HABITAT PATH TIMES and KEPLER'S LAWS

Consider a possible path of a "cycler",

an asteroid transformed into a space habitat

with a convenient orbit.

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As an orbiting object, cycler must obey

Laws of Kepler

I. Solar orbit is an ellipse with the Sun at a focus.
II. Line from Sol sweeps equal areas in equal times.
III. Square of period relates to cube of semimajor axis.

 T² = a³

BACKGROUND: In his excellent book, Mining the Sky, Dr. John S. Lewis describes concept of cycler orbits with 2 year period.(pg. 115, "formerly a Near Earth Asteroid (NEA) whose orbit has been slightly modified to assure a metastable resonant relationship to Earth. Small amounts of propulsion will be expended from time to time to keep the orbit at resonance....Each cycler passes by Earth every two years...though transport to the Belt is the main function, it frequently passes near Mars...")

This chapter uses Kepler's Three Laws to describe any given cycler path.

RESONANCE If the asteroid's orbital period is an exact multiple of the Earth's orbit (i.e., integer number of years); then, asteroid will arrive at nearest position to Earth at same date every cycle. Consider such an asteroid with a period of exactly two years. As an example, let asteroid's closest approach to Earth happen on September 25, year X. For every two years thereafter (X+2, X+4, .....), that resonant asteroid would approach Earth at same position on September 25. Humanity could leverage that resonance by transforming asteroid into a habitable habitat and hitching a ride for the two year orbit from Earth to beyond orbit of Mars and return back to Earth.

Per Kepler's Third Law,  2 year Solar orbit has certain semimajor axis, a. a = ∛T2 = ∛(22) = 1.59 AU.

Kepler's First Law

The orbit of each Solar satellite is an ellipse with the Sun at one focus.

	Pythagorean Properties Semimajor axis (a), semiminor axis (b) and focus (c) have a Pythagorean relationship: a2 = b2 + c2 a and c define range of distance from Sol. Aphelion, Q, is orbit's furthest point. q = a + c =  (1.59 + 0.53) AU = 2.12 AU  Perihelion, q, is closest. q = a - c =  (1.59 - 0.53) AU = 1.06 AU

Focus and Eccentricity Since Kepler's first law places Sol at a focus, define focus as c, distance from center to Sun. Eccentricity, e, is the "flatness" of an ellipse. It is defined e = c ÷ a. Common usage often substitutes term, ae, for c. EXAMPLE: Arbitrarily pick e=1/3 which produces c = ae = 1.59 AU × .3333 = 0.53 AU c = 0.53 AU	

	Major/Minor Axii All features of an ellipse apply to an orbit,. Thus, major axis and minor axis intersect at the center. Orbit positions can be described by the Cartesian elliptical equation x2 a2 + y2 b2 = 1 Origin (0,0) of X,Y coordinates could be at ellipse center where axii intersect.

Auxiliary Circle Auxiliary Circle (AC) circumscribes the elliptical orbit. Semimajor axis, a, is also the AC radius. While orbit radius varies with angle from major axis. AC radius remains the same. Orbit and AC share two endpoints of the major axis, -a and +a. Next diagram shows further relationship: For every point, x, between -a and +a, there are two corresponding y points directly above. YE is on the elliptical orbit. YC is on the auxiliary circle.

Exactly midway between the major axis endpoints (-a and +a), there is an unique X point at orbit center, XCenter. Directly above XCenter are two Y points. 1) On orbit, YE is distance, b (semiminor axis). 2) On AC, YC is distance, a (semimajor axis). For every point x between -a and +a, ratio a:b applies to corresponding YC and YE.  Thus, a / b =  YC / YE Semilatus Rectum Inside Sol at an orbit focus, there is an unique X with corresponding YC and YE. As shown in diagram, Y point on circle (YC) is distance, b, and Y point on orbit (YE) is distance  ℓ, semilatus rectum. ℓ is formally defined as segment from Sol to orbit; perpendicular to major axis. a / b =  YC / YE=  b / ℓ EXAMPLE:  Let a = 1.59 AU and b = 1.499 AU. ℓ = b2/a. =(1.499AU)2/1.59AU = 1.41 AU

Kepler's Second Law

The line joining an object to the Sun sweeps out equal areas in equal times.

Positional Vector R = ℓ 1+ e×cos(ν) Angle Radius ν  R 0° 1.06 AU 45° 1.14 AU 90° 1.41 AU 135° 1.85 AU 180° 2.12 AU e = 1/3 ℓ=1.41 AU	Variable Vectors First, draw the line: Vector, R, connects Sol to the orbiting object. Positional vector, R, ranges from minimum at perihelion, q, to maximum at aphelion, Q. Variable range increases difficulty of computing segment areas.

Equal Areas Indicate Equal Times

For time, t1, let orbiting object proceed counter clock wise (CCW) from point q to next position as shown in diagram.  The associated positional vector would then sweep an area shown by the green sector, A1. Similarly, for time, t2,  positional vector would sweep out another sector,A2, of different shape but same size area. Thus, for same duration, asteroid travels greater orbital distances when closer to the Sun and a lesser distance when further away. Per Isaac Newton's work in gravity, Sun's closer satellites have greater speeds then those further away. Thus, if we can readily determine sector areas, then we can readily determine satellite flight times throughout the orbit.

True Anomaly Area True Anomaly, ν (Greek "nu"), is angle from X-axis to object's current position. It uses Sol as the apex; recall that Sol is an orbit focus and is not the orbit center. Compute Partial Areas AT∠ = triangular area from apex. ATΔ = edge area, remainder of ecliptic sector. ν R XR YR AT∠ ATΔ AT deg AU AU AU AU 2 AU 2 AU 2 45° 1.14 0.808 0.808 0.325 ? 0.325 + ? Given ℓ 1+e×Cos(ν) RCos(ν) RSin(ν) 0.5XRYR Unk. AT∠+AT∇ Thus, determining True Anomaly sector area depends on value of edge area which is unknown due to variable range of position vector.

To approximate True Anomaly area, AT, divide into: 1) Triangle on left, A∠, easily determined. 2) Edge segment on right, AΔ, not so easy (due to variable range).

Value of A_E is easily determined. As a circular sector with apex at circle's center, use following:
Sector area is same percentage of circle area as sector angle is percentage of circle (360°).
A_EASector  / A_Circle = ∠_EASector  / ∠_Circle 
A_EASector  / πa²  = 32.6° / 360°
A_EASector  = 3.14 × (1.59 AU)² ×  .09055 = .7188 AU²

Use trigonometric identities to convert ν to Eccentric Anomaly (E). For this particular orbit, ν converts from 45° to an E of 32.6°.

Eccentric Anomaly (E) is similar to the True Anomaly (ν). It is an angle (∠E ) from X-axis (reference ray),  and it depends on object's current position in elliptical orbit.  Unlike ν, E uses the orbit center as the apex.  Also, ∠E measures from ref. ray to a ray from orbit center to AC point directly above object (see YE in figure). Convert ν to E: Use trig identities. ν R YT YE E Deg AU AU AU Deg 45° 1.14 0.808 0.857 32.6° Given ℓ 1+e×Cos(ν) R × Sin(ν) YT × a/b Sin-1(YE/a) To determine YE on AC, use (a:b) proportionality, a / b = YE  / YT

Compute Eccentric Anomaly Areas ν EA AE AE∠ AEΔ Deg Deg AU 2 AU 2 AU 2 45° 32.6° 0.719 0.574 0.146 Given Sin-1 [ b a Sin(ν) (1+e×Cos(ν)) ] π a2 ∠E 360° a2 ×Sin(2∠E) 4 AE- AE∠ Like the ecliptic TA section, EA triangle area  (AE∠) can be easily calculated: AE∠ = 0.5 (x + c) YE = 0.573 AU2 Fortunately, EA edge segment area (AE∇) can also be easily calculated: AE∇ = AE - AE∠  AE∇ = (0.719 - 0.573) AU2 = 0.146 AU2

Subdivide EA Sector Like the ecliptic section of the True Anomaly (TA), Eccentric Anomaly's (EA's) circular sector can also divide into a triangle (AE∠) and an edge segment (AE∇) .

Determine Edge Area of True Anomaly

 COMPARE TWO EDGE AREAS Sadly, True Anomaly (TA) edge area (AΔ) is difficult to compute because TA is an offset elliptic sector. Gladly, Eccentric Anomaly (EA) edge area (A∇) is easily determined because EA is a circular sector.

Compute AΔ with  a : b Ratio Ratio of  semiminor axis (b) to semimajor axis (a) applies to several ellipse : circle  values. Ellipse Area Circle Area = a b π a a π = a b Orbit's Y coord  AC's Y coord  = R Sin(ν) a Sin(E) = b a b:a ratio also applies to corresponding edge areas: True Anomaly Edge Area Eccentric Anomaly Edge Area = b a  = AΔ A∇ Determine True Anomaly edge area: True Anomaly Edge Area (AΔ) = b a × A∇ Determine True Anomaly area AT = A∠ + AΔ AT = 0.33 AU2 + 0.14 AU2 AT = 0.47 AU2

 

Kepler's Third Law The square of the Solar orbit's period (time2) is proportional to the cube of its semimajor axis (distance3). Also known as the Harmonic Law, this is one of the most powerful statements of physical law in astronomy. NOTE: Above has been rephrased for simplicity. For example, Kepler said "average distance from Sun" vs. "semimajor axis". There's no intent to change Kepler's content.

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