Orbits and Propagators

Transcription

DVE1‐T07‐1
DVE1‐T07‐2011 Orbits & Propagators
Diseño de Vehículos Espaciales 1
[email protected]
All active Sun‐synchronous satellite orbits crossing at the North Pole
Contents
• Part I – Orbits revision
 Historical perspective
Historical perspective
 Fundamentals of orbits
 Basic perturbations
Basic perturbations
 Basic maneuvers
 Part II – Propagators
 Why
Wh are orbits
bit important
i
t t for
f design
d i and engineering?
d
i
i ?
 STK propagators
 Part III – STK basics
 An example scenario
 STK training DVE1‐T07‐2
DVE1‐T07‐3
Kepler observation
of Mars’ motion
relative to Earth (1609)
Part I – Orbits revision
Claudio Ptolome
Diseño de Vehículos EspacialesIsaac Newton
1
DVE1‐T07‐4
Tycho Brahe
Johannes Kepler
Ni l
Nicolaus
C
Copernico
i
Claudio Ptolomeo
All active Sun‐synchronous satellite orbits crossing at the North Pole
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Milestones
Geocentrism (Claudio Ptolomeo, s.II aC)
Heliocentrism (Nicolás Copérnico, 1543 dC)
Observations (Ticho Brahe, ~1600 dC)
Empirical Laws (Johannes Kepler, ~1600 dC)
Analytical Laws (Isaac Newton, 1666dC)
2‐body equations of motion
of motion
PLANETARY MOVEMENT
BINARY STARS
BINARY STARS
SPACECRAFT
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Kepler’ss Laws
Kepler
Laws
• In a 2‐body universe :
‐ K‐1: The orbit of one mass around another is a conic section with one b d
body at a focus
f
‐ K‐2: The radius vector of the motion sweeps out equal areas in equal times
K 3: The square of the period of an
‐ K‐3: The square of the period of an orbit is proportional to the cube of the p
mean separation between the masses
• Found only by observation. Newton’s laws (1643‐1727) allow to prove them
DVE1‐T07‐7
Newton’ss Laws
Newton
Laws
• Newton’s Laws:
‐ N‐1:
‐ N‐2:
m2
r = r2 ‐ r1
m1
‐ N‐3:
N 3:
Newton’ss Law of Universal Gravitation:
of Universal Gravitation:
• Newton
F12
GMm
ur
2
r
GMm
F21   2 u r
r
ur  r / r
F12 
11 Nm2kg‐2
2
•G
G = 6.67x10
= 6 67x10‐11
F21
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2‐body problem
2‐body problem
2‐body equation
2
b d
i
of motion
Pos. of m1 and m2 wrt center of mass
DVE1‐T07‐9
Conservation of the
of the specific angular momentum
angular momentum
•
Specific angular momentum of the system:
Uθ
y
Ur
•
Considering polar coordinates:
r
θ
x
•
Deriving the momentum:
‐ h constant and perpendicular to r
‐ m1 and m2 confined to moving in a plane perp to h
perp. to h
‐ Orbit is planar
Kepler’ss 2nd Law
Kepler
2nd Law
•
Specific angular momentum:
DVE1‐T07‐10
Conservation of the specific energy
Conservation of the specific energy Elementary work made by F (exerted by P1 on P2):

dW  F  dr  d mv 2 / 2

GM
F  dr   m

r
 dW  m  dr  md

= potential energy per unit mass

 d mv 2 / 2  md
v2
m  m  C
2
C
v 2 GM
E 
m
2
r
Conic Section
Total Specific Energy (C)
Major Axis a
Circle
<0
r
Ellipse
<0
>0
Parabola
0
∞
Hyperbola
yp
>0
<0 (o
( > 0))
= total specific energy = constant
DVE1‐T07‐11
Polar equation of the conic sections
Polar equation of the conic sections
• Solve for position by integration: • Constants of integration = e, θ0
• Final result:
general • Eccentricity e determines the kind of conic section:
Conic Section
Conic Section
Eccentricity (e)
Eccentricity (e)
Circle
0
Ellipse
0<e<1
Parabola
1
Hyperbola
>1
DVE1‐T07‐12
2D curve, locus of points for 2D
curve locus of points for
which the ratio of the distances from a point F and a line d is constant and = e
Conic section
d
F
F1
d1
d1
F1
F2
d2
d2
F2
DVE1‐T07‐13
Ellipse
ELLIPSE
x2 y2
 2 1
2
a
b
c  ae
b  a 1  e2
θ = true anomaly. Counted from pericenter (θ =0)
E 

2a
0
n a    GM
2
3
A  ab
h
A 


P
P
2
Kepler’s 3rd law
2 a 2 1  e 2
 P  2
h
a3

P
P b l
Parabola
e 1
E0
v  0
h2
r 

2
2
v 
r
h2
r ( ) 

1  cos 
DVE1‐T07‐14
DVE1‐T07‐15
H
Hyperbola
b l
e 1
E
r 


2a
2a
h2
0

1 e
 a(e  1)

a (e 2  1)
r ( ) 
1  e cos 
h2

 a (e 2  1)
DVE1‐T07‐16
Exercise – Escape velocity
Exercise –
Escape velocity
Escape velocity is the required instantaneous velocity to ‘break’ away from orbit and reach infinite distance from centre of attraction with zero velocity. Calculate escape Δv from:
(a) LEO, 200km
(b) Geostationary orbit
( ) 400k M
(c) 400km Mars orbit
bi
You have 5 minutes... [μE = 4.e5 km3/s2; RE = 6378 km; μM = 4.2e4 km3/s2;RM = 3396 km]
DVE1‐T07‐17
%clear all;
muE = 4e5;
muM = 4.2e4;
RE = 6378.0;
63 8 0
RM = 3396.0;
sq2m1= sqrt(2.0)-1;
% LEO h = 200 km:
r = 200
200.0+RE;
0+RE;
dv = sqrt(muE/r)*sq2m1
>> 3.23 km/s
% GEO:
r = 42290.0;
dv = sqrt(muE/r)
sqrt(muE/r)*sq2m1
sq2m1
>> 1.27 km/s
% Mars orbit h = 400.0:
r = RM + 400.0;
400 0
dv = sqrt(muM/r)*sq2m1
>> 1.38 km/s
Orbital Elements – Making orbits useful
Orbital Elements
DVE1‐T07‐18
Orbital Elements – Making orbits useful
Orbital Elements
DVE1‐T07‐19
DVE1‐T07‐20
• Orbits are represented by a number of parameters, the most common method i th K l i element set:
is the Keplerian
l
t t
– Inclination i
• Angle between normal to orbit plane (h) and z axis;
Angle between normal to orbit plane (h) and z axis;
0° ≤ i ≤ 180°
– Longitude of Ascending Node (RAAN in eq. coords.) Ω
• angle, measured at the centre of the earth, from the vernal equinox to the ascending node;
0° ≤ Ω < 360°
– Argument of perigee ω
• Angle from ascending node to perigee;
= angle between line of apsides and line of nodes;
= angle between line of apsides
and line of nodes;
0° ≤ ω < 360°
– Eccentricity e
• Measure of how elliptical orbit is;
0 ≤ e < 1; e = 0  circle
– Epoch T0
• Pericenter passage
• Time at which this set of elements is valid
The ellipse in time
in time
How long does it take to go from one point to another on a keplerian orbit?
p
Mean anomaly M = fraction of an orbit period which has elapsed
since pericenter, expressed as an angle:
DEF: M‐ Mo = n(t‐t0) with Mo = M (t=t0)
n = 2π/P The link between M and θ is the The link between M and θ
is the
eccentric anomaly E
e  cos 
1  e 2 sin 
; sin E 
cos E 
1  e cos 
1  e cos 
....a bit of geometry  algebra give :
M  E  e sin E
M  M 0  n(t  t0 )
t  t0  ( M  M 0 ) / n
DVE1‐T07‐21
DVE1‐T07‐22
TLEs (Two Line Elements Sets)
TLEs (Two Line Elements Sets)
• A standard text format for sharing orbit information
• Updated and distributed regularly by NASA GSFC
[http://celestrak.com/NORAD/elements/]
1 AAAAAU YYLLLPPP BBBBB.BBBBBBBB .CCCCCCCC DDDDD-D EEEEE-E F GGGGZ
2 AAAAA HHH.HHHH
HHH HHHH III.IIII
III IIII JJJJJJJ KKK.KKKK
KKK KKKK MMM
MMM.MMMM
MMMM NN
NN.NNNNNNNNRRRRRZ
NNNNNNNNRRRRRZ
• Example
p TLE for the International Space
p
Station:
SOYUZ-TMA 13
1 33399U 08050A
08291.52340378 .00013791 00000-0 10622-3 0
156
2 33399 51.6416 109.1653 0003546 229.8945 178.2709 15.72317173567788
DVE1‐T07‐23
TLEs – Line 1
TLEs –
Line 1
1 AAAAAU YYLLLPPP BBBBB.BBBBBBBB .CCCCCCCC
• [1] ‐ Line #1 label • [AAAAA] ‐ Catalogue number
• [U] ‐ Security Classification
• U = Unclassified
• C = Classified
C = Classified
• S = Secret
DDDDD-D EEEEE-E F GGGGZ
• [YYLLLPPP] ‐ International Designator
• YY = 2‐digit Launch Year
• LLL = 3‐digit Launch day of the Year
• PPP = up to 3 letter Sequential Piece ID
for launch
• [BBBBB.BBBBBBBB] ‐ Epoch Time
• 2‐digit year, 3‐digit day of the year
• time represented as a fraction of 1 day •[.CCCCCCCC] ‐
•[
CCCCCCCC] Drag Parameter (rev/day
Drag Parameter (rev/day2)
• 0.5 x dN/dt
DVE1‐T07‐24
TLEs – Line 1 ctd
TLEs –
Line 1 ctd
1 AAAAAU YYLLLPPP BBBBB.BBBBBBBB .CCCCCCCC
DDDDD-D EEEEE-E F GGGGZ
• [DDDDD‐D] ‐ Drag Parameter (rev/day3)
• 1/6 x d2N/dt2
•"‐D" is the tens exponent (10‐D).
• [EEEEE‐E] –
[EEEEE E] B* Drag Parameter (1/Earth Radii)
B* Drag Parameter (1/Earth Radii)
• Pseudo Ballistic Coefficient
• The "‐E" is the tens exponent (10‐E)
p
(
)
• [F] ‐ Ephemeris Type
• 1‐digit integer (0 = SGP or SGP4)
• [GGGG] ‐
[GGGG] Element Set Number
Element Set Number
assigned sequentially
B* = C
CDρ0A / 2m
A/2
• [Z] ‐ Checksum (1‐digit integer)
• Computed from sum of all integer characters plus 1 for each “‐”
• Modulo
Modulo‐10
10 of the result is of the result is ‘Z’
Z on on
each line
DVE1‐T07‐25
TLEs – Line 2
TLEs –
Line 2
2 AAAAA HHH.HHHH III.IIII JJJJJJJ KKK.KKKK MMM.MMMM NN.NNNNNNNNRRRRRZ
• [HHH.HHHH] ‐ Orbital Inclination ‐ [0, 180]
• [III.IIII] ‐ Right Ascension of the Ascending Node ‐ [0, 360]
• [JJJJJJJ] ‐ Orbital Eccentricity
• implied leading decimal point ‐ [0.0, 1.0)
• [KKK.KKKK] ‐ Argument of Perigee ‐ [0, 360]
• [[MMM.MMMM] ‐
] Mean Anomalyy ‐ [[0, 360]
,
]
• [NN.NNNNNNNN] ‐ Mean Motion (revolutions per day)
• [RRRRR] ‐ Revolution number
DVE1‐T07‐26
Inclination regimes
Inclination regimes
and also by inclination i …
i = 0°
Equatorial
i = 90°
Polar
0° ≤ i < 90°
90° < i ≤ 180°
Posigrade,
P
i d
Direct,
Prograde
Retrograde,
R
t
d
Indirect
DVE1‐T07‐27
Ground Tracks
Ground Tracks
• Application orbits are determined by the users’ desired ground track
• Ground track is the path the sub‐satellite point traces over the Earth’s surface during its orbit
h’
f
d
b
DVE1‐T07‐28
Ground Tracks
Ground Tracks
Equatorial
orbit
G
Geosynchronous
h
HST orbit (i=28.5 deg)
Geosynchronous
DVE1‐T07‐29
Earth‐synchronous orbits
Earth‐synchronous orbits
• Orbit whose period is contained an integer number of times in one Earth rotation. Therefore, the ground track closes onto itself.
• Examples:
– Geosynchronous (P=24h), semi‐synchronous (P=12h)
– P = 4 h, P = 2 h, P = 6 h
DVE1‐T07‐30
The real Universe is not so simple
The real Universe is not so simple
• There are many more forces acting on a spacecraft than just a single gravity pull of e.g. Earth 
ll f
E th the 2‐body equations are solvable, but useless for real th 2 b d
ti
l bl b t
l
f
l
mission design.
• We can make the model more realistic (and harder to solve) in 2 main ways: – Add more gravity sources
– Add other forces
• Solving
Solving a general n‐body problem is analytically difficult, and in general a general n‐body problem is analytically difficult and in general
contains 6n parameters for which we do not have sufficient independent integrals.
• For spacecraft, we can simplify the n‐body problem to the restricted 3‐body problem, by setting the mass of the 3rd body (spacecraft) to be negligible problem, by setting the mass of the 3
body (spacecraft) to be negligible
compared to other 2 masses (e.g. Earth, 61024 kg, and Moon, 71022 kg)
DVE1‐T07‐31
Perturbations
• So far, we consider gravity to be a field that behaves in a purely 1/r spherical way
• It should be obvious that the Earth is not spherical, and even if it was, there is no reason why the gravity field should be spherical. y
g
y
p
Gravity Map of Earth;Credit: JPL, NASA
DVE1‐T07‐32
Perturbations overview
• No closed‐form analytic solution – also time‐varying (e.g. with solar cycle)
• For design, instead take average values or guidelines – this will be a key lesson for many subsystems!
Source
Acceleration (ms‐2)
@500km
@GEO
Atmospheric Drag
6 x 10‐5 A/M
1.8 x 10‐13 A/M
Radiation Pressure
4.7 x 10‐66 A/M
4.7 x 10‐66 A/M
Sun (mean)
5.6 x 10‐7
3.5 x 10‐6
Moon (mean)
Moon (mean)
1.2 x 10
1
2 x 10‐6
7.3 x 10
7
3 x 10‐6
Jupiter (mean)
8.5 x 10‐12
5.2 x 10‐11
A
A = projected area; M = mass
j t d
M
DVE1‐T07‐33
DVE1‐T07‐34
Non‐spherical gravity
Non‐spherical gravity • An representation in spherical harmonics is used to define a general gravity fi ld f
field of a massive body:
i b d
 
n
n
 RE  n
 
 RE 
U r , ,     1   
 Cn ,m cos m  S n ,m sin m  Pn ,m cos   
 J n Pn , 0 cos     
r 
n2 
m 1  r 
 
 r 

• J, S and C are dependant upon the mass distribution of the central body, 
and  are latitude and longitude, and Pn,m are Associate Legendre Functions.
J2 dominates by FAR
DVE1‐T07‐35
Regression of Line of Nodes
Regression of Line of Nodes
• The bulge produces a torque that rotates the angular momentum vector. For prograde
d (i < 90°), the
90°) th orbit
bit itself
it lf rotates
t t to
t the
th west.
t
• If we ignore everything except J2, the regression is approximated as:
 t  t   O[ J ]
  0  
J2
0
2
2

   3 nJ  RE  cos i 1  e 2

J2
2
2
 a 
n   a3
= mean motion

2
DVE1‐T07‐36
Sun‐synchronous orbits
• LEO orbits whose line of nodes p
precesses (eastward, hence
retrograde orbit) so as to mach the apparent angular motion of the Sun around the Earth
  0.9856o /day

DVE1‐T07‐37
Precession of Line of Apsides
Precession of Line of Apsides
• If you are sitting in your spacecraft, as you cross the plane of the equator, you ‘f l’
‘feel’ more mass than the mean value
th th
l
•  Your orbit curves more rapidly (faster fall)
of energy leads to
leads to a rotation
a rotation of the
of the orbit within the plane
• Conservation of energy
• This can only happen if the major axis rotates, so the line of apsides (joining
perigee and apogee) must rotate.
  0   J t  t0   O[ J 2 ]
2
2


3
R 
 J 2  nJJ 2  E  4  5 sin 2 i 1  e 2
4
 a 

2
• For i = 63.4°,  = 0. This means that apogee for orbit at this inclination
always stays in the same place over the Earth. This was discovered by the Soviet Union in 1960s, and the Molniya orbit was invented.
Molniya orbit
Approx 12h period,
period very high
elliptical orbit
First successful launch: Aug 1965.
Semi‐synchronous orbit
DVE1‐T07‐38
DVE1‐T07‐39
Atmospheric Drag
Atmospheric Drag
• For low‐orbit spacecraft, the very thin edge of the atmosphere creates drag and lift forces, as it would for any flying object. We are going to ignore lift.
d lift f
it
ld f
fl i
bj t W
i t i
lift
FD 
  Vr
1
 SC DVr2 
2
 Vr




• CD is hard to estimate; there is a large mean free path, so free molecular flow is most suitable model (no shock wave, no particle interactions). Typical values are around CD = 2,5.
• S is usually taken as the cross
is usually taken as the cross‐sectional
sectional area projected in the direction of the area projected in the direction of the
velocity vector.
• By inspection, drag will reach a peak at perigee, because both velocity and d it
density are at their maximum values. Intuitively, the drag force can be t th i
i
l
I t iti l th d
f
b
considered a negative, impulsive velocity increment at perigee.
• For an ellipse, this reduces the semi‐major axis; for a circular orbit, drag occurs continuously and reduces the radius.
DVE1‐T07‐40
Solar Radiation Pressure (SRP)
Solar Radiation Pressure (SRP)
• Incident radiation from the Sun imparts momentum to illuminated objects.
• Reflecting light from a surface represents a momentum exchange. There is therefore a small, but measurable, ‘pressure’ effect: P  F c
• F* is the solar energy flux at the spacecraft, c
is the solar energy flux at the spacecraft, c is speed of light. is speed of light.
• At Earth orbit, F* = 1400Wm‐2 so solar radiation pressure is about P = 4.710‐6 Nm‐2
• This is very small, but the effect it has depends on the surfaces of the space vehicle.
• The perturbing effect of the SRP depends upon the area
The perturbing effect of the SRP depends upon the area‐to‐mass
to mass ratio A/m, ratio A/m,
and is inversely proportional to the square of the distance from the Sun.
• The size of the force per unit spacecraft mass (the perturbation), is empirically assembled:
bl d
f SRP
A a 
 (1  k r ) P   
m  r 
2
 Kr is a parameter representing ‘reflectivity properties’: kr = 0.0 for full absorption, kr = 0.4 for diffuse reflection, kr = 1.0 for specular reflection
DVE1‐T07‐41
One‐impulse maneuvers
One‐impulse maneuvers
Consist in applying an impulse at some point (r1,θ1) of the orbit
characterized by velocity and flight path angle (v1 ,γ1).
The result is a new orbit with true anomaly θ2 and velocity and
flight path angle (v2 ,γγ2).
) The radius vector r1 does not
change: the point of application of the maneuver does not change,
Final orbit
the velocity does.
ΔV
V2
V1
Flight path angle γ = angle from local
horizon to velocity vector
Local
horizon
Initial orbit
DVE1‐T07‐42
Two‐impulse maneuvers: Hohmann
Two‐impulse
maneuvers: Hohmann

v1 
Initial velocity (r1)
r1
v2 

Final velocity (r2)
r2
r  r1
a
v 
r  r2
r  r
r r
e   
r  r
2
2 r
r  r r
T 
2 r
v 
r  r r
a3

2r2

v1  v  v1 

r1  r2 r1 r1
v2  v2  v 

r2

2r1
r1  r2 r2
Part II ‐ Propagators
DVE1‐T07‐43
DVE1‐T07‐44
Why do orbits matter for design?
• Orbit size determines period; they both determine lighting levels and directions…
• Lighting determines potentially
determines potentially available solar power…
solar power
• Distance from Sun determines direct thermal load; distance from Earth or
other body determines indirect IR and albedo lighting and heating…
• Night‐side
Ni ht id flight
fli ht and eclipses determine worst‐case thermal
d li
d t
i
t
th
l rates…
t
of nodes and inclination
and inclination determines visibility
determines visibility and duration
and duration of of
• Geometry of nodes
overflights for communications…
• Overflight duration determines required memory size, on‐board processing
power, and communications constraints…
DVE1‐T07‐45
• Communication constraints determine the frequency, coding, and modulation
of link… • Communication link design
link design drives power
drives power requirement for amplifiers and and
antennas…
• Power requirement drives solar array size…
• Solar array
S l
size
i drives attitude
di
i d control requirements…
l
i
place limits on thruster/actuator size, which
size which determines determines
• Rate requirements place limits
fuel requirement, which sizes the structures… • and so on…
DVE1‐T07‐46
Introduction to propagators
• Predicting the orbits of satellites is an essential part of mission analysis and h i
has impacts on the power system, attitude control and thermal design. t
th
t
ttit d
t l d th
ld i
• It is the starting point in planning whether a proposed mission is feasible and how the satellite(s) need to be designed. g
• The computation of the orbits of artificial satellites around planets such as the Earth has been studied in detail in the last century. • The problem of computing the orbits of satellites, however, is not Th
bl
f
ti th
bit f t llit h
i
t
straightforward. The main factors affecting the orbit of a satellite are: – the non‐spherical Earth, – atmospheric drag, – perturbative effects from the gravitational pull of the Sun and other planets
l
t
– radiation pressure. Diseño de Vehículos Espaciales 1
DVE1‐T07‐47
Introduction to propagators
These effects are important in different levels for different types of satellite orbits. For example satellites in LEO are strongly affected by the non‐
spherical nature of the Earth and even atmospheric drag. Satellites out in geostationary orbit however are sufficiently far from the Earth for these
geostationary orbit, however, are sufficiently far from the Earth for these effects to be ignorable. The gravitational pull of the Sun and Moon, however, does play a significant role in the evolution of their orbits. Effects such as atmospheric drag and radiation pressure are also very dependent upon the shape, size and mass of the satellite.
DVE1‐T07‐48
Propagators
• The purpose of satellite orbit propagators is to provide high accuracy in predicting the position of a satellite. di ti th
iti
f
t llit
• Orbits are simulated numerically for future time steps to predict where a satellite/vehicle will be in the future (necessary in order to plan y
p
communications, manage operations, etc.)
• Analytical solutions apply a time‐limited rule about how position and velocity will change in a small time step and apply them to the previous state
will change in a small time step, and apply them to the previous state
• Starting from a set of initial conditions at t=0, the evolution of an orbit is propagated into the future
• In order to achieve precision, a very short time step is required. However, the calculation of the forces acting on a satellite at each time step slows down the computation which makes it prohibitive to propagate anything but crude
computation which makes it prohibitive to propagate anything but crude force models on the satellite itself. • This means that it is important to identify the forces that mainly affect a given orbit in order to propagate an adequate and appropriate force model and bit i
d t
t
d
t
d
i t f
d l d
avoid spending computing time with complete and detailed force models. Diseño de Vehículos Espaciales 1
DVE1‐T07‐49
2‐body propagators
2‐body propagators
• 2‐body propagator assumptions: – satellites travel high enough above the atmosphere so that the drag force is small
– the satellite won
the satellite won’tt perform manoeuvres (we can ignore any thrust)
perform manoeuvres (we can ignore any thrust)
– we only consider the motion of the satellite close to the Earth
– we can ignore the gravity of the Sun, the Moon or anything else
– compared to Earth’s gravity, other forces e.g. solar radiation pressure, electromagnetic fields, etc. are negligible
– the mass of the Earth is much, much larger than the mass of the the mass of the Earth is much much larger than the mass of the
spacecraft
– the Earth is spherically symmetrical with uniform density (so assume it is a point mass)
• A 2‐body propagator does not need numerical integration in a force model The model allows for closed‐form
model. The model allows for closed
form solution, with the only need for solution with the only need for
a numerical technique to solve Kepler’s equation Diseño de Vehículos Espaciales 1
DVE1‐T07‐50
Kepler’ss equation with NR
Kepler
% From time to true anomaly with the solution of Kepler's equation by
Newton Raphson
Newton-Raphson
clear all;
% constants
t
twopi
i = pi
i + pi;
i
% data:
e = 0.4; % eccentricity
t0 = 0
0;
% epoch
h of
f pericenter
i
t
passage
P = 5800.0; % period (sec)
t = 35.0;
% time (min)
tol = 1.e-10;
% from min to sec
t = t*60.0;
% mean motion
n = twopi/5800.0;
% mean anomaly at t
M = n*(t-t0);
% NR loop
dE = 1.0;
E = M
% first guess for E
iter = 0; % iteration counter
while(dE > tol)
iter = iter + 1;
Eold = E;
E = E + (M+e*sin(E) - E)/(1.0-e*cos(E));
dE = abs(Eold-E);
end
iter
>> 4
E*180/pi
>> 143.9 deg
% from E to true anomaly:
cE = cos(E);
sE = sin(E);
cth = (e-cE)/(e*cE-1.0);
sth = sqrt(1.0-e*e)*sE/(1.0-e*cE);
th = atan2(sth,cth)*180/pi
p
% true anomaly
y (deg)
g
DVE1‐T07‐51
>> 155.9 deg
g
DVE1‐T07‐52
Propagators
• However, propagators can include terms that correct for some of the model assumptions and hence introduce some additional force.
ti
dh
i t d
dditi
lf
• This requires propagation equations (either analytical or numerical)
• We will not be concerned with deriving the propagation equations, but will We will not be concerned with deriving the propagation equations, but will
learn about the main propagators and their differences
[New geometric integration schemes have been developed recently which exploit the geometric properties of the orbital dynamics of a satellite. These are called symplectic methods. Explicit schemes have been developed which also enable methods Explicit schemes have been developed which also enable
very fast computation of the satellite orbit for a given required level of accuracy. As well as the accuracy, symplectic schemes also provide a method for splitting time steps for different types of force, and this can be exploited for satellites to great effect, greatly reducing the expensive force calculations. The resulting propagation schemes are now able to compute accurate orbit trajectories onboard the satellite and therefore enable real time accurate orbit determin.]
DVE1‐T07‐53
STK Propagators
STK Propagators
• When using STK to get useful engineering values, remember to choose a suitable propagator.
suitable propagator. • STK uses 2 basic types: numerical and analytical
Analytical propagators use a closed‐form solution of the time‐dependent propagators use a closed form solution of the time dependent
motion of a satellite to produce ephemeris or to provide directly the position and velocity of a satellite at a particular time. N
Numerical
i l propagators numerically integrate the equations of motion for the t
i ll i t
t th
ti
f
ti f th
satellite.
http://www.stk.com/resources/help/online/stk/source/stk/vehsat_orbitprop_ch
oose.htm
DVE1‐T07‐54
STK Propagators
STK Propagators
• Main propagators to use are:
– J2 Perturbation (1st Order)  LEO
• The J2 Perturbation (first‐order) propagator accounts for variations in the orbital elements due to Earth oblateness
the orbital elements due to Earth oblateness. • Does not model atmospheric drag
• Does not model solar or lunar gravity
– J4 Perturbation (2nd Order)  LEO
• The J4 propagator includes the 1st & 2nd order effects of J2
• Includes the 1st order effects of J4
• J4 is approximately 1000 times smaller than J2
• Little difference between orbits propagated with J2 and J4
DVE1‐T07‐55
STK Propagators
STK Propagators
– HPOP (High Precision Orbit Propagator)
• HPOP can handle orbits up to and beyond the Moon
HPOP can handle orbits up to and beyond the Moon
• Includes 1st order effects of J4
– SGP4
SGP4 
 LEO, GEO, HEO
LEO GEO HEO
• Simplified General Perturbations (SGP4) propagator is used with TLEs
• Includes J2 and J4
• Also includes Solar and Lunar gravity and gravitational resonance
• Includes orbit decay using a simple drag model
– LOP
• Long‐term (months, years) propagator
– Astrogator
• Includes thrusts, manoeuvres, and targeting
DVE1‐T07‐56
STK Propagators
STK Propagators – StkExternal propagator: allows to read the ephemeris for a satellite from a fil Th fil
file. The file must end in an .e extension. t di
t i
SPICE: reads ephemeris from binary files that are in a standard format reads ephemeris from binary files that are in a standard format
– SPICE:
produced by the Jet Propulsion Laboratory (JPL). They are intended for ephemeris for celestial bodies but can be used for spacecraft (see instruction in STK page)
instruction in STK page)
– Real‐time: allows to propagate vehicle (all types) ephemeris using near‐
real‐time data received using a connect socket
DVE1‐T07‐57
STK Propagator summary
STK Propagator summary
Earth Gravityy
J2
J4
Sun Gravityy
Lunar Gravityy
Solar Wind
Other Planets
Earth Atmos’
Thrusts
2‐Body
J2
J4
SGP4 (TLE)
HPOP
Astro‐
gator
LOP
The remaining three (StkExternal, SPICE and
Th
i i
h
(S kE
l SPICE d Real‐time) are based on R l i )
b d
Ephemerides (they do not use models), hence they do not appear in the table
Part III - STK Basics
DVE1‐T07‐58
DVE1‐T07‐59
What is STK? ?
• STK is a physics‐based software geometry engine that accurately displays and analyzes land, sea, air, and space assets in real or simulated time.
l
l d
i
d
t i
l
i l t d ti
• Users can model the time‐dynamic position and orientation of vehicles via various propagation algorithms or external inputs.
p p g
g
p
• Given these dynamic positions and orientations, users can model the characteristics and pointing of – Sensors S
– Communications
– Other payloads aboard the asset.
Other payloads aboard the asset.
• STK can then determine spatial relationships (e.g. line of sight) between an asset of interest and all of the objects under consideration.
• STK's technically‐accurate 2D and 3D visualization and analytical data outputs can help enhance situational awareness and understanding. • Users can share their results with others via snapshots, movies, and even VDF Users can share their results with others via snapshots, movies, and even VDF
files for use in AGI Viewer.
Creating tutorial scenario
Creating tutorial scenario
• Open STK, close the Launchpad
• File → New
DVE1‐T07‐60
DVE1‐T07‐61
General configuration
General configuration
• Edit→ Preferences→ Save/Load Prefs
– verify that Save Vehicle Ephemeris is enabled and Binary Format is disabled
– verify that Save Accesses is disabled
– verify that Auto Save is enabled, and that the Directory field is pop lated ith the location here o ant to sa e STK files
populated with the location where you want to save STK files. Set the Save Period to 5 min (300 sec)
• Object Browser:
Object Browser:
– Select the scenario, right‐click, open the Properties browser
Basic proper es
es → Time Period → → Time Period →
– Basic proper
• Start: 6 Oct 2011 8:00
Start: 6 Oct 2009 14:00
• Start: 6 Oct 2009 14:00
→ [Apply]
DVE1‐T07‐62
Scenario configuration
Scenario configuration
• Check the start time on the ‘animation’ settings page also
→ [OK]
• Select 2D map view, and edit properties
• Select Overlays. In the Animation Time frame, turn on the Show option, set X to 20 and Y to –20, set Text Color to white
• Disable the background
• In Details page, remove International Borders from the selected li t d di bl L t/L li
list, and disable Lat/Lon lines ‘Show’ (i.e. hide them)
‘Sh ’ (i hid th )
• Disable background image, and set color to black
• In Projection page, verify that the Type is set to Equidistant I P j i
if h h T
i
E idi
Cylindrical, and check Center Lon is 90 deg.
Initialise 2D display
Initialise 2D display
• Open the Map Annota ons page → [Add…]
• Enter your name as ‘String’, set Position Type to X,Y
• Set X = ‐160, Y = ‐60
→ [OK]
• Reset the animation
reset
DVE1‐T07‐63
DVE1‐T07‐64
Configuring the scenario
Configuring the scenario
• In Object Browser, rename the scenario to DVE1 (right‐click)
• ‘Right menu’ also provides links
to many features of the tools
• Add a satellite:
Insert → New…
• Satellite → [Insert]
• Cancel
C
l the th
orbit wizard
Configure the satellite
Configure the satellite
•
•
•
•
Maximise the 2D map view
Rename the satellite to ‘Molniya’ (right‐click → Rename)
Molniya → Right‐click → Proper es → Basic → Orbit
Set:
– Propagator → TwoBody
– Semi‐major axis → 26554 km
Semi major axis → 26554 km
– Eccentricity → 0.72
– Inclina on→ 63.14
on→ 63.14°
– Argument of Perigee → 270°
– RAAN → 0°
– Orbit Epoch: 1 Jul 2007 00:00:00.000 UTCG
→ [OK]
• Reset the animation, then play it
R
h
i
i
h
l i
DVE1‐T07‐65
Creating a ground station
Creating a ground station
• Select the DVE1 scenario
• Insert → New…
Facility → [Insert]
• Rename the facility to ‘Terrassa’
• Open the properties browser, and set: – Latitude: 41:33:45.2700 DMS
– Longitude: 02:01:24.7500 DMS
– Select ‘Use Terrain Data’
– Local time offset from GMT: 7200 sec
→ [OK]
→ [OK]
DVE1‐T07‐66
DVE1‐T07‐67
Add another satellite
Add another satellite
•
•
•
•
•
Select the DVE1 scenario
Insert → Satellite from database
Select ‘Common Name’ → ISS
Select SSC=25544 → [OK]
Reset, start simulation, and explore the map views and settings Diseño de Vehículos Espaciales 1
DVE1‐T07‐68
Calculate access (times)
Calculate access (times)
•
•
•
•
Select Terrassa in the scenario
Right click → Access
Select ISS (leave other se ngs) → [Compute]
Reports panel → [Access]
DVE1‐T07‐69
Calculate Access (AER)
Calculate Access (AER)
• → [AER] (Azimuth, Eleva on, Range)
DVE1‐T07‐70
Graphing Results
Graphing Results
DVE1‐T07‐71
DVE1‐T07‐72
Using sensors
Using sensors
• Remove the ISS from the scenario
• Add a sensor to Molniya:
– Select Molniya
– Insert → New… → Sensor
– rename the sensor ‘horizon’
• Open sensor properties:
– Definition: Simple conic, cone angle = 90°
– Pointing: Fixed, elevation = 90°
→ [OK]
• Reset and run the scenario
• Change the cone angle to 45°, 25°, 5° and re‐run the scenario
• Adjust 3D options of the sensor display
DVE1‐T07‐73
Using sensors
Using sensors
• Add a sensor to Terrassa: – Name = ETSEIAT
– Sensor Type = Complex Conic
– Inner Half Angle = 0°
– Outer Half Angle = 85°
– Minimum Clock Angle = 0°
– Maximum Clock Angle = 360°
– Pointing = Fixed
– Elevation = 90°
• ETSEIAT → 2D graphics projec on proper es → projec on
– Extension distances → maximum al tude = 785
– Project to: constant altitude, step count = 1
Using sensors
Using sensors
• Reset and run the scenario
• Ground sensor projects a fixed region of ‘possible reach’ projected onto the map, up to a fixed altitude
DVE1‐T07‐74
DVE1‐T07‐75
STK Training
STK Training
• Repeat this scenario for yourself
• In STK → Help → Desktop Applica on Help → Tutorials
– Introducing STK
– 3D graphics
• Objective: get familiar with the tool, with the way to set parameters, add objects, visualise results.
dd bj
i li
l
• STK will be used in later topics, as a ‘simulation engine’ that gives results about: lighting thermal communications conditions etc
results about: lighting, thermal, communications conditions… etc.
• It is important that you become happy with it. It is powerful
and rather easy to use.
d th
t
Some ready‐to‐use scenarios
Some ready‐to‐use scenarios
•
•
•
•
sim‐geo.vdf
ExampleOrbits.vdf
sim‐hohmann‐transfer.vdf
DVE1_STK_Examples.vdf
DVE1‐T07‐76
DVE1‐T07‐77
To be included in the report of “Proyecto
To be included in the report of Proyecto de Equipo
de Equipo – parte 1
parte 1”::
• First of all: download & install software, apply for licence and install it
• Follow installation instructions available in Atenea
Follow installation instructions available in Atenea
• Very few people have completed this step, the rest are “warmly” invited to proceed...
THEN:
• Take your study satellite and create a scenario for its orbit, and one ground station (or all if it has more than one and you feel like it)
station (or all if it has more than one and you feel like it). • Set the time frame for your scenario
• Change any display parameters
• Run the scenario and obtain:
– A 2D ground track plot
– An access report for the ground station‐to‐satellite
f h
d
ll
(if you have many accesses – give only max, min, mean values)
• It is not a bad idea (and will cause you no harm) that of the three (
y
)
members of each group each performs the exercise independently and then exchanges and discusses results with the group...

Orbits and Propagators

Transcription

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