Mission Analysis and Tools - Summer School Alpbach · Mission Analysis and Tools Alpbach...

transcript

Mission Analysis and Tools

Alpbach Summerschool 2014

Marcus Hallmann (DLR)

Marcus.Hallmann@dlr.de

Content

Mission Analysis in the Context of System Engineering

Keplerian Elements

Hohmann Transfer / Inclination Change

Lambert Problem

Arriving and Leaving a planet

Launcher Performance

Gravity Assist

Low Thrust

Special Orbit Types

Quelle: TSTI

The Orbit Design Process

Establish orbit types

Determine orbit related requirements

Assess launch options

Create ΔV budget

Perform orbit design trades

Determine orbit related requirements

Temperature gradient

Absolute temperature

Straylight reduction

Max eclipse time

Communication requirements ( volume, timeliness )

Sun-Spacecraft-Earth angle ( >5deg for communication )

Scan strategy

Attitude disturbance reduction

Radiation (total dose over mission time)

Content

Keplerian Elements

Lambert Problem

Gravity Assist

Low Thrust

Special Orbit Types

Equation of motion:

Energy:

Orbit angular momentum:

Two-Body System

3· 0r r

mx = -

h r v= ´

Conic Sections

Equation of a conic section (trajectory equation):

Slicing the cone with a plane:e>1 Hyperbola0<e<1 Ellipsee=0 Circlee=1 Parabola

1 cos( )

Ellipse

Focus F and F‘

Semimajor axis a

Semiminor axis b

Semiparameter p

Eccentricity e

True anomaly n

Apoapsis rmax = ra

Earth = Apogee; Sun = Aphelion; Moon = Aposelen

Periapsis rmin = rp

Earth = Perigee; Sun = Perihelion; Moon = Periselen

Ellipse

2PF PF a¢ + =

2 2 2 2( ) 1b ae a b a e+ = = -

(1 )1 cos(0) 1

(1 )1 cos( ) 1

p pr a e

e ep p

r a ee ep

= - = =+ ⋅ +

+ = ==+ ⋅ -

2 2(1 ) /p a e b a= - =

2a pr r a+ =

Hyperbola

2PF PF a¢- = -

1b a e= -

(1 )pr a e= -

2 2(1 ) /p a e b a= - = -

Hyperbola

1 cos( )

1cos( ) lim

n¥ ¥

æ ö÷ç= - ÷ç ÷ç ÷è ø

1arccos( )

en¥ = -

Two-Body Motion

Orbital period:

Orbital energy:

Vis-Viva equation:

Eccentric anomaly E:

Mean anomaly M:

·2·a

2 2 1v

r amæ ö÷ç= - ÷ç ÷ç ÷è ø

m mx = - = -

3sin( ) ( )pM E e E t t

m= - = -

cos( )cos( )

1 cos( )

Orbital Elements

a semimajor axis

e eccentricity

I inclination

Ω ascending node

ω argument of periapsis

ν true anomaly

0 0 0 0

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

r t v t

r t r t v t v t

r t v t

æ ö æ ö÷ ÷ç ç÷ ÷ç ç÷ ÷ç ç÷ ÷ç ç= =÷ ÷ç ç÷ ÷ç ç÷ ÷÷ ÷ç ç÷ ÷ç çè ø è ø

Ecsape Velocity from a Circular Orbit

Circlevr

2Parabolav

2CiParabol ca r lev v v

m m⋅D = = --

( )2 1D = ⋅ -vr

Content

Keplerian Elements

Lambert Problem

Gravity Assist

Low Thrust

Special Orbit Types

Hohmann Transfer

1D = -p p Kv v v

2D = -a K av v v

D = D + DHohmann p av v v

+ += =p ar r r r

( )1 1

2Hohmann

m mæ ö æ öæ ö⋅ ÷ ÷ç ç ÷ç÷ ÷ç çD = - + - ÷ç÷ ÷ç ÷ç ç÷ ÷÷÷÷ çç è ø+ +è øè

Solar System Parameters

Planet e i[°] r[AU] vF[km/s] v[km/s] Period[yr]

Mercury 0.205 7.0 0.39 4.3 47.9 0.241

Venus 0.007 3.4 0.72 10.4 35.0 0.615

Earth 0.017 0.0 1.00 11.2 29.8 1.000

Mars 0.094 1.9 1.52 5.0 24.1 1.88

Jupiter 0.049 1.3 5.20 59.5 13.1 11.9

Saturn 0.057 2.5 9.58 35.5 9.7 29.4

Uranus 0.046 0.8 19.20 21.3 6.8 83.7

Neptune 0.011 1.8 30.05 23.5 5.4 163.7

Pluto 0.244 17.2 39.24 1.1 4.7 248.0

Hohmann Transfer, Δv Magnitude in the Solar System

0 5 10 15 20 25 30 35 400

Zielorbit [AU]

Hohmann Transfer von der Erde

Hohmann Transfer

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

Zielorbit [AU]

Merkur dvdvp

Hohmann Transfer, Time-of-Flight

P at p

mD = =

Time-of-Flight:

( )1 23

1.52 1 1

æ ö+ ÷ç= = ⋅ ÷ç ÷ç ÷è ø

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.25

Zielorbit [AU]

VenusMerkur

0 5 10 15 20 25 30 35 400

Zielorbit [AU]

Phase Angle γ

( )1 23

D = D =

Mars Mars Marsn t nn pm

( )1 23

Mars Marsnn pm

( )1 23

1.51 1

æ ö+ ÷ç ÷çD = ⋅ ÷ç ÷ç ÷÷çè ø

kMarsn p

1.51 11

æ öæ ö ÷ç + ÷÷çç ÷÷çç= - D = ⋅ - ÷÷çç ÷÷çç ÷÷÷çç ÷è ø ÷çè ø

kMarsg p n p

> >< <

outbound, target planet must be in front of Earthinbound, target planet must be behind Earth

0 5 10 15 20 25 30 35 40-300

Zielorbit [AU]

Phase Angle γ

1.51 11

æ öæ ö ÷ç + ÷÷çç ÷÷çç= - D = ⋅ - ÷÷çç ÷÷çç ÷÷÷çç ÷è ø ÷çè ø

kMarsg p n p

Synodic Period

Mean angular velocity:

( )1 1.51

pt = = ⋅

2 1n n nD = -Relative angular velocity between two planets:

( )1.51

Pt Å= ⋅-

Synodic period:

Synodic Period

0 5 10 15 20 25 30 35 400

Zielorbit [AU]

( )1.51

1Å= ⋅

Δv for a change in Inclination of an Elliptical Orbit

1 2 cos= = ⋅ v v v f

cos sin2 2

æ öD D ÷ç= ⋅ ⋅ ÷ç ÷ç ÷è øv i

2 cos sin2

æ öD ÷çD = ⋅ ⋅ ⋅ ÷ç ÷ç ÷è øi

Inklination [deg]

[km/s] für r=7000 km

0 20 40 60 80 100 120 140 160 180

Node Change

( )2 212

2 1 cos ( ) sin ( ) cos( )D = ⋅ ⋅ - - ⋅ DWv v i i sin( )

sin hd

r adt h i

w n+W= ⋅

Content

Keplerian Elements

Lambert Problem

Gravity Assist

Low Thrust

Special Orbit Types

Lambert Problem

1 2 2 1, , ,r r t t We know :

1 2r r

We are looking for the ellipse or hyperbola which connects und

In the real world the orbits of the planets are neither coplanar nor circular.

2 1t t t- = DIf we specify the time-of-flight( ), only one soultion exists.

Earth-Mars Transfer Lambert

Launch Date: 61344 ModJDate 31 Oct 2026Transfer Time: 309 daysArrival Date: 61654 ModJDate 6 Sep 2027

Earth-Venus Transfer Lambert

Launch Date: 60676 ModJDate 1 Jan 2025Transfer Time: 127 daysArrival Date: 60803 ModJDate 8 May 2025

Earth-Mercury Transfer Lambert

Launch Date: 60806 ModJDate 11 May 2025Transfer Time: 100 daysArrival Date: 60907 ModJDate 20 Aug 2025

Content

Keplerian Elements

Lambert Problem

Gravity Assist

Low Thrust

Special Orbit Types

„Hyperbola Elements“

Hyperbola elements are useful for describing an orbit when leaving or approaching a planet.

6 elements are needed:

C3 energy = (v∞)2

Right ascension α and declination δ of theoutgoing asymptote

Velocity azimuth at periapsis

True anomaly

Example Mars-Transfer in 2003

Solve Lambert problem:

Launch Epoch [ModJDate]

Cost Function = dv1 + dv2 [km/s]

5.6675 5.6675 5.6675 5.6675 5.6675 5.6675 5.6675 5.6675 5.6675 5.6675

5.265 5.27 5.275 5.28 5.285 5.29 5.295 5.3

3 8.786

æ ö÷ç ÷ç ÷ç ÷ç= = - ÷ç ÷ç ÷÷ç- ÷çè ø

= - = -

In spherical coordinates:

Credits: OFOISR, Proton Users Manual

1 1.1452pr ve

⋅= + =

1acos 150.83

æ ö÷ç= - ÷ = ç ÷ç ÷è ø

sin sin sin( )

sin sinasin 180 asin

sin sin

d w nd d

w n w n

= +æ ö æ ö÷ ÷ç ç= ÷ - = - ÷ -ç ç÷ ÷ç ç÷ ÷è ø è ø

200ParkingOrbit

ParkingOrbit

Start from Baikanour:

2 solutions:

0 50 100 150 200 250 300 350-1

1sin(i)*sin( + ) ; i=64.8

0 50 100 150 200 250 300 350-1

1sin(i)*sin( + ) ; i=3

( ) ( )tan cos( )sin ; cos

tan cosi

d w na a

d¥ ¥

¥ ¥¥

+-W = -W =

For the ascending node we can derive:

tan cos( )atan ,

tan cosi

d w na

d¥ ¥

Wæ ö+ ÷ç ÷W = - ç ÷ç ÷çè ø

goes from 0..360°, quadrant check must be performed:

351.84

346.66

W = W =

is controlled via the time spent in the parking orbit

is controlled via the daily launch time

the launch date comes out of the solution to the Lambert problem

Approaching the Target Planet, B-Plane

Definition B-Plane:

zT S e= ´

ze can be the North Pole or e.g. the ecliptic

R S T= ´

forms a right-handed orthogonal system:

cos cos cos i

qq d¥ =

We can derive a relation for the B-Plane angle :

.i d¥³Out of this we can see that the inclination is constrained:

0 50 100 150 200 250 300 3500

Content

Keplerian Elements

Lambert Problem

Gravity Assist

Low Thrust

Special Orbit Types

VEGA: S-ClassSSO@800 ~= 1300 kg

Soyuz: M-ClassSSO@800 ~= 4400 kg

Ariane 5: L-ClassSSO@800 ~= 10000 kg

Ariane 5ME: XL-ClassSSO@800 ~= 30000 kg

Launcher Performance, VEGA

Launcher Performance, Soyuz

Launcher Performance, Ariane

Credits: ESOC

Launchers

Content

Keplerian Elements

Lambert Problem

Gravity Assist

Low Thrust

Special Orbit Types

Gravity Assist 2D Case

Planet

⋅= +

12 arcsin

æ ö÷ç= ⋅ ÷ç ÷ç ÷è ø

Deflection angle:

( )v va¥- ¥+= ⋅Rot

12 ¥D = ⋅Satv v

In front of the planet => Heliocentric velocity decrease

Behind the planet => Heliocentric velocity increase

Gravity Assist, 3D Case

( ) ( ( ))pv r vq a¥- ¥+= ⋅ ⋅Rot Rot

Sat Planetv v v¥- -= -

Sat Planetv v v+ ¥+= +

You can choose θ and rp for free(without spending any deltaV), because the asymptote can bemoved around the B-plane.

Gravity Assist, Tisserand‘s Graph

( )22 1 cos( )SatSat Sat

r aT e i

= + ⋅ - =

constant for sequential fly-by's at a planet

constant (it's just a rotation of the velocity vector relative to the planet )

Can be used to find possible GA sequences

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

rp [AU]

r a [AU

E 3E 4E 5E 6V 3V 4V 5V 6

0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.780.8

rp [AU]

r a [AU

E 3E 4E 5E 6V 3V 4V 5V 6

Gravity Assist e.g. Rosetta

Gravity Assist e.g. Cassini

Content

Keplerian Elements

Lambert Problem

Gravity Assist

Low Thrust

Special Orbit Types

Low thrustEngine performance Fregat Upper stage

0 50 100 150 200 2501000

Time [s]

Isp = 320s m1 = 2000kg Thrust = 14000 N

0 50 100 150 200 2500

Time [s]

Isp = 320s m1 = 2000kg Thrust = 14000 N

Low thrustEngine performance ion engine

0 1 2 3 4 5 6 71000

Time [year]

Isp = 2000s m1 = 2000kg Thrust = 0.1 N

0 1 2 3 4 5 6 70

Time [year]

Isp = 2000s m1 = 2000kg Thrust = 0.1 N

Ion thrusters survey

ESA (Qinetiq T6):ISP 4200 sFmax 0.2 NP 5-6 kWMissions:

• Bepi Colombo

ESA (PPS 1350 SNECMA):ISP 1660 sFmax 0.09 NP 1.5 kWMissions:

• SMART

Credit: ESA

NASA(NSTAR):ISP 3100 sFmax 0.09 NP 2.6 kWMissions:

• Deep Space 1• Dawn

Credit: NASA/JPL

Credit: NASA/JPL-Caltech

Credit: NASA/JPL

JAXA(NEC):ISP 3200 sFmax 0.008 NP 0.35 kWMissions:

• Hayabusa

Lambert Solver for Low ThrustShape-Based Approach

Exponential sinuoid used by Petropoulos:

Inverse polynominal used by Wall and Conway:

Example Mars Transfer

tof = 700 days

m0 = 2100 kg (Soyus)

v_inf = 1.0 km/s

m_final = 1765kg

Isp = 4000 s

F_max = 0.2 N

Example Mars Transfer

ESA cornerstone mission to MercuryLaunch: 9 July 2016

MPO: Mercury Planetary Orbiter (480 x 1500 km)

MMO: Mercury Magnetospheric Orbiter (JAXA)(600 x 11800 km)

Bepi Colombo (Credits: Rüdiger Jehn, ESOC)

Low thrust trajectory optimization

Specific impulse 4022 sMaximum thrust 0.290 NSEP availability 90%Fuel consumption for navigation 5%

4.1 tons

Bepi Colombo

Optimization ConstraintsConstraint ValueEarth flyby altitude > 300 kmVenus 1 flyby altitude < 1500 kmVenus 2 flyby altitude > 300 kmMercury flyby altitudes > 200 kmLast Mercury flyby altitude > 300 kmDuration of coast arc before flybys > 30 days

Duration of coast arc after flybys > 7 days

Duration of coast arc after launch > 90 days

Duration of coast arc before MOI > 60 days

Solar aspect angle (outside 0.8 AU) 77° < SAA < 112.5°

Solar aspect angle (inside 0.8 AU) 77° < SAA < 94.5°

Bepi Colombo

Departure Launch Date 9 July 2016Escape velocity 3.475 km/sEscape declination -3.8°Initial mass 4100 kg

Cruise ∆V SEP 4.254 km/sTotal impulse 16.6 MNsCruise time 7.5 years

Arrival Date 1 Jan 2024Mercury true anomaly

67.8°

Velocity at periherm

3.793 km/s

Ωarr 67.7°ωarr (South pole) -2°

Content

Keplerian Elements

Lambert Problem

Gravity Assist

Low Thrust

Special Orbit Types

Sun Synchronous Orbit (SSO)

A SSO maintains the same orientation with respect to the sun all year round. This natural phenomen is due to the irregular shape of the earth.

Imagined by Jean-Pierre Penot (CNES) and Bernard Nicolas, illustrated by Bernard Nicolas

Sun Synchronous Orbit (SSO)

23( ) sin

2(3 1)Å Å- ⋅

⋅= -

Potential:R

U r Jr

24(sin2 sin( ))

2h ia u

mÅ Å= -⋅

⋅⋅

sin( )

W» ⋅ h

du h i

9.96 cosÅ

æ ö÷ç ÷çDW = - ⋅÷ç ÷÷ç +è øday

0.9856 9.96 cosÅ

æ ö÷ç ÷ç = - ⋅÷ç ÷÷ç +è ø

acos 0.0989 Å

æ ö÷ç æ ö ÷ç + ÷÷çç ÷÷çç= - ÷÷çç ÷÷÷çç ÷è ø ÷ç ÷çè ø

SSOSSO

0 1000 2000 3000 4000 5000 600090

Height (km)

Geostationary Orbit

2 23h56min

42162km

Orbital Period:

1 Jan 26 Feb 12 Apr 29 Aug 14 Oct 31 Dec22

Sunlight GEO

Highly elliptical orbit

Elliptical orbit:Different regions can be studied

Two satellites:Time derivations can be determined

Literatur

Interplanetary Mission Analysis and Design, Stephen KembleISBN 3-540-29913-0

Fundamentals of Astrodynamics and Applications, David ValladoISBN 978-1881883142

Space Mission Engineering: The new SMAD, James R. WertzISBN 978-1-881-883-15-9

http://nssdc.gsfc.nasa.gov/planetary/planetfact.html

http://naif.jpl.nasa.gov/naif/spiceconcept.html

http://gmat.gsfc.nasa.gov/

Thanks for your attention

Time for questions

Mission Analysis and Tools - Summer School Alpbach · Mission Analysis and Tools Alpbach...

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