 years = 1.41 years for the complete orbit

The actual transfer period is one-half of this, or 0.71 years (8.5 months) each way


A Hohmann transfer requires that the target planet be at a position 180o from the point of spacecraft launch from Earth, therefore the launch can only take place during the correct alignment of Mars and Earth for the spacecraft to reach the target planet Mars which repeats with the 25.6 month (2.13 yr) synodic period


Phasing and synodic period

To calculate the approximate time of flight to Mars and back and the time needed for the two planets to realign correctly for the return trip, begin by finding the mean motion between the two planets in their orbit around the Sun in degrees/day, or similar units

The Earth's mean motion is simply 360o/year x 1 year/365.24days, or 0.986 deg/day

For Mars, this is 360 degrees/1.88 years x 1 year/365.24 days or 0.524 deg/day

Next, calculate the transfer period flight time to Mars using the circular orbit approximation. This was calculated earlier with a Hohmann transfer period of 0.71 years, or 259.3 days



The next step is to find how far Mars will travel in its orbit during the spacecraft transfer period from Earth to Mars. This is just Mars' mean motion times the transfer period, or 259.3 days x 0.524 deg/day = 135.87o

Mars must arrive at the 180o position during the transfer orbit; therefore, the spacecraft, which is launched from Earth at the 0o point, would be behind Mars by 180o - 135.87o, or 44.13o (alternatively, Mars would be 44.13o ahead of Earth)

This alignment can be approximated from the Earth's and Mars' orbital elements, being careful to not confuse the 2-dimensional exercise with the 3-dimensional orbital elements


Phase angles for the Earth-Mars-Earth interplanetary mission



To calculate the time needed for the phase difference for the two planets to move into a correct alignment for the return trip, use the difference in mean motion between the two planets

This relative motion is just the difference in the mean motion of the two planets, or 0.986 deg/day - 0.524 deg/day = 0.462 deg/day

During the spacecraft transfer that took it to Mars, the Earth has traveled 255.66o in its orbit. The difference in positions would therefore be 255.66o - 180o, or 75.66o, with the Earth ahead of Mars by this value



For the return trip, Mars must be ahead of the Earth by the same 75.66 degrees (the launch from Mars can be visualized at the 0o position with the Earth-arrival 180o from that launch position, which would begin -255.66o from that point, hence the angular difference between the Earth and Mars would be 255.66o -180o, or 75.66o)

For an estimate of the time required for the Earth to catch up to this position, the angle needed is just 360o minus the current lead of 75.66o minus the lead angle of 75.66o

This is a total of 208.68o (360o - 75.66o - 75.66o). At a closing rate of 0.462 deg/day, this makes the return realignment period = 208.68o/0.462o/day = 451.69 day = 1.24 years


The important number here is the turnaround or realignment period of 1.24 years. With a transfer period of 0.71 years (times two for return travel time), the total mission time would be 2.7 years for this 2-dimensional, circular orbit, Hohmann transfer approximation

Although the transfer orbit period can be decreased by increasing vehicle propulsion thrust, the realignment period is a critical element that can be as long or longer than the travel time to and from Mars


A nearly three year minimum mission period to Mars and return, of which nearly 1½ years is in zero-g transit, means that the critical human space flight exposure problems must be researched and solved before humans can reach Mars

If the total mission duration were reduced to a minimum with an immediate return - a turnaround time of zero duration using a high-thrust, Type I direct transfer orbit - the crews would encounter reduced space exposure, but could contribute little, if anything, to a Mars exploration mission

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