ORBITS & V AND SOME SPACECRAFT PARAMETERS · Alpbach 07 Rules of thumb 1 SOME RULES OF THUMB FOR...

Alpbach 07 Rules of thumb 1

SOME RULES OF THUMB

FOR DOING FIRST ORDER COMPUTATION

ON

ORBITS & ∆V

AND

SOME SPACECRAFT PARAMETERS

Alpbach - July 2007

D.J.P.Moura

Issue 1


TABLE OF CONTENTS

1. ORBIT GENERALITIES............................................................................................................................. 3

1.1 MASSIVE BODY (SUN, PLANET ...) .............................................................................................. 3

1.2 GEOMETRICAL DESCRIPTION OF THE ORBIT IN ITS PLANE ..................................................... 4

1.3 POSITIONING OF THE ORBIT COMPARED TO THE BODY........................................................... 5

1.4 POSITIONING OF THE SPACECRAFT ON THE ORBIT & DYNAMIC ASPECTS............................. 6

2. DELTA V COMPUTATION ........................................................................................................................ 7

3. PROPELLANT(S) MASS ............................................................................................................................ 9

4. SOLAR ARRAY .......................................................................................................................................... 10

5. BATTERIES ............................................................................................................................................... 11

6. THERMAL RADIATORS .......................................................................................................................... 12

7. SPACECRAFT ANTENNA ....................................................................................................................... 12

8. LINK BUDGET (TELEMETRY AND TELECOMMAND RATES) ....................................................... 14

___________________ APPENDIX 1: LINK BUDGET EXAMPLE.............................................................................................. 16

APPENDIX 2: MOON ENGINEERING PARAMETERS ....................................................................... 18

APPENDIX 3: MARS ENGINEERING PARAMETERS......................................................................... 20

APPENDIX 4: LAGRANGE POINTS ........................................................................................................ 21


1. ORBIT GENERALITIES

1.1 Massive body (Sun, planet ...) Before computing and describing an orbit, you need to have the characteristics of the body around which the spacecraft will orbit. The most important ones are: - the gravitational constant (called µ = universal constant of gravitation * mass of the body) µ for the sun # 1.33 1011 km

3s

-2

µ for Earth # 3.99 105 km

3s

-2

µ for the moon # 4.90 103 km

3s

-2 µ for Mars # 4.29 10

4 km

3s

-2 - the mean radius Rm Rm # 695 500 km for the sun, Rm # 6 400 km for the Earth, Rm # 1 738 km for the moon Rm # 3 398 km for Mars

- a reference direction such as the sun direction at the vernal equinox for the Earth

- a reference plane ecliptic plane for orbits around the sun equator plane for orbits around a planet Remarks - The galactic pole is defined by α = 192.25 deg and δ = 27.4 deg in the

equator plane. - For more accurate computations, it is necessary to introduce additional

parameters (such as J2) to take into account the fact that the body is not spherical and not homogenous. These parameters drive the orbit evolution in the medium/long terms.

- In the case there is a need of phasing the spacecraft with the body (such as geostationary spacecraft), the knowledge of the body rotation parameters is necessary.

- Appendixes 2 & 3 give engineering parameters for moon and Mars missions - Appendix 4 gives the definition of the Lagrange points which remain fixed

compared the 2 bodies driving the gravity field.


1.2 Geometrical description of the orbit in its plane In the case of an elliptical orbit around a body, the key parameters to describe it geometrically in its own plane are (see drawing 1): - the eccentricity (e) which describes the shape, - the semi major axis (a) which describes the length. Two interesting points of an elliptical orbit are: - the apoapsis (also called apogee for Earth orbit): most distant point from

the body, Ra = a * (1+e) - the periapsis (also called perigee for Earth orbit): most close point

from the body, Rp = a * (1-e) The eccentricity is given by: e = (Ra-Rp)/(Ra+Rp) The line defined by the apoapsis and periapsis is called the apsides line.

Ra Rp

2 c

2 b

r

2 a

ν

Φ

V

a p o a p s i s

peri aps i s


1.3 Positioning of the orbit compared to the body The next step is to characterise the orbit in respect to the body references thanks to the following parameters (see drawing 2):

- the inclination (i) compared to the body reference plane, - the right ascension (called Ω) of the ascending node defining the longitude of the orbit point which is also in the body reference plane and corresponds to a crossing from South to North. The line defined by the descending and ascending node is called the lines of nodes, - the argument of the periapsis, (called ω) which defines the position of the periapsis from the nodes line.

Periapsis direction

Ω

Vernal equinox direction Line of

nodes

ω

i

ν0

X

Y

ZNormal to orbit plane


1.4 Positioning of the spacecraft on the orbit & dynamic aspects The position of a spacecraft on the orbit is then defined by the true anomaly (ν) and the distance from the body is:

r = a * (1 – e2

) / (1 + e * cos(ν))

Addressing the dynamic aspects is quite complex since there is no explicit relations between the time and the true anomaly. To make it easier, it is desirable to start computation with the true anomaly and then introduce the following successive parameters (all angles in radians): Eccentric anomaly (Ea) given by cos(Ea) = (e + cos(ν)) / (1 + e * cos(ν)) or also

tang(Ea/2) = tang(ν/2) * √(1-e)/(1+e) Mean anomaly: Ma = Ea - e * sin(Ea)

Time is then given by: t = to + (Ma - Mao) *√ (a3/µ)

The velocity modulus at one given point on the orbit is given by: V

2 = 2 * µ * (1/r - 1/(2*a))

Remarks - The period of the elliptical orbit is given by: T

2 = 4 * π2

* a3 / µ

- The velocity modulus at the 2 apside points which are very interesting for

performing manoeuvres are (see § 2):

velocity at the apoapsis: Va2 = 2 * µ * (1 / Ra - 1/ (2*a))

velocity at the periapsis: Vp2 = 2 * µ * (1 / Rp - 1 / (2*a))

- The velocity slope (angle Φ in § 1.2 figure) is given by: tang(Φ) = e * sin(ν) / (1 + e*cos(ν))


2. DELTA V COMPUTATION A spacecraft orbit around a given body is defined by 6 parameters (position: 3 coordinates and velocity: 3 coordinates). So, to change the orbit characteristics, it is sufficient to change the velocity (modulus and/or direction), thanks to a propulsive manoeuvre (that is why we refer to "delta V" for describing an orbit manoeuvre). As a consequence, the initial and final orbits must have at least one common point which is the point where the manoeuvre is performed. In practise, several manoeuvres are necessary to reach the operational orbit from the launcher injection orbit . Optimisation (minimum delta V) shows that for changing the semi major axis (and also the orbit period), impulsive manoeuvres have to be done at the apoapsis or periapsis (Hohmann manoeuvres). Indeed, to modify the apoapsis altitude, an impulsive manoeuvre will be performed at the periapsis. In the same manner, to modify the periapsis altitude, a manoeuvre will be performed at the apoapsis. An increase of the velocity at one of the apsides gives an increase of the altitude to the opposite apside. For co-planar orbits, the computation is very easy: if Vi is the velocity at the considered apoapsis of the initial orbit and Vf is the velocity at the same place of the final orbit (ref. § 1.4 for computation), delta V = Vf-Vi since the two velocities have the same direction. In a more general case, the initial and final orbits are not in the same plane (Di = difference of inclination) and the previous relation becomes:

(delta V)2 = Vi

2 + Vf

2 - 2 * Vi * Vf * cos (Di)

Example ARIANE injects communications satellites in a Geostationary Transfer Orbit (GTO) defined by: perigee altitude: 300 km, apogee altitude: 36000 km, inclination: 7 deg and ω # 180 deg (means that the ascending node is the apogee).


The final operational orbit is a circular orbit with an altitude of 36000 km and an inclination of 0 deg. Thus, to reach this final orbit from the GTO, a manoeuvre is required at the apogee to increase the perigee altitude and decrease the inclination. Computation gives the following values: initial velocity at the apogee: 1600 m/s final required velocity: 3067 m/s delta V to provide: 1491 m/s. Remark Planetary gravity assist manoeuvre changes the direction of the spacecraft relative velocity, but not its modulus (however, the heliocentric velocity modulus is changed !), with very limited propellant consumption (only for navigation).

Relative velocity at departure

Spacecraft velocity / Sun

at arrival

θ

Relative velocity at arrival

Planet velocity / Sun

Planet velocity / Sun

Spacecraft velocity / Sun at departure


3. PROPELLANT(S) MASS The required mass of propellant for performing a manoeuvre modifying the orbit is given by the following formulas:

Ma = Mb * exp ( - dV / ( Si * g ) ) Mp = Mb - Ma

where

Ma = Mass of the spacecraft after the manoeuvre Mb = Mass of the spacecraft spacecraft before the manoeuvre Mp = Mass of propellants dV = Amplitude of the manoeuvre (m/s) g = standard free fall = 9.81 m/s

2

Si = Specific impulse of the used spacecraft propellant (s) * liquid bipropellants (MMH+N2O4) Si # 310 s (the most generally used technique) * liquid monopropellant (hydrazine) Si # 220 s (only for missions requiring a small total dV) * solid propulsion (powder) Si # 290 s (only for missions requiring a single fixed dV manoeuvre) * electrical propulsion > 3000 s (but needs kW !) Remark You can compute that a total delta V of 2000 m/s requires to have about 50 % of the initial spacecraft mass devoted to conventional propellants (Si = 300s).


4. SOLAR ARRAY The solar array is used to provide power during sun shining conditions at spacecraft level. The required surface and mass of the solar array are given by the following relations:

Ssa = P / Sf / Ceff / Cra Msa =P / Mesa

Ssa = Surface of the solar array (m

2)

P = Power required at spacecraft level (W) Sf = Solar flux # 1350 W/m

2 @ 1 A.U. (150 000 000 km) from the sun

Ceff = Cells efficiency # 20 % (depends on the cell type, temperature and the

received radiation dose) Cra = Coverage ratio # 0.9 (the solar array surface has about 10% of its surface not covered by cells due to attachment points, connectors, wires …) Msa = Mass of the solar array (kg) Mesa = Massic efficiency of the solar array # 30 W/kg (@ 1 A.U.) Remark For distances greater than # 5 A.U., Radio-isotopic Thermal Generators (RTG) have to be used due to the too low level of the solar flux.


5. BATTERIES Rechargeable batteries are used to provide energy during sun shade conditions at spacecraft level (eclipse) or specific mission phases (before solar array deployment or phases such as manoeuvre without sun orientation …). They have to be recharged after use. The required mass of the batteries is given by the following relation:

Mb = E / Me / Dod Mb = Mass of the batteries (kg) E = Energy required at spacecraft level (W.h) = required power * duration of the batteries supply phase Dod = Depth of discharge (to have a long lifetime, the batteries are discharged only at a fraction of their accumulated energy) # 0.6 for NiCd batteries (limited demands) # 0.8 for NiH2 batteries (large demands) Me = massic energy # 40 W.h/kg (NiCd) # 60 W.h/kg (NiH2) Remark In some specific case (short duration missions …), primary batteries (not rechargeable) may be used. Their mean massic energy is 300 to 500 W.h/kg.


6. THERMAL RADIATORS Usually, the main part of a spacecraft external surface is covered by « super insulation » blankets decoupling the spacecraft from its external environment. However, in order to evacuate the heat dissipated by the spacecraft instruments and equipments (they mainly transform the consumed electrical power into heat), some parts of the surface are covered by thermal radiators which dissipate the energy by radiative exchange with the visible part of space. Of course, the areas which do not see the sun at all (or at least do not face the sun) are selected. The formula used for sizing the surface of these radiators is the following one:

Srad * Rf = Srad * (Afs + Afp *Vf ) + Qh

Srad = Surface of the radiators

Rf = Radiative flux exchange (W/m2) = ε * σ * T4 , with

ε= emissivity of the radiator # 0.9

σ = Stephan Boltzman constant = 5.67 10-8

W/m2/K

4

T = required temperature (K) Afs = Absorbed flux from the sun (W/m

2)

Radiator in sun shade => Afs = 0

Radiator in sun light => Afs = α * Sf * cos(θ) , with

α = absorptivity # 0.3 Sf = Solar flux # 1350 W/m

2 on the Earth orbit

θ = angle between the normal of the radiator and the direction of the sun beams

Afp = Absorbed flux from the planet (if any) = ε * planet IR flux + α * reflected Solar Flux

IR flux from Earth = 220 W/m2

Reflected solar flux from Earth = 0.3 * Sf W/m2 (over the day side)

Vf = View factor function of the altitude and the orientation of the radiator compared to the planet (< 1) Qh = Quantity of heat to be dissipated


7. SPACECRAFT ANTENNA In addition to its size, an antenna is characterised by two key linked parameters: its gain and its half power beam width. The antenna maximum gain characterising its capacity to concentrate the energy is given by:

Gmax (dB) # 17.9 + 20 * log(Da) + 20 log(F) F = frequency of the link (in GHz) Da = diameter of the antenna (m) Remark: it is usually assumed that the link may be done within the half power beam (see below), so for computations we consider Gmax – 3 dB. The antenna half power beam width (cone angle across which the antenna gain is ≥ 50 % (= 3 dB) of the maximum gain along its symmetry axis) is given by the following relation:

θ3 (deg) = 21 / (F * Da)

F = frequency of the link (in GHz) Da = diameter of the antenna (m)


8. LINK BUDGET (telemetry and telecommand rates) Radio-frequency links are used to send orders from the Earth to the spacecraft (telecommand or uplink) or from the spacecraft to the Earth (telemetry or down link). A high frequency carrier is modulated by a signal containing the information. Typically 2 GHz (uplink and sometimes down link) or 8 GHz (down link) are used for scientific satellites. The quality of a radio frequency link is defined by the Carrier power to Noise density ratio (C/No) which is given by the following relation (in dB):

C/No = EIRP + Fsl + G/T – k EIRP = Emitted Isotropic Radiated Power which represents the effective power emitted by the emitter (in dBW, includes the amplifier output, emitting losses and the antenna gain) Fsl = Free space loss = 20 log (wavelength/(4 * π * path length)) G/T = the figure of merit of the receiver (in dB/K), which includes the receiving antenna gain (depends of its size, see § 7) and the temperature noise of the receiver system (# 1300 K for receiving system at spacecraft level) k = Boltzmann constant = - 228,6 dBW/Hz/K Remark In some case, additional losses (attenuation) are considered for the Earth atmosphere traverse (some tenth of dB to some dB, depending of the wavelength). The Carrier power has to be decreased by the various losses due to the modulation/demodulation and the remaining power is shared between the transmitted bits. Each bit must have a given quality (measured by a ratio Energy per bit / Noise density) to meet the specified bit error rate.


The associated data rate is thus given by the following relation (in dB):

C/No = Dr + Ml + E/No + Margin Dr = data rate Ml = modulation/demodulation loss (depends of the used modulation technique, usually phase shift keying, sometimes frequency shift keying) Ml # 4 dB for telecommand, Ml # 1 dB for telemetry Eb/No = ratio of the required Energy per bit to Noise density; linked with the specified bit error rate (typically 10

-6 for telecommand and 10

-4

for telemetry), the used modulation technique and the used error correction code Eb/No # 11 dBHz for telecommand Eb/No # 4 dBHz for telemetry Margin is introduced to overcome the various uncertainties and simplifications made Margin > 6 dB for telecommand Margin > 3 dB for telemetry Appendix 1 gives an example of link budgets with associated inputs.


APPENDIX 1: LINK BUDGET EXAMPLE

---

Inputs

Spacecraft Antenna characteristics UPLINK DOWNLINK

Frequenc y (GHz) 2 8

Diameter (m) 1 1

Effic ienc y 0,7 0,7

Theta3 (deg) 10,5 2,625

Max Ga in (dB) 24,9 36,9

Edge of Coverage Ga in (dB) 21,9 33,9

Ground Segment Stations Characteristics UPLINK DOWNLINK

Classic a l sta tion (for Ea rth orb iting spac ec ra ft)

EIRP (dBW) 58 @ 2 GHzG/ T (dB/ K) 12 @ 2 GHzG/ T (dB/ K) 22 @ 8 GHz

Deep spac e sta tion (for fa r spac ec ra ft)

Kourou (15 m) EIRP (dBW) 81 @ 2 GHzEIRP (dBW) 81 @ 8 GHzG/ T (dB/ K) 30 @ 2 GHzG/ T (dB/ K) 38 @ 8 GHz

New Norc ia (35 m) EIRP (dBW) 97 @ 2 GHzEIRP (dBW) 107 @ 8 GHzG/ T (dB/ K) 37,5 @ 2 GHzG/ T (dB/ K) 50 @ 8 GHz

DSN (70 m) EIRP (dBW) 117 @ 2 GHzEIRP (dBW) 115 @ 8 GHzG/ T (dB/ K) 47 @ 2 GHzG/ T (dB/ K) 56,5 @ 8 GHz


OUTPUT (data rates)

Link Budget

UP LINK DOWN LINK

Frequenc y (GHz) 2 Frequenc y (GHz) 8

Distanc e (km) 180000000 Distanc e (km) 180000000

Required da ta ra te (kb / s) 1 Required da ta ra te (kb / s) 6

Required b it error ra te 1,00E-06 Required b it error ra te 1,00E-04

Spac ec ra ft Amp lifier Output (W 10

Ground Sta tion PIRE (dBW) 97 Spac ec ra ft Amp l. Output (dBW) 10,0

Free Spac e Losses (dB) -263,6 On boa rd loss (dB) -1

Spac ec ra ft Antenna Ga in (dB) 21,9 Spac ec ra ft Antenna Ga in (dB) 33,9

Rec eiver noise tempera ture (dBK 31,1 Free Spac e Losses (dB) -275,6

On boa rd losses (dB) -0,5 Ground St.Figure of Merit (dB/ K) 50

Boltzmann Constant (dBW/ Hz/ K) -228,6 Boltzmann Constant (dBW/ Hz/ K) -228,6

Tota l C/ No (dBHz) 52,3 Tota l C/ No (dBHz) 45,9

Modula tion loss (dB) -4 Modula tion loss (dB) -1

Da ta ra te (dBHz) 30,0 Da ta ra te (dBHz) 37,8

Required E/ No (dBHz) 11 Required E/ No (dBHz) 4

Marg in (dBHz) 7,3 Marg in (dBHz) 3,1


APPENDIX 2: MOON ENGINEERING PARAMETERS ----------

Body characteristics

Mean diameter 3476 km

Mass 7.353 1022

kg

Rotation period 27.3 days

Surface Gravity 1.62 m/s2

Escape velocity 2.4 km/s

J2 - 1.996 10-4

---

Orbit characteristics

Revolution period 27.3 days

Inclination of orbit plane / ecliptic plane 5.2 deg

Mean distance to the Earth 384 400 km Precession period 18.6 years

Inclination / Earth equator between 18.5 and 28.5 deg (due to precession – cf drawings hereafter)


ROTATION OF THE MOON ORBIT PLANE

4.65 years later

4.65 years later

Moon orbit

Ecliptic

23.5 deg

5 deg Equator

Rotation axis of the moon orbit plane

23.5 deg5 deg


Ecliptic Equator

23.5 deg

5 deg

EquatorEcliptic


Moon orbit


APPENDIX 3: MARS ENGINEERING PARAMETERS ----------

Body characteristics

Mean diameter 6758 km

Mass 6.418 1023

kg

Rotation period 24h 37mn 23s

Inclination of equator plane / orbit plane 25 deg

Surface Gravity 3.72 m/s2

Escape velocity 5.0 km/s

J2 1.96 10-3

---

Orbit characteristics

Revolution period 686.98 days

Inclination of orbit plane / ecliptic plane 1.85 deg

Eccentricity 0.0934

Semi major axis 227.94 106 km

Argument of the periapsis 253 deg


APPENDIX 4: LAGRANGE POINTS ----------




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ORBITS & V AND SOME SPACECRAFT PARAMETERS · Alpbach 07 Rules of thumb 1 SOME RULES OF THUMB FOR...

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