Earth-Moon-S/C: Star A, B, Planet1 Earth, Moon & Spacecraft - Stars A, B & Planet 17 November 2006...

Earth-Moon-S/C: Star A, B, Planet 1

Earth, Moon & Spacecraft - Stars A, B & Planet

17 November 2006

Wes Kelly

Triton Systems, LLC www.stellar-j.com

[email protected](281) 286-3680 17000 El Camino Real – Suite 210AHouston, TX 77058

PART I: INTRODUCTION


Moon with Respect to Earth and SunEcliptic, Equatorial and Lunar Orbit Planes



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.


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.

As features of celestial mechanics Equilibrium or Libration Points Occupy 5 positions around 2 finite gravitating bodies Revolving about their Common Barycenter.

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.


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.


What’s 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


Alpha Centauri A&B: Nearest Binary Star System (e = 0.52)

Two Stars Orbit about BarycenterTerrestrial Planet traces about each star in Habitable Zone

Larger, brighter (blue) starTraces smaller ellipse

Inversely proportional to Mass ratio

RA/RB = MB/MA


-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0 20 40 60 80 100 120 140 160 180

time (years)

R/R

o

-200

-150

-100

-50

0

50

100

150

200

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Time (years)

R/R

o*10

00 &

p

Long

itude

(de

g)

Alpha Centauri A: 400, 350, 300 K lines

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Time (years)

R/R

o

-12.5

-10

-7.5

-5

-2.5

0

2.5

5

7.5

10

12.5

-17.5 -15 -12.5 -10 -7.5 -5 -2.5 0 2.5 5 7.5 10 12.5 15 17.5 20

Orbit Plane AUs

Orb

it P

lane

AU

s

Terrestrial Planet Orbiting Alpha Centauri AIn Alpha Centauri Stellar Binary

Every 78 years at Stellar Pericentron Passage, Terrestrial PlanetAt 1.246 AU Semi-Major Axis (400 K T-local)

Increases Eccentricity Measured by R/R0

Over 8000 Earth Years Planet Perihelion Rotates 360 degrees As eccentricity cycles between near 0 and 0.07

Planets Positioned Further from A experience faster cycles andHigher eccentricities until they become unstable.


Doppler Velocity Data Spectral Line Shifts of First Few Stars

Determined to Possess Large Planets ~1995

Astrometry DataSun Position w.r.t. System Barycenter

Scale: +/- 1 Milli-Arcsec Trace from 1960-2025

Large Circles Radius 4 Million kmTangential Velocity (for Doppler) ~65m/s

View from 10 Parsec Distance“North Pole of Ecliptic”

LGM “Evidence for Jupiter & Saturn”1 Parsec = 3600*180/ = 206,264.8 AUs

51 Pegasi: +/-60 m/s

47 Ursae Majoris: +40 -60 m/s

70 Virginis: +400 -220 m/s


By pocket calculator, Mars; the planetary SOIs involved more smoothly connect small, local “patched conic” regions. Lunar SOI is embedded within the Earth’s 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.

Lunar Sphere of Influence (SOI) Earth at Origin

X-Rel: Earth Moon Line (Earth Radii) +11.2, -12.6 E-Radii

Which is easier? Calculating trajectories to Mars or to the Moon?


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).


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.





Contours of Zero Velocity for =1/82Valid for: Earth-Moon, If Eccentricity Near or Equal 0.

G2 Star with Brown Dwarf or “ “Red Dwarf (M9) with Jupiter “ “

Development of late 1990s:

Extra-Solar Planets, Brown DwarfsFrequently Detected with High EccentricityWhere Triangular Libration Points Vanish

“Meta-Stable” L1 & L2 Become Less Stable Derivation of Lagrangian Equations Shows How


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 SequenceMass-Luminosity

Relation

L*/L0 =(M*/M0)3.5

HERTZSPRUNG – RUSSELL DIAGRAM

Solar EffectiveSurface Temperature

~5800 o K (G2)Surface Radius~700,000 km.

L= 4R2 Teff4

For T= f(R)

1 AU “Flux Temperature”400o K ( Teff @ R=1AU )

Simple Defining Line of Habitability Zone



Hierarchy of Modeling Methods for Lunar Applications---------------------------------------------------------------------------------------------------------------------------------------2-Body & Patched Conic model provides:

Estimates of times, geometries, delta velocitiesWorks 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 & BStability 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 ConicAssumptions do not address angular and translational accelerations of primary bodies in elliptic orbits------------------------------------------------------------------------------------------------------------------------------------Restricted Elliptic 3-Body model provides

Two principal bodies in inversely proportional to mass ellipses about barycenterRefined 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-Body (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.


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) involvingSub-stellar objects (Brown Dwarf or Jovian Planets) where Terrestrial PlanetsIn 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 studiesGravity model converted form solar units to planetary units & “second” time steps.

Future Direction:- Re-integrate finite burns and entry atmospheric drag models.Typical issue: In elliptic binary system, what are suitable navigation coordinates?

-Additional perturbing bodies (e.g., Ballistic Captures with Solar assistance such as Japanese Hiten mission)



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 Moon’s angular rate as well as distance changes – andIts gravity distorts Earth-based conic paths.

Shoot for the Moon with Naïve AssumptionsParabolic / Escape Trajectory along Initial Line of Nodes.

Moon Moves 26 degrees in 2 DaysConic Bends in Opposite Direction


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/)

r ( lunar orbit) = p / ( 1 + e cos f) ~ 59R0

--------------------------------------------------------------------------------------------------------------------cos (f ) = [( 1 + e)( rp / r) – 1] / e

Offset angle : = 180 – f -------------------------------------------------

parabola : ~ 15.4 degrees hyperbolas : >15.4 apogee of an ellipse : 0 larger ellipses : 15.4 > > 0

--------------------------------------------------------------------------------------------------------------------Two-Body Departure Angle for Parabolic Trajectory to Moon:

Transit ( 2-days): 26.66 o

Offset: 15.42 o -----------------------------------------Departure Angle: 41.08 o


Restricted Elliptic 3-Body Simulation ResultsClose Lunar Flyby Results form 45o Departure Angle


Earth-Moon Parameters Departure Angle 45 Degrees, Escape VelocityInertial 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.)


Lunar Flyby: Continuously Calculating Orbital ElementsWatch Eccentricity Shift from Parabolic to Elliptic Value for Return Leg

But what does this mean for return to Earth?

<<Transformation 2 Days Out


Nice shot, but…

Departure at V-Escape with 45o Departure AngleReturn 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


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)


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 BarycenterRe-Entry between Red Plots at Far Left

Retrograde Lunar Orbit Departure at -31 Degrees


Revised V for Lunar Perigee Exit

Anticipate Return Trajectory Partials for Orbit PhaseOperationally, residual entry errors also corrected

With mid-course maneuvers

Circle: Earth Surface


Earth Injection with 2821.5 fps Velocity from Low Lunar OrbitAltitude, True Anomaly and Flight Path Angle w.r.t. Earth


“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 atDeparture Angles of 60 and 70o Fail to escape and crash


Barycentric Inertial Coordinates:Moon and Escape Trajectory V= 7549 fps or V = 2211 fps

Initiated at 60 o Departure Angle (Prograde Orbit)


Stability of Orbits and Lagrangian PointsStability


Lurking Instability…


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.

As features of celestial mechanics Equilibrium or Libration Points

Occupy 5 positions around 2 finite gravitating bodies

Revolving about their Common Barycenter.

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.


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.


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.



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.]


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.


StabilityThe 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).



Y AxisAccelerations(ft/sec**2)

Acceleration Level at X distance for Moon atPerigeeApogeeMean

Mean, Perigee & Apogee Equilibrium Points from Eq. 12a






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 (fps): -67.5 0 -65.0 0 -65.0 10-67.0 -10


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 aroundLagrange 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.


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.

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.


THIS IS THE END… For Now

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?


Celestial Mechanics References


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

http://exoplanet.eu/

http://www.obspm.fr/~schneider

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Earth-Moon-S/C: Star A, B, Planet1 Earth, Moon & Spacecraft - Stars A, B & Planet 17 November 2006...

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