a, i.e. on the great circle A at z1, z2, z3, etc. The scans intersect at a large angle, which allows the determination of two-dimensional object coordinates from one-dimensional measurements. Image courtesy Lennart Lindegren.]

ment z axis. Second, the spin phase Ω(t), which is the angle between the plane containing the Sun and the instrument z axis and the instrument zx plane; the fundamental xy plane is the plane through the two viewing directions (see Lindegren et al. (2016) for a definition of the scanning reference system). The governing equation for ν(t) equals

[Figure: Fig. 6. Illustration of the scanning law of Gaia, showing the path of the spin axis (z), and the corresponding path of the preceding viewing direction, during four days. For clarity, the path of the following viewing direction is not shown. Image courtesy Lennart Lindegren.]

S 2 −cos2 ν + ˙λ⊙cos ξ sin ν, (1)
˙ν sin ξ = ˙λ⊙ √
– There is a fixed spin rate ωz = 60′′ s−1 around the spacecraft spin axis (z) to ensure that the optical stellar images move over the detector surface with the same speed as the electrons are transferred inside the CCD during the TDI operation (Sect. 3.1).
– The solar-aspect angle, ξ = 45◦, between the Sun and the instrument z axis, is fixed to ensure maximum parallax sensitivity (because the measurable, along-scan parallax displacement of an object is proportional to sin ξ ; Sect. 3.1) and maximum thermal stability of the payload and basic angle in particular.

In practice, the scanning law is defined with respect to a fictitious, nominal Sun: this allows unambiguous specification of the scanning law with respect to the ICRS, independent of the orbital motion of Gaia around L2.

The difference between the actual and nominal Sun is never larger than a few arcminutes.
– A slow precession of the spin axis around the Sun results in a series of loops around the solar direction (Fig. 7).

A side effect of the precession is an across-scan speed of stellar images during their focal plane transits.

The speed of the precession is as small as possible to limit the across-scan smearing of images when they transit the focal plane yet (just) large enough to ensure that subsequent loops overlap, in which case there are at least six distinct epochs of observations per year for any object in the sky.

For Gaia, this is achieved with 5.8 revolutions per year (4◦day−1 precession relative to the stars), which means that the precession period is 365.25/5.8 = 63 days and that the across-scan speed of images transiting the focal plane varies sinusoidally with time, with a nominal period of six hours and an amplitude of
where λ⊙denotes the ecliptic longitude of the (nominal) Sun and S = |ω × z| ˙λ⊙ −1 = 4.220745 is a dimensionless constant (corresponding to 5.8 revolutions per year).

The spin phase Ω(t) then follows from ˙Ω= ωz −˙λ⊙sin ξ sin ν −˙ν cos ξ. (2) The above two equations have only two free parameters: the initial spin phase and the initial precession angle, at the start of science operations.

Both angles have been initialised to observe the most favourable passages of bright stars very close to the limb of Jupiter, aiming to measure the light deflection owing to the quadrupole component of the gravitational field of this planet (de Bruijne et al. 2010b).

In practice, the scanning law (i.e. the intended orientation of the scanning reference system with respect to the ICRS as a function of time) is generated by Runge-Kutta integration of the above equations, converted from heliotropic angles ν and Ωto celestial coordinates, and finally approximated by piecewise polynomials using the Chebyshev representation.

During the first weeks of nominal science operations (between 25 July and 21 August 2014), Gaia was operating with a special scanning law, known as the ecliptic poles scanning law (EPSL).

In this mode, ν(t) stayed constant at 180◦, which means that the spin axis followed the Sun on the ecliptic (the alternative EPSL solution, with ν(t) = 0◦, was not used).