Time measurement: Solar Time

Meccanica 

del 

volo spaziale

Space system phases 

Sistemispaziali+ 

Meccanica Volo Spaziale 

Design and development 

Technology development 

Component procurement 

Manufacturing 

Assembly & Integration 

Verification & test 


Launch 


Operations 

Harness 

Thermal control 

Electrical 

Telecomms 

power 

Data handling 

Payload 

AOCS 

Structure 

Propulsion 

Mechanisms

Design Process 

Mission 

Requirements & 

Constraints 

Software 

Objectives 

Environment 

Lifetime 

Payload 

Reliability 

Schedule 

Technology 

Budget 

Attitude 

determination 

& control 

Mission 

analysis 

Thermal 

control 

Instruments 

Electrical 

power 

Dry 

mass 

Data 

handling 

Telemetry 

tracking & 

command 

Operations 

& ground 

systems 

Study 

Results 

S/C Design 

S/C Configuration 

Launcher 

Risk 

Cost 

Simulation 

Study Requirements 

Products 

Study Level 

Planning 

Resources 

Propellant 

mass 

Propulsion 

Structure 

Adapter 

Wet 

mass 

Launch 

mass 

Programmatics 

Options 

Conceptual model 

of mission & 

spacecraft 

design process

Trajectory classes 

Mission Analysis Trajectory and trajectory control design 

Meteosat 

Ulysses 

Olympus 

ERS 

Hipparcos 

Soho 

Huygens (Cassini) 

Spacelab 

Ariane 4 Ariane 5 

Envisat 

ECS 

International 

Space Station 

ISO 

Artemis 

Giotto

Mission phases: Earth Satellite Example 

Launch –Low Earth Orbit Phase (LEOP) -Transfer – Commissioning – 

Operations – De-commissioning - End of Life 

AGILE (04/07) 

Astro-rivelatore Gamma 

a Immagini Leggero

Mission phases: 

interplanetary transfer trajectory example 

Ulysses (10/90) 

Plane inclination 

variation:Gravity assist 

maneuver: 80.2 deg 

Mass at launch: 370 kg 

Payload mass: 55 kg 

Fuel mass: 33 kg 

Power: 285 W



Cassini-Huygens (10/97) 

Gravity Assist : Venus-Venus- 

Earth-Jupiter 

Launch Mass: 2523 kg 

Fuel Mass: 1027 kg 

Power: 640 W


interplanetary transfer trajectory 

example 

Messenger (08/04) 

Mass: 1100 kg 

Fuel mass 592 kg 

Payload mass 50 kg 

Power: 450 W



Messenger 1° GA Earth 

2005 

Messenger 2° GA Venus 

2006 

Messenger MOI 

2011

Trajectory design 

Rosetta (03/04) 

Launch Mass: 3000 kg 

Fuel mass 1800 kg 

Power: 850 W

Programma del Corso 

Sistemi di riferimento e misura del tempo 

Sistemi di riferimento: Eliocentrico-Eclittico; Geocentrico-Equatoriale; Topocentrico- 

Orizzontale. Fenomeno della precessione degli equinozi. Misura del tempo: tempo solare e 

tempo siderale. La data Giuliana. (Navigazione interplanetaria)


Meccanica Celeste 

Il problema degli n corpi. Potenziale gravitazionale. Il problema dei due corpi. Richiami di proprietà 

geometriche delle coniche. Legami sussistenti tra le costanti del moto e la geometria dell'orbita. Analisi 

energetica orbitale. Calcolo del periodo orbitale e leggi di Keplero. 

Orbite kepleriane 

Definizione dei sei elementi orbitali classici. Trasformazioni di coordinate tra i vari sistemi di riferimento. 

Calcolo dell'orbita dati gli stati iniziali. Tempo di volo: Equazione di Keplero. Metodi numerici per il 

calcolo dell'anomalia eccentrica. 

Facoltativo: Problema di Lambert e sua soluzione nel caso di orbita ellittica. Traiettorie interplanetarie. 

Sfere di influenza. Metodo delle coniche raccordate.


Manovre orbitali 

Manovre impulsive ad un impulso e a due impulsi (trasferimenti alla Hohmann e manovre biellittiche). 

Manovre impulsive a tre impulsi: coplanari e con cambiamento di piano orbitale. Trasferimento orbitale 

tra orbite ellittiche. Trasferimenti orbitali con impulso fissato. Finestra di lancio. Trasferimenti con 

spinte basse. Perdite di gravità. Manovre di rifasamento. 

Documento Acrobat


Progetto e dinamica dei lanciatori 

Dinamica impulsiva. Dinamica di un monostadio: lancio sulla verticale; lancio balistico; 

Ottimizzazione di un pluristadio. 

Facoltativo: dinamica bidimensionale di lancio

Programma del corso 

Perturbazioni orbitali 

Orbite particolari ottenute attraverso l’effetto perturbativo: sunsincrone, Molnya 

Moto di assetto di un satellite (facoltativo) 

Richiami di meccanica dei corpi rigidi. Terne di riferimento. Angoli di Eulero. 

Equazioni di Eulero. Stabilità delle rotazioni attorno agli assi principali di inerzia. 

Dinamica di satelliti a tre gradi di libertà attorno al baricentro. Calcolo del gradiente di 

gravità. Dinamica dei satelliti stabilizzati a gradiente di gravità.

Sistemi di riferimento e misura del tempo 

V.M.Blanco “Basic physic of the solar system”, Addison Wesley Ed. 

Dispense del Corso 

.Progetto e dinamica dei lanciatori 

K.J.Ball, G.F.Osborne, “Space Vehicle Dynamics”, Oxford Press 

Dispense del Corso 

Problemi di Meccanica Celeste 

M.H.Kaplan, Modern Spacecraft Dynamics and Control”, J.Wiley & Sons 

R.Bate,D.Mueller,J.White,”Fundamentals of astrodynamics”, Dover Pubbs, NY 

G.Mengali,”Meccanica del volo spaziale”, Ed.PLUS 

Orbite kepleriane 

H.D.Curtis, Orbital Mechanics for Engineering Students 

V.A.Chobotov, “Orbital Mechanics”, AIA education series 



Manovre orbitali 

H.D.Curtis, Orbital Mechanics for Engineering Students 


C.Brown,”Space Mission Design”,AIAA Education series 

Perturbazioni orbitali 


C.Brown,”Space Mission Design”,AIAA Education series 

Moto di assetto di un satellite 


W.Viesel, “Spaceflight Dynamics”, Mc Graw-Hill 

References

Exam rules 

Modalità di valutazione 

Tipologia delle prove 

Le prove necessarie ai fini della valutazione del corso consistono in: 

a) 3 compitini scritti ciascuno relativo ad un’area tematica affrontata nel corso e composto da alcuni esercizi numerici e 

domande teoriche 

b) interrogazione orale sul programma svolto 

I compiti scritti vengono svolti durante il corso, al termine del periodo di lezione/esercitazione inerenti all’argomento 

trattato; indicativamente: 

1. Prima metà doi Novembre: trasformazioni di coordinate 

2. Fine Dicembre: Meccanica Orbitale 

3. Fine Gennaio: lanciatori 

La valutazione dei compiti scritti è la seguente: 

Valutazione 

gravemente insufficiente 

insufficiente 

Sufficiente/discreto/buono/ottimo 

Voto 

E* 

E 

D/C/B/A 

Esito 

impedisce la partecipazione all’orale; richiede lo 

svolgimento in aula di uno scritto 

Richiede lo svolgimento a casa del tema del compito 

Consente la partecipazione all’orale il cui voto di 

partenza è definito dalla media dei voti acquisiti nei 

compiti 

• L’insufficienza grave in 3 scritti preclude l’ammissione all’orale e implica la re-iscrizione e frequenza del corso 

• L’insufficienza grave in 2 scritti richiede lo svolgimento in aula di uno scritto su tutto il programma svolto 

• L’insufficienza grave in 1 scritto implica lo svolgimento in aula di uno scritto relativo alle tematiche insufficienti 

Regole dell’esame 

Sono previsti appelli d’esame a Febbraio, Giugno/Luglio Settembre per sostenere la prova orale.

Reference systems

Reference systems 

Goal: to describe the state vector of an object in space 

Coordinate system: cartesian/spherical 

Coordinate 

Name 

Celestial 

Fixed with 

or respect 

to 

inertial space 

Centre 

Earth (GCI 

Geocentric 

Inertial) or 

body 

Z-axis 

X-axis or 

Ref.Point 

Application 

Celestial Pole Vernal Equinox orbit analysis, astronomy 

Ecliptic inertial space Sun/Earth Ecliptic Pole Vernal Equinox 

solar system orbits, planet 

ephemeris 

Planet-fixed Planet Planet Planet Pole 

Greenwich or 0 

meridian 

apparent satellite motion 

Perifocal Orbit Planet 

Angular 

momentum 

Eccentricity 

vector 

Orbit analysis 

Spacecraft 

fixed 

RPY, RθB, 

NTB 

Spacecraft 

s/c c.m. 

Dependent 

on the 

design 

Body 

(spacecraft 

or other) 

Spacecraft axis 

toward nadir 

Yaw in the nadir 

or R in the radial 

direction 

spacecraft axis in 

the direction of 

velocity 

perpendicular to 

nadir toward 

velocity 

position and orientation of 

instruments 

attitude manoeuvres, planet 

observation and encounter 

Main plane + reference axis to be defined 

Orbital 

Mechanics

Revolution: angular motion of the c.m around the Sun 

Reference systems: planets motion 

Rotation: angular motion of the planet around its own axis 

Revolution axis 

equator 

23°27’ 

Rotation axis 

Revolution axis 

equator 

Earth Orbit 

plane 

Venus 

Orbit plane 

Rotation axis 

177°

Rotation motion effects 

• daylight and night experienced 

• apparent daily motion of the Sun and night motion of the stars in the 

sky 

• polar bulginess 

• different escape velocity with latitude 

• deviations of objects moving along meridians because of the Coriolis 

apparent acceleration, depending on the latitude: 

– null at poles, maximum at equator (N S displacement = boreal 

hemisphere West deviation)

Revolution motion effects 

• Seasons existence (inclined rotational axis) 

• Different height of the Sun above the horizon along the year 

• Different rising and sunset points on the horizon along the year (at 

East-West at Equinoxes; toward North in summer and South in winter 

for a boreal observer) 

• Apparent rotation of the sky 

• Apparent annual motion of the Sun (Zodiac band rotation)

planets motion: relevant events 

Two particular illumination conditions can be identified: 

Equinox: Sun rays are perpendicular to the Earth (planet) equator; daylight 

and night duration is equal all over the planet 

Solstice: Sun rays are oblique to the surface, different latitudes experience 

different daylight/night duration

planets motion: relevant events 

Solstice line 

Equinoxes line

planets motion: relevant events\points 

linea 

equinozi 

Perielio 

3 gennaio 

Afelio 

7 luglio 

linea 

apsidi 

linea 

solstizi 

11°

Reference systems: Astronomy 

Spherical coordinates: 2 angular+1 radial information r=[r;ξ,ζ] 

Rectangular coordinates: 3 components information r=[r x 

; r y 

;r z 

] 

Spherical coordinates 

Radial information 

far objects: r∞ 

spacecraft: r=|r| 

origin: Sun ; Earth ⊕; Planet 

Celestial Sphere 

r 

Reference planes 

•Planet’s orbit: defined by the planet’s revolutionary angular velocity vector 

•Planet’s equator: Defined by the planet’s rotational angular velocity vector 

In-plane reference vector 

•Inertial e.g. Aries point 

•Non-Inertial e.g. Prime meridian

Reference systems: Aries point 

Eliocentric reference 

Normal to the Earth Orbit = Celestial North Pole 

Earth revolution 

motion 

Summer solstice 

(≈ 21/06) 

γ 

Aries point 

Autumn Equinox 

(≈ 21/09) 

Ω 

Vernal Equinox (≈ 21/03) 

Winter solstice (≈ 21/12) 

Earth Orbit 

Geocentric reference 

Celestial North Pole 

Earth revolution motion 

Winter solstice (≈ 21/12) 

Aries point 

γ 

Vernal Equinox 

(≈ 21/03) 

Ω 

Autumn Equinox (≈ 21/09) 

Summer solstice (≈ 21/06) 

Ecliptic

Reference systems: Aries point 

Alternative definitions: γ point (Aries point) 

•Earth-to-Sun alignment points to the Aries Constellation (Vernal Equinox) 

• Line that has: 

•direction: E∩Π E= ecliptic plane; Π=Celestial Equator plane; 

•versus: positive whenever the Sun rises from the bottom to the top of the Π 

plane 

Ecliptic North 

Pole (K) 23. 

5° 

Boreal Pole 

(PNC) 

Ecliptic 

North Pole 

23.5° 

Boreal Pole 

(PNC) 

Earth Orbit 

Sun Apparent Orbit 

Austral Pole 

γ 

Celestial 

sphere 

Celestial 

Equator 

Austral Pole 

γ 

Celestial 

Equator 

Celestial 

Sphere 

Eliocentric reference 

Geocentric reference 

Ecliptic plane: plane of the apparent motion of the Sun around the Earth in one year

Reference systems: Spherical 

Ecliptic Reference 

geocentric – [K γ] 

Ecliptic North Pole K 

E 

β 

Celestial 

Sphere 

Ecliptic 

λ 

γ 

Ecliptic North Pole: ⊥E ∩Celestial Sphere 

Coordinates: Ecliptic Longitude; Ecliptic latitude: -90°


Celestial Equatorial Reference 

geocentric – [γ PNC] 

Celestial North Pole (Boreal Pole) PNC 

Celestial Sphere apparent rotation 

γ 

E 

α 

δ 

Celestial 

Sphere 

Celestial Equator 

Austral Pole 

Celestial North Pole: Celestial Sphere ∩ Earth rotation axis 

Coordinates: Right Ascension; Declination: 0°

Body Fixed Reference 

geocentric – [prime meridian - PNC] 

Northern Celestial Pole 

Reference systems: Geographical 

Relative Celetial sphere rotation 

Prime 

Meridian 

E 

L 

φ 

Celestial 

Sphere 

Southern celestial Pole 



Coordinates: Longitude; latitude: -12 h


Horizontal (topocentric) Reference 

Celestial North 

Pole PNC 

Z zenith-local vertical 


Local horizon 

Celestial 

Sphere


geocentric –[ENZ] 

Topocentric Reference 

Celestial sphere 

relative rotation 

Zenith 

Zenith 

PNC 

ϕ= latitude 

East 

Az 

Celestial 

sphere 

North 

Celestial 

equator 

Horizon 

West 

Celestial 

sphere 

South 

N 

E 

Nadir 

h 

Local horizon 

Zenith≅ Geometric normal to the Earth Ellipsoid 

Nord: Z - Pb ∩ Horizon plane = N (Pb∩horizon plane


Topocentric Reference


Time dependence:examples 

Vernal Equinox 

Sun Coordinates 

δ=0°; α=0° 

Summer Solstice 


δ=23.5°; α=90°


Time dependence:examples 

Annual Sun trace on the celestial sphere 

Winter Solstice 


δ=-23.5°; α=270°

Time measurement: Solar Time 

Terrestrial reference: Prime meridian Long=0° 

Different observer’s time measurement geometric reference frame rotation 

Polo Boreale 

Zenit 

N 

E 

ϕ = latitudine 

Long 

Σ 

W 

sfera 

celeste 

Prime 

meridian 

Σgreenwich 

S 

Long East0


Galactic Reference 

heliocentric – [galactic center –galactic plane (69.2°)]

Reference systems: Rectangular- ECI 

geocentric – [I;J;K] 


Coordinates: 

[X;Y;Z] I≅γ; K≅PNC 

Notes: 

Time and observer independent 

On wide time windows γ=γ(t) ⇐Equinoxes precession and nutation

Reference systems: Rect.Heliocentric-ICRF 

Heliocentric – [I;J;K] 

International Celestial Reference 

Frame (1998 IAU)standard 

reference system 

Centre: Solar system baricentre 

Ecliptic Pole: Celestial Sphere ∩ Normal to Ecliptic 

Coordinates: 

[X;Y;Z] 

I≅γ; K≅Ecl.Pole 

Notes: 

Time and observer independent 

On wide time windows γ=γ(t) ⇐Equinoxes precession and nutation

Reference systems: Rect. ECEF-ITRF 

Geocentric – [Greenwich;J;North Pole] 

Because of plate tectonic motion (cm/yr) Greenwich Meridian is not planetfixed 

therefore the ITRF is used 

International Terrestrial Reference Frame standard reference system for 

accurate orbit determination 

Coordinates: [X;Y;Z]

Reference systems: Horizontal Rectangular 

Horizontal Reference

Perturbations 

Precession: Solar perturbation due to the Earth oblateness, Aries point γ moves 

counterclockwise (T=26000 years) 

Nutation:Lunar perturbation (T=18.6 years ~ Metonic cycle) 

⇒γ= γ(t) 

Periodical motion of the Earth rotational axis around the angular momentum vector 

of the ecliptic

Perturbations effects 

•All significant points rotate: 

equinoxes, solstices, perihelion, 

aphelion 

•Celestial Poles changes 

•Zodiac band rotation: 

constellations are visible earlier 

in time precession 

•Season chances: 20 min earlier 

every year 

The star indicating the 

Geographic North changes

Time measurement 

[..]The main purpose of time is to define with precision the moment of a 

phenomenon [..]. 

Time occurrence of the event= EPOCH (date) 

The definition of time interval is needed. 

Time interval definition asks for: 

• Origin to count from Begin of Christian Era 

• repeatable time interval based on some physical phenomenon 

At the time being four scales are defined: 

1. Sideral time 

2. Solar time 

3. Dynamical time 

4. Atomic time 

Earth rotation based

Time measurement:Astronomical clocks 

Significant intervals: 

Rotation of the Earth 

Revolution of the Earth 

Revolution of the Moon around the Earth 

¼ Moon cycle 

day 

year 

month 

week

Time measurement: Astronomical clocks 

The time interval definition and measurement are accomplished by taking into account the 

relative position of objects whose motion is known: 

For a terrestrial observer: 

Reference may be: the Sun (☼) 

the fixed stars 

North celestial pole 

Zenit 

ϕ = latitude 

Unit (day) = time span between 2 consecutive 

passages of the reference body on the observer 

meridian (celestial equatorial reference) 

N 

E 

Celestial 

sphere 

W 

S 

Σ 

t =time 

angle 

Time angle = celestial equator arc measured 

counterclockwise starting from the observer 

meridian (Σ point) up to the meridian passing 

through the observed object

Time measurement: Astronomical clocks 

Solar versus Sidereal time 

By changing the observed body the interval time changes 

not uniform definition of day


Solar Time unit: reference target = ☼ 

Apparent Solar Day: time interval between two consecutive passages of the real Sun 

on the local meridian 

Local Apparent Time(LAT): time angle of the real Sun anti-meridian 

Main Issues 

Earth orbit eccentricity ⇒ variable Solar Day duration 

Earth Orbit inclination ⇒ variable Solar Day duration

Time measurement:Solar Time 

GoalUniform Solar Time unit definition 

Mean Sun Definition: 

•Circular Orbit (e=0) 

•Zero inclination of Earth orbit on the Equator (i=0°) 

Mean Solar Day= time interval between two consecutive passages of the Mean Sun 

on the local meridian 

(Proposed by Newcomb in 1895) 

Mean Solar 

Day 

Local Mean Time(LMT): local hour angle of the mean Sun anti-meridian 

Universal Time(UT): Greenwich hour angle of the mean Sun anti-meridian

Time measurement 

Giorno solare: corpo di riferimento = ☼ 

Giorno solare vero: tempo intercorso tra due 

passaggi consecutivi del sole vero sul medesimo 

meridiano 

Local Apparent Time(LAT): angolo 

orario dell’antimeridiano 

del sole vero 

Ellitticità orbita terrestre ⇒ durata 

giorno solare variabile 

Inclinazione orbita terrestre ⇒ durata 

giorno solare variabile 

Giorno solare medio= tempo intercorso tra due 

passaggi consecutivi del sole medio sul medesimo 

meridiano 

1 giorno 

solare 

medio 

moto medio del sole: 

•orbita circolare (e=0) 

•inclinazione sull’eq.celeste nulla (i=0°) 

Local Mean Time(LMT): angolo 

orario dell’antimeridiano del 

sole medio


Mean Solar Day= constant duration=24h 

Observed Sun (apparent)≠measured Sun (mean) ⇒ Equation of time (E.o.T.) 

LAT-LMT=E 

E= Actual sun position error 

E ecc =error from e≠0 

Actual motion≅ mean motion at: 

apocentre (July) and pericentre (January) 

⇒T=1 year 

~Pericentre + =v↑ LAT0 

~Apocentre=v↓ LAT >LMT ⇒E>0


Actual motion≅ mean motion at: 

equinoxes(March-September) & solstices 

⇒T=0.5 years 

E incl =error from i≠0° 

Equinox SVLMT E>0 

solstice SV>SM ⇒LAT


Effects sum= cumulative error: E=E ecc + E incl 

Listed in the Ephemeredes tables


Reference UT=00 h 00 m 00 s


Terrestrial reference: Prime meridian Long=0° 

Different observer’s time measurement geometric reference frame rotation 

Polo Boreale 

Zenit1 

Zenit2 

N 

E 

ϕ = latitudine 

W 

sfera 

celeste 

S 

Σ 

angoli 

orari 

Long East0 

LMT Greenwich =GMT = Universal Time (UT)


Solar Time: corrections 

Error (E): measured according to an Earth meridian = Greenwich meridian at 00h 

LMT greenwich 

=GMT=UT (universal time) 

Greenwich meridian-observer meridian link: terrestrial longitude (L) 

Therefore: 

• Goal: mean angular position of the Sun into actual (apparent) sun position conversion: 

• Procedure for error computation: 

LMT→E → LAT 

1. Transfer to Greenwich: 

E(LMT) →E(GMT) ⇒ LMT →L →GMT 

2. Listed time translation: 

E 

E: E=E| 

h i 

00 UT 

se GMT≠00h E 

→lineear interpolation on 24h 

E i+ 

1 

00hUT, g i 

00hUT, g i+1 

a h b m c sUT , g i 

days 

Ei 1 - 

E (GMT) = 

+ Ei 

(GMT - GMT i ) + 

24 

h 

Ei


Analemma: 

Apparent Sun Trajectory with respect to the mean Sun within a year

Analemmas 

Milano 

Bergamo


Civil Time: LCT 

LMT ≠ for different longitude on Earth→rule = Earth sphere partitioning into 24 orange 

slices 

1 slice =1h=15° 

Reference time in the slice: LMT of the central meridian 

Reference Origin: Greenwich (GB) 

GMT-LMT=L EST green L0 

From: LCT to LMT: LCT-LMT=∆L 

Ex: Etna (central meridian): LMT=LCT=16.00h 

Milan LCT=16.00h ≠ LMT ∆L= 15°-9°=6°=24min 

LMT=15h 26min


24 slices 1h each (15 deg) 

Each location assumes its fuse central meridian time Local Civil Time 

LCT+∆long = LMT

Time measurement: CivilTime 

CET = Central Europe time: UT+1h 

CEST = Central Europe Saving time: UT+2h 

EET = Eastern European time: UT+2h 

EEST = Eastern European Saving time: UT+3h 

MST = Moskow Standard Time: UT+3h 

MSD = Moskow Saving Daylight: UT+4h 

EST = Eastern Standard time: UT-5h 

EDT = Eastern Daylight saving time UT-4h 

PST = Pacific Standard time: UT-8h 

PDT = Pacific Daylight saving time UT-7h 

… … … …

Time measurement: Sideral Time 

Sideral Time: reference body = Stars 

∆t≈1 

° 

Sideral day: 

1 sideral day 

Time span betweeen two consecutive passages of the vernal point on the local meridian: 

23h 56m =360° 

LST=α star +t star 

Local Sideral Time (LST): 

Hour angle of γ 

Sideral Day

Time measurement: sideral time 

Reference UT=00 h 00 m 00 s

Time measurement: Calendars 

• Calendar date : Gregorian time representation DD/MM/YYYY 

uncomfortable for computation 

• Julian date: no. of days since 01/01/4713 B.C. , noon 

(the Julian date for January 1, 2000, 12.00h p.m. is 2451544.5) 

•Modified Julian Date: no.of days since November 17,1858 (-2400000.5 JD) 

midnight January 1, 2000, 00.00h am=51544 days 

•Modified Julian Date 2000(MJD2000): no. of days since midnight January 1,2000 

•Common time reference dates are: 

• 1950 coordinates mean value at epoch 01/01/1950 

• 2000 coordinates mean value at epoch 01/01/2000 

Gregorian to JD 

Y + (M + 9) /12 275 M 

J = 367 Y − 7 

+ + D + 1721013 

4 

9 

.5

Time measurement: Years 

Year: time interval between two consecutive passages of the Earth on a same 

point on its orbit 

Tropic Year: vernal point passage = 365.2422 solar days 

366.2422 Sidereal days 

Anomalistic year: pericentre passage: 365.2564 solar days 

Sideral year: fixed star alignment: 365.2596 solar days 

Tropical and anomalistic years depend on time dependent reference frames

Time measurement: Calendars 

Calendars: long period reference 

-Gregorian calendar: ex. January 31, 2007 Wednesday 

Period: year 

reference: tropic year (365,2422dd~365dd) 

decimal correction: +1 day every 4 years: leap: 0.25x4 

excess: .25-.2422=0.0078 d/yr~0.01dd/yr 

⇒ every 100>1d ∃ leap 

due: 0.0022 d/sec 

⇒ every 400 ∃ leap 

-Julian Calendar: (introduced by Giulio Cesare in 45 BC up to 1500) 

Period: year 

Reference: tropic year (365.25)

Reference systems: summary

Spherical reference systems transformation: basics 

Spherical trigonometry 

Hp: arcs= geodetics = maximum circle arcs 

Sine theorem 

sin b 

sin Bˆ 

sin b 

sin Bˆ 

= 

= 

sin c 

sin Ĉ 

sin a 

sin Â 

Cosine theorem 

cos c = cos a ⋅cos b + sin a ⋅sin b ⋅cos Ĉ 

cos a = cos b ⋅cos c + sin b ⋅sin c⋅cos Â 

cos Cˆ =−cos Aˆ ⋅ cos Bˆ 

+ sin Aˆ 

⋅sin Bˆ 

⋅cos c 

cos Aˆ 

=−cos Bˆ ⋅ cos Cˆ 

+ sin Bˆ ⋅sin cˆ 

⋅cos a 

Cotangent theorem 

sin a ⋅ctgb 

= cos a ⋅cos Ĉ + sin Ĉ⋅ctgBˆ 

sin c⋅ctga 

= cos c⋅cos Bˆ + sin Bˆ ⋅ctgÂ

Rectangular reference systems transformation: basics 

⎡1 0 0 ⎤ 

ROT1( ϑ ) = 

⎢ 

0 cos sin 

⎥ 

⎢ 

ϑ ϑ 

⎥ 

⎢⎣ 

0 −sinϑ cosϑ⎥⎦ 

⎡cos ϑ 0 −sin 

ϑ⎤ 

ROT2( ϑ ) = 

⎢ 

0 1 0 

⎥ 

⎢ 

⎥ 

⎢⎣ 

sin ϑ 0 cos ϑ ⎥⎦ 

⎡ cos ϑ sin ϑ 0⎤ 

ROT3( ϑ ) = 

⎢ 

sin cos 0 

⎥ 

⎢ 

− ϑ ϑ 

⎥ 

⎢⎣ 

0 0 1⎥⎦

Reference systems transformation: Rectangular 

ECEF 

IJK 

ECIECEF 

r = ROT3( ϑ )r 

GST 

r = ROT3( −ϑ )r 

GST 

IJK 

ECEF 

ECEFTopocentric ENZ 

r = ROT2( − 90 °+ϕ) ROT3( λ) 

r 

ENZ 

LST 

ECEF 

= ROT2( − 90 °+ϕ) ROT3( ϑ = λ+ ϑ ) r 

GST 

IJK 

ECITopocentric ENZ 

ρ= 

r 

ENZ 

IJK 

−r 

site−IJK 

ρ = ROT2(-90°+ ϕ)ROT3( λ+ 

ϑ ) ρ 

GST 

IJK

Reference systems transformation: Spherical/Rectangular 

Geocentric inertialECEF 

r = ROT3( ϑ )r 

ECEF 

-1 ⎛rJ 

⎞ 

λ=tan 

⎜ ⎟ 

⎝ rI 

⎠ 

ϕ= sin 

⎛r 

⎜ r 

⎝ 

−1 k 

⎞ 

⎟ 

⎠ 

GST 

IJK 

⎡rcosϕcosλ⎤ ⎡rI 

⎤ 

r = 

⎢ 

rcosϕsinλ ⎥ 

= 

⎢ 

r 

⎥ 

ECEF 

⎢ ⎥ ⎢ J ⎥ 

⎢⎣ rsinϕ 

⎥⎦ ⎢⎣r 

K 

⎥⎦ 

Geocentric Inertial (equatorial)ECI 

⎡rcosδcosα⎤ ⎡rI 

⎤ 

r = 

⎢ 

rcosδsinα ⎥ 

= 

⎢ 

r 

⎥ 

IJK 

⎢ ⎥ ⎢ J ⎥ 

⎢⎣ rsinδ 

⎥⎦ ⎢⎣r 

K 

⎥⎦ 

2 2 2 -1 ⎛rK 

⎞ 

r = r 

ECEF I + rJ + rK 

δ=sin 

⎜ ⎟ 

⎝ r ⎠ 

⎛r 

-1 J 

λ=α−ϑGST 

α=tan 

⎜ ⎟ 

rI 

⎝ 

⎞ 

⎠

Reference systems transformation: Spherical/Rectangular 

HorizontalENZ 

ρ 

ENZ 

( ) 

( ) 

sin ( el) 

⎡ρcos el sin Az ⎤ 

⎢ 

⎥ 

= ⎢ 

ρcos el cos Az ⎥ 

⎢ ρ ⎥ 

⎣ 

⎦ 

K 

( ) cos( el) 

sin el 

E 

( ) cos( Az) 

sin Az 

ρ 

= = 

ρ 

2 2 

+ ρ 

ρ 

ρ 

= = 

ρ ρ ρ ρ 

N 

2 

+ 

2 2 

+ 

2 

ρ 

E N E N 

E 

ρ 

N

Northern sky

Southern Sky

Time measurement: Solar Time

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