Appunti 2 - Space Missions and Systems

Space missions and systems

AppuntiMartina SantoianniMay 2023

Time scales, reference frames

Precise space navigation requires a precise definition of time scale and frame. From Einstein’s relativity theory, we know that time and space are deeply related and cannot be treated separately, but we’ll use a Newtonian point of view.

Reference system

Is meant to set the prescriptions and conventions together with the modelling required to define at any time a triad of axes and the coordinates of a point.

• Is a spatial reference frame rotating with the Earth in its diurnal Terrestrial Reference System (TRS) motion in space. Positions of points anchored on the Earth’s solid surface have coordinates that undergo only small variations with time due to geophysical effects (tectonic or tides).
• Is a set of physical points with precisely determined coordinates in a specific coordinate system (cartesian, geographic etc.) attached to a TRS. TRF is the realisation of TRF.
• Is defined as a 3D reference frame close to Earth and co-rotating with it. Euclidean space is considered as a 3D affine space (O,E), where O is the origin and E_j is a set of points defining the frame, a vector base of the associated vector space. E_j are orthogonal and have the same length, the common length of the vectors defines the scale of the TRS. The set of unit vectors collinear to the base defines its orientation: ‖E_i‖λ_i= 1,2,3.

For IERS (International Earth Rotating Service) we consider the geocentric systems where the origin is close to the geocenter, and the orientation is equatorial (z-axis in the north pole direction). Under these hypotheses, the general transformations of the cartesian coordinates of any point close to Earth from a TRS1 to a TRS2 is given by a 3D similarity:

(2) X⁽²⁾ = T_1,2 + λ_1,2R_1,2X⁽¹⁾

Where T is the origin’s translation vector, λ a scale factor, and R a rotation matrix.

Inertial reference system

Are due to the torques exerted by the sun and the moon on the spinning Earth.

• Precession and nutation: The nutation is due to the perturbations of the lunar orbit (18.6 years of precession of the node). The best realisation of an inertial frame is the realised ICRF (International Celestial Reference Frame) using point-like radio sources located at cosmological distances, their proper motion is undetectable. The ICRS complies with the conditions specified by the 1991 and 2000 IAU recommendations: it is located at the barycenter of the solar system, its origin is defined by the IAU models for precession and nutation. Its origin of right ascension is defined by fixing the right ascension of quasar 3C273B to the value of the FK5 optical catalog transferred at epoch J2000.0 - we can measure radial velocity in an accurate way because quasars move far away from us but the angle between them doesn’t change.

Figure 1: Definition

• Right ascension and declination: Positions of a celestial object as might be seen from the center of the Earth. RA is measured in hours (24 hrs stands for 360 degrees), measured eastwards from the Vernal axis. Declination is the elevation above or below the equator, ranges from 90° north to 90° south.
• Hour Angle: A sidereal day lasts about 23 hrs and 56 mins, the hour angle is the angle at the celestial pole measured clockwise (westwards) from 0° to 360° from the observer’s meridian to the celestial meridian of the body. The local hour angle of the Vernal Equinox is the same as the local sidereal time. HA = LST - α.

Old and new systems

Old systems derive from Newton mechanics with relativistic corrections due to perturbations, reference axes were based on the position of two moving planes (equatorial and ecliptic), but this required the equinox J2000.0 to be placed despite an erroneous constant of precession and an insufficient nutation/precession theory. There were three different equinoxes. The reference system was only theoretically dynamical, defined using the solution of planetary motion with no Coriolis acceleration.

Since not every antenna can detect weak quasar signals, some stars can be used as reference since they’re strong radio emitters. These stars become fiducial points for the positioning of all other celestial bodies, and we have obtained the ICRF coordinates of catalogued stars, such as those used by star sensors.

• Dynamical definition of the planetary reference frame: In a non-inertial frame, the equations of motion of a body subject to only gravitational interactions are:

dv/dt = -∑(GMi/|r_i-r_j|³) × (r_i-r_j) + 2ω × v + ω × (ω × r)

Relative motion is irrelevant for defining inertial frames, so Newton needed to define rotation with respect to absolute space. IERS publishes EOP every one/two days, they’re five numbers that specify:

• Deviation in longitude and in obliquity of the celestial pole with respect to its position defined by the conventional prec/nut models.
• UT1-UTC or UT1-TAI from which we can derive the variations of the length of the mean solar day (with respect to its nominal value of 86400 s).
• The polar motion of the Celestial Intermediate Pole with respect to the terrestrial crust. It has for terrestrial coord (x,-y,1). In addition to precession and nutation, the spin axis undergoes a quasi-periodic motion of amplitude 0.4 arcsec and period 430 days. This is caused by the displacement of masses in the interior and the atmosphere.

Terrestrial reference frame

Tides and tectonic produce variations in the relative positions between points on the crust, hence the Earth-fixed reference frame is not well defined (also the origin moves in response to the redistribution of fluid masses).

• The origin is set in the geocenter.
• Z-axis: coincides on average with the Earth’s vectorial angular velocity, not precisely due to polar motion.
• X-axis: along the direction of a conventional point on the equator (intersection with Greenwich meridian).

TRF-ICRF transformations

Figure 2: ECEF to ECI

The reference frame used for GPS positioning is called WGS84, it is a 3D coordinate system and associated ellipsoid. Its positions can be described in XYZ cartesian coordinates or latitude, longitude, and ellipsoid height (the origin is the geocenter). WGS84 is a convention and includes:

• It is geocentric, masses include oceans and atmosphere.
• Scale of axes equal to the local earth frame.
• Orientation coincides with the equator and prime meridian of the Bureau Internationale de L’Heure at epoch 1984.0.
• Since 1984.0, the orientation is such that the average motion of the crustal plates relative to the ellipsoid is 0. Z axis coincides with International Reference Pole and x with International Reference Meridian.

Figure 3: WGS84 defining constants

Figure 4: Ref. surfaces

• Topography: Represents the physical surface.
• Geoid: Defined as the level surface of gravity field with the best fit to mean sea level (max difference geoid-mean sea level is 1 m), it is an equipotential surface.
• Ellipsoid: Defines the mathematical surface approximating the physical reality.

Time

A clock is composed by:

• Oscillator: Electrical device that delivers frequency that can be controlled by inputs.
• High Q resonator: Another oscillator which gets in input the frequency coming from the first oscillator, delivering a forced response: F₀ (2ẍ + 2λẋ + ω₀x = mcosγt).
• Feedback loop: Checks output from resonator, if the response value is above or below the max frequency γ.

Figure 5: Clock

The output signal is: x(t) = ae^-λtcos(ω₀t + α) + |B|cos(γt + δ).

If the forcing action is turned off, the system loses energy at rate e^-2λt, so, with damping τ = λ/2, E = E₀e^-λt. Ideal resonators are non-dissipative (λ = 0), E being the energy stored in the resonator, the lost energy in a cycle, the quality of the resonator can be measured as: Q = (2πE)/ΔE = ω₀/2λ.

The lower the frequency, the higher the quality. Best resonators need little energy to be kept oscillating and suffer minimal perturbations. The width of the resonance curve is proportional to 1/Q. Having high quality makes the resonance curve (|B|, γ) sharper, in this way the feedback can better find the maximum on the curve.

Types of clocks

• Electro-mechanical: Tuning fork Q = 10³, Quartz wristwatch Q = 10⁴, OCXO Q = 10⁶.
• Atomic: Cesium beam Q = 10⁸, Cesium fountain Q = 10¹⁰, Mercury ion optical standard Q = 10¹⁴.

Cesium clocks

The SI second is defined from a frequency of the radiation emitted when a cesium atom (isotope 113) undergoes a hyperfine transition: 9 billion Hz. These clocks are very stable in the long run. Commercially available oscillators use cesium beams: inside, we have cesium atoms heated to a gaseous state in an oven, atoms leave the oven in high-velocity beam that travels through a vacuum tube towards a pair of magnets. The magnets work as a gate that allows particular energy level atoms to pass in a cavity where they’re exposed to a microwave frequency (derived from the quartz oscillator), that is, this frequency matches the resonance frequency of cesium, the atoms change their magnetic state. Atoms that have not changed their state are deflected away. The detector produces a feedback signal that continually tunes the quartz oscillator in a way that maximizes the number of state changes so that the greatest number of atoms reaches the detector.

Hyperfine transition: Cesium has 1 unpaired electron (1/2 total spin, nucleus 7/2), the total spin can be F = 4 or F = 3. The F = 4 state corresponds to slightly larger energy: every time an electron reverses its spin, a hyperfine transition occurs, with the emission of radiation of frequency f = ΔE/h. The spin is associated with the magnetic moment, so nucleus and electron behave like magnets. The larger potential state energy occurs when the spins are parallel (F = 4).

Time scales

• Atomic time: Being the unit second of SI defined as the duration of 9 billion cycles of radiation corresponding to the two hyperfine levels of the ground state of cesium 133. TAI is a statistical timescale based on a large number of atomic clocks, an average.
• Universal time (UT): From 0 hours at midnight with a unit of duration of mean solar day (86,400.002 seconds). Based on an imaginary mean sun that moves along the celestial equator at a constant rate that matches the real Sun’s average rate over the year.
• UT0: Rotational time of a particular place of observation, diurnal motion of stars.
• UT1: Correcting UT0 for the effect of polar motion on the longitude (irregularities of rotation).
• Coordinated Universal Time (UTC): Differs from TAI for an integral number of seconds. UTC is kept within 0.9 seconds of UT1 by the introduction of the "leap second". It is a discontinuous time scale since we decide to add or subtract a second once or twice a year (June 30 or December 31) when UTC differs from atomic time. The length of the day generally shows annual, semiannual, random, and systematic variations.
• Sidereal time: Unit of duration of the Earth’s rotation period with respect to a point nearly fixed to the stars.
• Delta T: The difference between Earth rotational time UT1 and UTC, predicted measure of UT1-UTC are provided by Earth Rotation Service.
• Apparent solar day: Time between two successive transits of the sun at the meridian (≈ 24h).
• Sidereal day: Time between two successive transits of a star at the meridian (year made by 366.25 sd), about 23 hours and 56 minutes.
• Julian day: Continuous calendar from 4713 BC onwards. Julian Day 0 is noon on Monday, January 1, 4713 BC, with a Julian day of 86,400 seconds.
• Julian century: Is 36525 Julian days.
• Julian epoch: First was J1900.0, which was Julian day 2415020.0 (January 0.5, 1900).
• Modified Julian day: Use a smaller number. EX: Julian epoch J2000.0, which is JD 2451545.0, omit the 24 and the .0 (since observations start at night), the modified is 51545. MJD = JD - 2400000.5.

Einstein clock

Two identical clocks, one at rest and one moving. The light blips in both travel at the same time speed relative to us, the one in the moving clock goes further, so must take longer between clicks.

w = w₀ 1/(1 - V²/c²)^1/2

Relativity is built on the fact that the speed of light is the same for all inertial observers. For an observer at rest, the light source is a spherical surface expanding at c.

x² + y² + z² = (ct)²

Considering an observer O’ moving at speed V along the x-axis of the reference frame of observer O, the light source at the origin is also a spherical surface, but we must use the Galilei transformations:

x’ = x - Vt, y’ = y, z’ = z, t’ = t

(x - Vt)² + y² + z² = c²t² -> not a sphere.

If we consider new transformation laws, we add the spatial term into the time:

x’ = x - Vt, y’ = y, z’ = z, t’ = t + fx

(x - Vt)² + y² + z² = c²(t + fx)² -> closer to a sphere.

However, we still need to scale the distances and time by a factor to get the Lorentz transformations.

(1 - V²/c²)^-1/2

These new transformations suggest a more intimate relation between space and time: the space-time of inertial observers is endowed with a metric that allows the computation of distance between two points in space-time.

c²(t₂ - t₁)² - (x₂ - x₁)² - (y₂ - y₁)² - (z₂ - z₁)² = s² (space-time interval between two events)

INVARIANT:

1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 -1

General relativity rests on two pillars: constancy of c and laws of physics are the same for all freely falling observers.

ds² = (cdt)² - (dx² + dy² + dz²)

ds² = g_ikdxⁱdx^k

Where xⁱ is any space-time coordinate and g_ik is the metric tensor (set of coefficients which provides the prescription on how to compute space-time differences between infinitesimally close events.

In general relativity, the metric tensor is determined by Einstein’s equations.