Earth, Moon

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Earth, Moon Spacecraft - Stars A, B
Planet 17 November 2006 Wes Kelly Triton
Systems, LLC www.stellar-j.com Desk1Triton_at_aol.c
om [phone] 17000 El Camino Real Suite
210A Houston, TX 77058 PART I INTRODUCTION
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Moon with Respect to Earth and Sun Ecliptic,
Equatorial and Lunar Orbit Planes
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What was missing from the previous illustrations
of the moon? A representation of the binary
nature of the Earth Moon System Revolution about
the System Barycenter.
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Crew Exploration Vehicle (CEV) operations to the
Moon elsewhere near-Earth could include
flight to the vicinity of Equilibrium Points
related to 1.) the Earth-Moon Binary
2.) the Earth/Moon-Sun Systems.
Also called Lagrangian Points, They can be
divided into two sets - 3 collinear points
between the 2 finite bodies L1, L2 L3 - 2
remaining equilateral points L4 L5
supplying the remaining vertices defining
equilateral triangles. The line between the 2
finite bodies forms their common base.
As features of celestial mechanics Equilibrium
or Libration Points Occupy 5 positions around 2
finite gravitating bodies Revolving about their
Common Barycenter.
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Applications for Co-Linear Points L1 Between
the 2 finite masses, nearer the smaller of the
two ( M2 if M1 M2). Sun-Earth L1 Ideal
for making observations of the Sun. Objects here
never shadowed by the Earth or Moon. The Solar
and Heliospheric Observatory (SOHO) is stationed
in a halo orbit around L1. Earth-Moon L1
Allows easy access to lunar and earth orbits
with minimal delta-v. Potential for a half-way
manned space station to transport cargo and
personnel to the Moon and back. L2 Lies on
the line defined by the two large masses, beyond
the smaller of the two. Moving out from Earth
away from the Sun, the orbital period of an
object would normally increase, but Earth's
gravity decreases orbital period, and locked at
the L2 point that orbital period becomes equal to
the Earth's. Sun-Earth L2 A good spot for
space-based observatories needing thermal
shielding ( e.g., Earth Shadow). The Wilkinson
Microwave Anisotropy Probe is already in orbit
around the Sun-Earth L2. The proposed James Webb
Space Telescope will be placed at the Sun-Earth
L2. Earth-Moon L2 A candidate location for a
communications satellite covering the Moon's far
side. Orbiting about the L2 point would allow
view of both back side and earth simultaneously.
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Whats still missing from the previous
slide? Representation of the ELLIPTIC Earth-Moon
System
Eccentricity influences the stability and
existence of Equilibrium Points, Escape
Capture Trajectories
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Alpha Centauri AB Nearest Binary Star System
(e 0.52)
Two Stars Orbit about Barycenter Terrestrial
Planet traces about each star in Habitable Zone
Larger, brighter (blue) star Traces smaller
ellipse Inversely proportional to Mass ratio
RA/RB MB/MA
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Every 78 years at Stellar Pericentron Passage,
Terrestrial Planet At 1.246 AU Semi-Major Axis
(400 K T-local) Increases Eccentricity Measured
by DR/R0
Terrestrial Planet Orbiting Alpha Centauri A In
Alpha Centauri Stellar Binary
Planets Positioned Further from A experience
faster cycles and Higher eccentricities until
they become unstable.
Over 8000 Earth Years Planet Perihelion Rotates
360 degrees As eccentricity cycles between near
0 and 0.07
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Astrometry Data Sun Position w.r.t. System
Barycenter Scale /- 1 Milli-Arcsec Trace from
1960-2025 Large Circles Radius 4 Million
km Tangential Velocity (for Doppler) 65m/s
Doppler Velocity Data Spectral Line Shifts of
First Few Stars Determined to Possess Large
Planets 1995
51 Pegasi /-60 m/s
47 Ursae Majoris 40 -60 m/s
View from 10 Parsec Distance North Pole of
Ecliptic LGM Evidence for Jupiter Saturn 1
Parsec 3600180/p 206,264.8 AUs
70 Virginis 400 -220 m/s
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Which is easier? Calculating trajectories to
Mars or to the Moon?
Lunar Sphere of Influence (SOI) Earth at Origin
X-Rel Earth Moon Line (Earth Radii)
11.2, -12.6 E-Radii
By pocket calculator, Mars the planetary SOIs
involved more smoothly connect small, local
patched conic regions. Lunar SOI is embedded
within the Earths so closely that border
calculations are significant efforts in
themselves. For Mars, V-infinity is half the
problem. CEV flights returning to the Moon imply
more complex missions than the Apollo series.
(e.g, Polar regions or the Lagrangian points of
Earth-Moon or Earth-Sun Systems). Yet even Apollo
flight plans require astrogation skills not used
during the Shuttle era. Apollo mission reviews
hold lessons for a new generation of flight
specialists. Example Determining launch angles
for trans-lunar or trans-Earth burns.
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While Earth-Moon system dynamics in the Apollo
era seemed unique, Not so since 1995, the Era of
Extra Solar Planets. New Star-Brown Dwarf or -
Jovian Planet binaries orbit in paths about
common centers of mass Like binary star
systems, newly discovered planets/binaries have
large eccentricities and Could influence smaller
unseen (e.g., terrestrial) planets Illustrative
cases provided based on integrations of the
Restricted Elliptic 3-Body Problem (RE3BP).
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Article in American Scientist ( Sept.-Oct.
2006, Vol. 94, No. 5) by Gregory P. Laughlin,
Astronomy Dept. UC, Santa Cruz. The Orion
Nebula, 1500 light years from Earth, a
well-known nursery for stars and the planets
that form around many of them. Optical images
increasing in magnitude show location of newborn
stars within the Orion constellation one of
many proto-stellar disks, at 17x Solar System
scale. Images from - European Southern
Observatory - Max Planck Institute - NASA Rice
University.
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Development of late 1990s Extra-Solar Planets,
Brown Dwarfs Frequently Detected with High
Eccentricity Where Triangular Libration Points
Vanish Meta-Stable L1 L2 Become Less Stable
Derivation of Lagrangian Equations Shows How
Contours of Zero Velocity for m1/82 Valid for
Earth-Moon, If Eccentricity Near or Equal
0. G2 Star with Brown Dwarf or Red
Dwarf (M9) with Jupiter
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Stellar Main Sequence Hydrogen Fusion into
Helium Energy Release Source of Luminosity
down to 0.08 Mass of Sun ( M0) Jupiter Mass
0.001 M0 or 330 Earths Brown Dwarf Deuterium
Fusion into Helium M 0.013M0 or M 13MJ
Main Sequence Mass-Luminosity Relation L/L0
(M/M0)3.5
HERTZSPRUNG RUSSELL DIAGRAM
Solar Effective Surface Temperature 5800 o K
(G2) Surface Radius 700,000 km. L 4s p R2
Teff4 For T f(R) 1 AU Flux Temperature 400o K
( Teff _at_ R1AU ) Simple Defining Line of
Habitability Zone
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Hierarchy of Modeling Methods for Lunar
Applications -------------------------------------
--------------------------------------------------
------------------------------------------------ 2
-Body Patched Conic model provides Estimates
of times, geometries, delta velocities Works
better with interplanetary trajectories owing to
sphere of influence sizes relative to free
trajectories. ------------------------------------
--------------------------------------------------
-------------------------------------------------
Restricted Circular 3-Body model
provides Uniformly rotating coordinate system
with two fixed bodies A B Stability analysis
based on integral solutions to equations of
motions zero velocity coefficient contours
Solution sets such as Lagrangian points,
special orbits Numerical integration involved in
some solutions vs. 2-Body Patched
Conic Assumptions do not address angular and
translational accelerations of primary bodies in
elliptic orbits ----------------------------------
--------------------------------------------------
------------------------------------------------ R
estricted Elliptic 3-Body model provides Two
principal bodies in inversely proportional to
mass ellipses about barycenter Refined solution
analysis of Lagrangian points (esp L1 L2),
system sensitivity to eccentricity Solutions
require numerical 3rd body integration amid
Kepler equation propagation ( position-time) of 2
primaries. ---------------------------------------
--------------------------------------------------
-----------------------------------------------
Beyond? -----------------------------------------
--------------------------------------------------
-------------------------------------------- 4-Bod
y (e.g., add Sun) or Solar System N-Body
Ephemeris provides Sun-Moon-Earth interactions
such as weak interactions and additional
Lagrangian points --------------------------------
--------------------------------------------------
--------------------------------------------------
--- -Collocation or Non-Linear Programming
Methods Address constraints difficult for
Calculus of Variations formulation. Caveat
Emptor Formulation is principally geometric vs.
physical.
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• Analysis Tool Origins Directions
• 1980s Bistar FORTRAN Code
• Derived from terrestrial orbital finite burn,
  re-entry and launch simulations
• 1985-1997
• Bistar used to examine terrestrial planet
  stability in habitable regions of
• Well known binary systems (Alpha Centauri,
  Procyon, Sirius)
• Gravity model shifted from terrestrial units and
  second times to solar units and day increments
• In inertial field, 2 bodies propagate on
  Keplerian orbits about barycenter
• Gravity forces attract third.
• 1995-1998
• Later studies include newly found binaries (70
  Virginis, 47 Ursae Majoris) involving
• Sub-stellar objects (Brown Dwarf or Jovian
  Planets) where Terrestrial Planets
• In habitable zones could be satellites or
  Perturbed by close-by massive companions
• ( Now over 200 such extra-solar planets).
• 2004 to Present
• Bistar Restricted Elliptic 3-Body Code adapted to
  lunar or planetary studies

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If the Moon were a target without mass, Then
2-body aim strategy reduces to time of arrival at
lunar orbit. For elliptic and parabolic
transfers of 116 and 48 hours And 27.32-day
Lunar period or 13.176o/day angular rate, LEO
departure angles advance from initial Earth-Moon
line by 63.6o and 26.35o. But A Hohmann
transfer delivers on node line All others (
higher energy ellipses and escape paths) have
off-sets. The Moons angular rate as well as
distance changes and Its gravity distorts
Earth-based conic paths.
Shoot for the Moon with Naïve Assumptions Paraboli
c / Escape Trajectory along Initial Line of
Nodes. Moon Moves 26 degrees in 2 Days Conic
Bends in Opposite Direction
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Two Body Offset Angle Calculation Solve for
True Anomaly by Combining Two Conic
Equations rp p / ( 1 e cos f) p /( 1 e
) R0 400 km ( where p is undefined by
parabola, substitute h2/m) r ( lunar orbit)
p / ( 1 e cos f) 59R0 ----------------------
--------------------------------------------------
-------------------------------------------- cos
(f ) ( 1 e)( rp / r) 1 / e Offset
angle y 180 f
-------------------------------------------
------ parabola 15.4 degrees
hyperbolas 15.4 apogee of an ellipse
0 larger ellipses
15.4 y 0 -------------------------------------
--------------------------------------------------
----------------------------- Two-Body Departure
Angle for Parabolic Trajectory to Moon
Transit ( 2-days) 26.66 o Offset 15.42 o
----------------------------------------- D
eparture Angle 41.08 o
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Restricted Elliptic 3-Body Simulation
Results Close Lunar Flyby Results form 45o
Departure Angle
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Earth-Moon Parameters Departure Angle 45
Degrees, Escape Velocity Inertial Velocity
(kfps) Distance (E-radii) to Moon
Inset vs. X-rel to Moon Center Time (minutes
reads to left) Radial Distance ( n.
mi.) Relative Velocity (fps) Y E-M Line Offset
Distance (n. mi.)
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Lunar Flyby Continuously Calculating Orbital
Elements Watch Eccentricity Shift from Parabolic
to Elliptic Value for Return Leg
But what does this mean for return to Earth?
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Nice shot, but Departure at V-Escape with 45o
Departure Angle Return Path Perigee
Insufficiently Low for Emergency Earth Return
Considerations for Lunar Trajectories
-Altitude of Passage, - Lunar Orbit coverage
- Time of Flight - Return Trajectory
Parameters Governed by - V departure (
Elliptic, Hyperbolic, Parabolic) - Departure
Angle - Inclination w.r.t. Earth-Moon
Orbital Plane
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Lower Altitude Return Trajectory
Obtained Control Adjustments are Velocity and
Departure Angle
Lunar Fly-Around Earth Departure Angle 47 o Vp
35,400 fps (vs. 35,580)
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Now what about getting back from the Moon
normally? Apollo 15 Departure Delta V 3000 fps.
Hyperbolic Exit Velocity
Blue Earth Track about Earth-Moon
Barycenter Re-Entry between Red Plots at Far Left
Retrograde Lunar Orbit Departure at -31 Degrees
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Revised DV for Lunar Perigee Exit Anticipate
Return Trajectory Partials for Orbit
Phase Operationally, residual entry errors also
corrected With mid-course maneuvers
Circle Earth Surface
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Earth Injection with 2821.5 fps DVelocity from
Low Lunar Orbit Altitude, True Anomaly and Flight
Path Angle w.r.t. Earth
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Houston, we have a problem Difficulties not
anticipated with 2-Body, Back of the Envelope
Calculations
Earth Departure on Escape Trajectory Bent Back
and Crashes
Lunar Elliptical Trajectories at Departure Angles
of 60 and 70o Fail to escape and crash
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Barycentric Inertial Coordinates Moon and Escape
Trajectory V 7549 fps or DV 2211
fps Initiated at 60 o Departure Angle (Prograde
Orbit)
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Stability of Orbits and Lagrangian
Points Stability
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Lurking Instability
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Crew Exploration Vehicle (CEV) operations to the
Moon elsewhere near-Earth could include
flight to the vicinity of Equilibrium Points
related to 1.) the Earth-Moon Binary
2.) the Earth/Moon-Sun Systems.
Also called Lagrangian Points, They can be
divided into two sets - 3 collinear points
between the 2 finite bodies L1, L2 L3 - 2
remaining equilateral points L4 L5
supplying the remaining vertices defining
equilateral triangles. The line between the 2
finite bodies forms their common base.
As features of celestial mechanics Equilibrium
or Libration Points Occupy 5 positions around 2
finite gravitating bodies Revolving about their
Common Barycenter.
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Wikipedia Definition The Lagrangian points,
(also L-point, or libration point) are the 5
positions in space where a small object can be
stationary with respect to two larger objects,
such as a satellite with respect to the Earth
and Moon). They are analagous to geosynchronous
orbits in that they allow an object to be in a
"fixed" position in space rather than an orbit in
which its relative position changes
continuously. More precisely, Langrangian
points are stationary solutions of the
Circular Restricted 3-Body Problem. Given
2 massive bodies in circular orbits around their
common center of mass, there exist 5 positions
in space where a 3rd body, of comparatively
negligible mass, could be placed which would
then maintain its position relative to the 2
massive bodies. As seen in a frame of reference
rotating with the same period as the 2
co-orbiting bodies, the gravitational fields of
2 massive bodies combined with the centrifugal
force are in balance at the Lagrangian points,
allowing the 3rd body to be stationary with
respect to the first 2 bodies.
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History and Concepts (Abridged and Adapted from
Wikipedia) In 1772, French mathematician Joseph
Louis Lagrange while working on the broad 3-body
problem obtained several interesting results.
Lagrange sought to calculate the gravitational
interaction between arbitrary numbers of bodies
in a system. Newtonian mechanics had concluded
that such systems result in bodies orbiting
chaotically until there is a collision, or a body
is thrown out of the system so that equilibrium
can be achieved. Including more than 2 bodies
in the system complicated mathematical
calculations considerably. Seeking to make
calculations simpler, Lagrange noted The
trajectory of an object is determined by finding
a path that minimizes the action over time. This
is found by subtracting the potential energy V
from the kinetic energy T L T - V (
and seeking zeros for systems of partial and
temporal derivatives related to position and
velocity WDK).
Lagrange thus re-formulated classical Newtonian
Mechanics into Lagrangian Mechanics.
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In the more general case of elliptical binaries,
there are no longer stationary points in the same
sense Points become more like Lagrangian
areas where the 3rd bodies make odd-shaped
orbits about the invisible Lagrangian points
these orbits are commonly referred to as halo
orbits. The Lagrangian points constructed at
each point in time as in the circular case form
stationary elliptical orbits which are similar to
the orbits of the massive bodies. This is due to
the fact that Newton's second law, p mv (p the
momentum, m the mass, and v the velocity),
remains invariant if force and position are
scaled by the same factor. Controversy A
body at a Lagrangian point orbits with the same
period as the 2 massive bodies in the circular
case, implying that it has the same ratio of
gravitational force to radial distance as they
do. This fact is independent of the circularity
of the orbits, and it implies that the elliptical
orbits traced by the Lagrangian points are
solutions of the equation of motion of the 3rd
body. Our analyses indicate that the last
statement does not apply to the co-linear points.
See our discussion.
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To the Rescue
Forest Ray Moulton 1872 - 1952 Member, then
Director, Dept. of Astronomy, Univ. of Chicago,
1898-1927. Research Associate at the Carnegie
Institution, 1908-1923. Director of Utilities
Power and Light Corporation, 1920-1938. AAAS
offices held Permanent Secretary, 1937 - 1946
Administrative Secretary, 1946 - 1948 Author
of Celestial Mechanics, 2nd Edition
(1914) Chapter 8, The Problem of 3 Bodies,
Articles 151-169 A Dover paperback.
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Stability The first three Lagrangian points are
technically stable only in the plane
perpendicular to the line between the two bodies.
This can be seen most easily by considering the
L1 point. A test mass displaced perpendicularly
from the central line would feel a force pulling
it back towards the equilibrium point. This is
because the lateral components of the two masses'
gravity would add to produce this force, whereas
the components along the axis between them would
balance out. However, if an object located at the
L1 point drifted closer to one of the masses, the
gravitational attraction it felt from that mass
would be greater, and it would be pulled closer.
(The pattern is very similar to that of tidal
forces.) Although the L1, L2, and L3 points are
nominally unstable, it turns out that it is
possible to find stable periodic orbits around
these points, at least in the restricted 3-body
problem. These perfectly periodic orbits,
referred to as "halo" orbits, do not exist in a
full n-body dynamical system such as the solar
system. However, quasi-periodic (i.e. bounded but
not precisely repeating) Lissajous orbits do
exist in the n-body system. These quasi-periodic
orbits are what all libration point missions to
date have used. Although they are not perfectly
stable, a relatively modest effort at
station-keeping can allow a spacecraft to stay in
a desired Lissajous orbit for an extended period
of time. At least in the case of Sun-Earth L1
missions, it is actually preferable to place the
spacecraft in a large amplitude (100,000 -
200,000 km) Lissajous orbit instead of sitting at
the libration point this keeps the spacecraft
off of the direct Sun-Earth line and thereby
reduces the impacts of solar interference on the
Earth-spacecraft communications links. Another
interesting property of collinear libration
points and their associated Lissajous orbits is
that they serve as "gateways" to control the
chaotic trajectories of the Interplanetary
Superhighway. By contrast, L4 and L5 are stable
equilibria (cf. attractor), provided the ratio of
the masses M1/M2 is 24.96. This is the case for
the Sun/Earth and Earth/Moon systems, though by a
smaller margin in the latter. When a body at
these points is perturbed, it moves away from the
point, but the Coriolis force then acts, and
bends the object's path into a stable, kidney
bean-shaped orbit around the point (as seen in
the rotating frame of reference).
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Mean, Perigee Apogee Equilibrium Points from
Eq. 12a
Y Axis Accelerations (ft/sec2)
Acceleration Level at X distance for Moon
at Perigee Apogee Mean
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Instability at Lagrangian Points Is it bad - or
a good thing? It depends on what you are selling.
Lunar Way Stations will requires orbit adjusts
every two weeks. But if you drop a hammer in the
right direction, you could send it off to Mars.
L2 Initial Condition Xdot0 Ydot0 Variations (fp
s) -67.5 0 -65.0 0 -65.0
10 -67.0 -10
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Web-Based Description of the The Interplanetary
Superhighway (IPSH) Denotes a set of transfer
orbits between planets and moons in the Solar
System. Based around orbital paths predicted by
chaos theory, leading to from the unstable
orbits around Lagrange points, these transfers
have particularly low delta-v requirements, Even
lower than common Hohmann transfer paths that
dominated past orbital trajectory analyses.
Although forces balance at these Lagrange
points, they are not stable equilibrium points.
If a spacecraft placed at L1 point is given
even a slight nudge towards the Moon, the Moon's
gravity will now be greater and the spacecraft
will be pulled away from the L1 point. The entire
system is in motion, so the spacecraft will not
actually hit the Moon, but will travel in a
winding path off into space. Semi-stable orbits
exist around each of these points. The orbits for
L4 and L5, are stable. But the orbits for L1
through L3 (i.e., near circular) are stable only
on the order of months.
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In the 1890s, Jules-Henri Poincaré first
noticed that paths leading to and from Lagrange
points would almost always settle, for a time, on
the orbit around it. An infinite number of paths
can take you to the point and back away from it,
and all of them require hardly any energy to
reach. When plotted, they form a tube with the
orbit around the point at one end (the IPSH). It
is very easy to transit from a path leading to
the point to one leading back out. Since the
orbit is unstable - it implies you'll eventually
end up on one of the outbound paths after
spending no energy at all. With careful
calculation you can pick which outbound path you
want. For example, for the low cost of getting
to the Earth-Sun L2 point, Spacecraft can travel
to a huge number of other interesting points,
almost for free.
These low energy transfers make travel to almost
any point in the solar system possible. On the
downside, these transfers are very slow, and only
useful for automated probes. They have already
been used to transfer spacecraft out of the
Earth-Sun L1 point, used in a number of recent
missions, including the Genesis mission. The
Solar and Heliospheric Observatory is also there.
The IPSH is also relevant to understanding solar
system dynamics Comet Shoemaker-Levy 9 followed
such a trajectory to collide with Jupiter.
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For every kilometer between Earth and the Moon
there is roughly one Astronomical Unit (AU)
between the Sun and the nearest Star System. For
centuries it was impossible to demonstrate the
validity of Copernican theory. Attempts to
discern angular measures of stellar shifts in the
heavens remained fruitless. Eventually
parallaxes to the stars yielded parsec
measures. At the beginning of the 19th century,
German philosopher Immanuel Kant asserted that
human kind would never determine the nature or
composition of the stars. A few decades later,
spectroscopy detected helium and hydrogen in the
sun and other stars. In the 20th century, using
the collapsing gas theory derived by Jeans, many
astronomers doubted that bodies smaller than the
faintest stars would naturally form in space.
If they did, they would not be seen or detected.
We now know of 200 of such objects. Some can
be seen and their motions are plotted. What
should we dare to predict?
THIS IS THE END For Now
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Celestial Mechanics References
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Astronomy or Astrophysical References on Stellar
Systems and Planet Formation
Quintana, E. V., Lissauer, J. J. , Duncan, M.
J., Terrestrial Planet Formation in the Centauri
System, Astrophysical Journal,Vol. 576, pp.
982-996, 10 September 2002. Butler, R.P.,
Wright, J.T., Marcy, G.W., et al.,Catalog of
Nearby Exoplanets, Astrophysical Journal, Vol.
646, pp. 505-522, 20 July 2006. Raymond, S. N.,
Mandell, A. M., Sigurdsson, S., Exotic Earths
Forming Habitable Worlds with Giant Planet
Migration, Science, Vol. 313, 08 September
2006. Jayawardhana, R., Ivanov, V. D.,
Discovery of a Young Planetary-Mass Binary,
Science, Vol, 313, 01 September 2006. Laughlin,
G. P., Extrasolar Planetary Systems, American
Scientist, Vol. 94, No. 5, pp. 420-429, September-
October 2006. Website http//exoplanet.eu/ The
Exoplanet Encyclopedia Jean Schneider  CNRS -
Paris Observatory