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Ground Tracks -1 AE 4310 Spaceflight Mechanics Ground Track

• Ground track: planetary surface overflight path • Consider a spherical planet – S/C orbit lies in plane passing through center of planet – Projection of this plane onto the surface is a great circle – For non-rotating planet, the s/c will retrace this same great circle each orbit r h i

2i !

• Inclination Ref: BMW – Higher inclination orbits result in larger amplitude ground tracks

max = i (for 0 i 90, posigrade)

max =180 i (for 90 i 180, retrograde) • Altitude – Altitude ↑ , Orbital period ↑, Longer ground track period – Orbital period in LEO ≈ 90 minutes, ~ 16 orbits/day

Ground Tracks -2 AE 4310 Spaceflight Mechanics Inclination Example

i A 10º B 30º C 50º D 85º

Ref: Hale

Ground Tracks -3 AE 4310 Spaceflight Mechanics Altitude Effects

• Altitude ↑, Orbital period ↑, Longer ground track period r r h ,h i 2 What’s wrong 1 2 with this 1 2 picture?

1

P1 P2

P2 > P1

• If ground track period known, can back out semi-major axis (a) 2 / 3 2"a3 / 2 $ Pµ1/ 2 ' P = # a = & ) µ1/ 2 % 2" (

• Orbital period in LEO ≈ 90 minutes, ~ 16 orbits/day !

Ground Tracks -4 AE 4310 Spaceflight Mechanics Rotating Planet

• Rotating planet effect – Orbital plane of s/c remains fixed in inertial space while planet rotates beneath it – Result, the ground track shifts westward (typically) over successive orbits

ψ (rad/time)

2 1

Δ • From successive ground tracks can determine orbital period P = (360º), where R = planet's rotational period R – LEO example ~ 1.5 hrs " = (360º) = 22.5º Is R > or < 24 hrs? ~ 24 hrs earth – Conversely, if Δ known, can solve for the orbital period P (and semi-major axis) • Instead of retracing previous ground track, a s/c eventually covers a swath around the ! planet between ± i latitude • a↑, Δ? Δ↑, NB: In some cases, Δ is sufficiently large (2.6DU < a < 4.2DU) that ground track appears to shift Eastward Ground Tracks -5 AE 4310 Spaceflight Mechanics Shifting Ground Track Example

Δ ≈ 23º P ≈ 1.53 hrs a ≈ 6750 km (1.06 DU)

Δ1Δ2

Ref: Hale

Δ2 Δ1

Note: ground tracks become more compact with increasing semimajor axis Δ ≈ 190º East, 170º West What about retrograde P ≈ 11.33 hrs orbit ground track shift? a ≈ 25512 km (4.0 DU)

Ref: Hale

Ground Tracks -6 AE 4310 Spaceflight Mechanics Ground Track Symmetry

• From the ground track we can infer something about the eccentricity of the orbit • Circular orbit (e=0) – Displays longitudinal (vertical) symmetry and equatorial (horizontal) symmetry • Eccentric orbit (0

Periapsis More movement over ground at periapsis (V↑) Less movement over ground at apoapsis (V↓) Ascending node? Apoapsis • Type of symmetry displayed also provides insight into argument of periapsis – Longitudinal symmetry only: ω = 90º or 270º (periapsis in Northern or Southern hemisphere) – Equatorial symmetry only: ω = 0º or 180º (periapsis along line of nodes) – No symmetry: ω ≠ 0, 90º, 180º, or 270º

Ground Tracks -7 AE 4310 Spaceflight Mechanics Symmetry Examples (Longitudinal)

Apoapsis (2)

(2) Ascending Node (1)

(3) (1)

Periapsis (3)

Ref: Hale • Observations – Longitudinal symmetry – Periapsis in Southern hemisphere (ω = 270º) What if retrograde? – Compressed ground track around apoapsis

Ground Tracks -8 AE 4310 Spaceflight Mechanics Symmetry Examples (Equatorial)

Ascending node (1) Periapsis (A3)poapsis (2) A

Ascending Node (1) (1), (3)

(2)

Apoapsis (2)

Periapsis (3)

• Observations Ref: Hale – Equatorial symmetry → line of apsides and line of nodes coincide – Apparent Westward motion along ground track – Descending node occurs during Westward motion → ω = 0º (for curve A) • Move quickest East at periapsis

Ground Tracks -9 AE 4310 Spaceflight Mechanics Coverage Contours

• Very high inclination orbits (i ~ 70º-110º) are necessary for significant global coverage • If time required for one complete planetary rotation is an exact multiple of s/c period (P), then eventually the s/c will retrace its ground path – Useful for reconnaissance and landing sites • Very high altitude orbits increase viewing angle of planets surface – Also increase P Increasing r $ r ' FOV% " 50&1 # earth )% % r ( In LEO, FOV ~ 2.4% In GEO, FOV ~ 42%

• At synchronous altitude (r = 42,240 km for earth), P = R – For all non-zero inclinations, gro!un d track appears as a “figure eight” – Stationary orbit implies zero inclination (ground track is a point)

Ground Tracks -10 AE 4310 Spaceflight Mechanics Synchronous Orbit Examples

A D B

C

• Observations Ref: Hale

– iA> iB> iC • With decreasing inclination, figure eights get narrower and shorter • The limit is when i = 0º and the figure eight becomes simply a point – Orbit D is an elliptical synchronous orbit ω?

Ground Tracks -11 AE 4310 Spaceflight Mechanics Synchronous Orbits

• For circular synchronous orbits with non-zero inclination

– Throughout orbit, Vs/c = Vc = Vsynch – At ascending and descending nodes, V = V cosi east c r u : argument r of lattitude – At u = 90º & 270º, Veast = Vc u Veast = Vc cosi + Vc (1 cosi)sinu s c i – Rotational speed of surface point = fn(λ, latitude) V = 2"r cos# /day r s ( e ) ! n • At ascending and descending nodes, λ = 0 Explains Westward motion on Vs = Vc = (2re )/day > Veast s c figure eight ground track !• At u = 90º & 270º

Vs = (2"re )cos# /day < Veast s c u =90º ! λ = +i Spacecraft Surface Point V c ! Eastward u =0º, 180º, 360º Equator Velocity

Vccosi λ = -i 0 90 180 270 360 u (deg) u =270º argument of latitude

Ground Tracks -12 AE 4310 Spaceflight Mechanics Questions?

Ground Tracks -13 AE 4310 Spaceflight Mechanics