In the following subsections, a set of parameters that describe the access features are studied.
Total Access Time
The first parameter to be analysed is the total access time (\(T_{total}\)) of an inertial direction in the sky. As mentioned in (10) this direction can be defined with its \(\varphi\) and \(\theta\) coordinates. The total access time is defined as the total amount of time that the deputy, i.e., the direction that defines its relative position, stays inside the instrument FoV. It is a feature of a given scan strategy.
Analytical Approach for Total Access Time
As explained before, if the rotation times (\(T_{spin}\) and \(T_{prec}\)) are properly selected and the analysis time is long enough, it can be assumed that the results will be axial-symmetric. This implies that \(T_{total}\) will not depend on \(\theta\) and only its variation with the \(\varphi\) coordinate of the direction is relevant. Thus, since \(T_{total}\) does not depend on \(\theta\), it will be equal for all the points in the sky with the same \(\varphi\). Furthermore, the precession speed does not influence the fraction of time that each point of the sky is seen as any increase in the speed will shorten the access duration but it will also make the number of accesses grow proportionally. Therefore, an arbitrary precession speed case can be analysed and the conclusions about \(T_{total}\) will be applicable to other velocities as long as the high combined period condition is met.
The instrument direction over the time, \(\mathbf {q}(t)\), can also be expressed in terms of \(\varphi\) and \(\theta\) coordinates, however, as the final value of \(T_{total}\) does not depend on \(\theta\), only the variation of the angle between the direction of the instrument and the \(X_{0}\)-axis is relevant. This angle, called here \(\varphi _v\), only depends on the spin motion and its variation is periodic every spin cycle, thus, the value of \(T_{total}\) obtained with the motion of the instrument, \(\mathbf {q}(t)\), will be equivalent to the one obtained by a vertical periodic motion \(\mathbf {q}^{\,*}(t)\) in the \(X_0Y_0\)-plane. Therefore, the analysis of this motion during one spin period will provide all the required information about \(T_{total}\).
Based on these considerations, let us study how much time the directions with the same \(\varphi\) coordinate are inside the instrument FoV for the simplest possible case: the one with almost zero precession (\(T_{prec}\longrightarrow \infty\)). Furthermore, it is only necessary to study one spin cycle, which, as previously said, can be represented as a vertical periodic motion with \(\varphi _v = f(t)\). A scheme of this analysis is shown in Fig. 7. It should be noticed that, although these schemes do not provide a rigorous projection of the trace, it helps to facilitate the understanding of the process. The angles regarding the precession axis are represented as in an azimuthal equidistant projection, i.e., directions with the same \(\varphi\) coordinate are displayed as a circumference with a radius proportional to \(\varphi\).
In order to calculate \(\varphi _v=f(t)\) it is necessary to define the direction of the instrument \(\mathbf {q}\). This direction can be expressed as a rotation around spin axis \(\mathbf {u}_{spin}\), which in this analysis is fixed and contained in \(X_{0}Z_{0}\)-plane, as follows
$$\begin{aligned} \mathbf {u}_{spin} = \begin{Bmatrix} \cos (\alpha )\\ 0\\ \sin (\alpha ) \end{Bmatrix}. \end{aligned}$$
(11)
The rotation matrix around the direction defined by this vector is
$$\begin{aligned} R = \begin{bmatrix} c_{\phi } + c_{\alpha }^2(1 - c_{\phi }) &{} - s_{\alpha }s_{\phi } &{} c_{\alpha }s_{\alpha }(1 - c_{\phi })\\ s_{\alpha }s_{\phi } &{} c_{\phi } &{} - c_{\alpha }s_{\phi }\\ c_{\alpha }s_{\alpha }(1 - c_{\phi }) &{} c_{\alpha }s_{\phi } &{} c_{\phi } + s_{\alpha }^2(1 - c_{\phi }) \end{bmatrix}, \end{aligned}$$
(12)
where \(\phi\) is the rotated angle. If the starting position of \(\mathbf {q}\) is
$$\begin{aligned} \mathbf {q}_0 = \begin{Bmatrix} \cos (\alpha + \beta )\\ 0\\ \sin (\alpha +\beta ) \end{Bmatrix}, \end{aligned}$$
(13)
also contained in \(X_{0}Z_{0}\)-plane, the value of \(\mathbf {q}\) depending on the spin angle would be
$$\begin{aligned} \mathbf {q}(\phi ) = R\cdot \mathbf {q}_0 = \begin{Bmatrix} \cos (\alpha )\cos (\beta ) - \sin (\alpha )\sin (\beta )\cos (\phi )\\ -\sin (\beta )\sin (\phi )\\ \cos (\alpha )\sin (\beta )\cos (\phi ) + \sin (\alpha )\cos (\beta ) \end{Bmatrix}. \end{aligned}$$
(14)
The angle between the instrument direction and the precession axis, \(\varphi _v\), is then obtained from the dot product between \(\mathbf {q}\) and \(X_{0}\)-axis direction as
$$\begin{aligned} \varphi _v = \arccos (\cos (\alpha )\cos (\beta )-\sin (\alpha )\sin (\beta )\cos (\phi )). \end{aligned}$$
(15)
Then, the equivalent vertical motion of the instrument, \(\mathbf {q}^{\,*}\), is given by
$$\begin{aligned} \mathbf {q}^{\,*} = \begin{Bmatrix} \cos (\varphi _v)\\ 0\\ \sin (\varphi _v) \end{Bmatrix}, \end{aligned}$$
(16)
where \(\varphi _v(t)\) can be calculated using (15).
Once the motion of the instrument has been characterised, the next step is to calculate the fraction of the circumference inside the FoV in each instant. A generic point of the circumference, \(\mathbf {p}\), is given by (10). The one where the access starts in a certain instant can be written as
$$\begin{aligned} \mathbf {p}_\varphi = \begin{Bmatrix} \cos (\varphi )\\ \sin (\varphi )\sin (\theta _\varphi )\\ \sin (\varphi )\cos (\theta _\varphi ) \end{Bmatrix}. \end{aligned}$$
(17)
For a given position \(\mathbf {q}^{\,*}\) of an instrument with FoV half-angle \(\delta\), the vector of the instrument \(\mathbf {q}^{\,*}\) will form an angle \(\delta\) with \(\mathbf {p}_\varphi\). Therefore
$$\begin{aligned} \mathbf {q}^{\,*}\cdot \mathbf {p}_\varphi = \cos (\delta ). \end{aligned}$$
(18)
From this equation, the half-arc angle \(\theta _\varphi\) of points with the same \(\varphi\) that are inside the FoV (see Fig. 7) can be obtained
$$\begin{aligned} \cos (\theta _\varphi ) = \frac{\cos (\delta )-\cos (\varphi _v)\cos (\varphi )}{\sin (\varphi _v)\sin (\varphi )}. \end{aligned}$$
(19)
Therefore, the fraction of points inside the FoV is \(2\theta _\varphi /2\pi\).
Averaging this fraction over a half spin period, the fraction of time that directions with the same \(\varphi\) spend inside the FoV during one period can be obtained. This averaged fraction is
$$\begin{aligned} f_t(\varphi ) = \frac{1}{\pi }\int ^{\pi }_{0}\frac{1}{\pi }\mathrm {Re}\left( \arccos \left( \frac{\cos (\delta )-\cos (\varphi _v)\cos (\varphi )}{\sin (\varphi _v)\sin (\varphi )}\right) d\phi \right) , \end{aligned}$$
(20)
as the second half of the period is symmetric to the first one.
The total access time for a point with angle \(\varphi\) can be calculated multiplying this fraction by the total time of analysis, \(T_{sim}\)
$$\begin{aligned} T_{total}(\varphi ) = f_t(\varphi ) \cdot T_{sim}. \end{aligned}$$
(21)
As can be deduced from equations (15) and (20), \(f_t\) depends solely on \(\varphi\), with \(\alpha\), \(\beta\) and \(\delta\) as parameters while the spin and precession speeds do not appear. An example of the solution of \(f_{t}(\varphi )\) is shown in Fig. 8.
Numerical Results for Total Access Time
The total access time has been calculated following the methodology explained in Sect. 2.1. To represent the numerical results, a Hammer projection [15] of the whole sphere has been chosen to depict them. The center of the image corresponds to the observation direction of the precession axis, while the lateral regions represent the opposite direction. The result for the same example on Fig. 8 is presented in Fig. 9, where the axial-symmetry is clear.
The main disadvantage of this approach is its computational cost, which grows proportionally with the simulation time and both the temporal and spatial discretization (nevertheless, more than one parameter can be calculated in the same simulation with small additional effort). For this reason, the analytical approach can be useful for validation of the numerical method. In Fig. 10 the analytical and numerical results for total access time are compared. The numerical results have been averaged over \(\theta\). Overall, the error is low except in the deep slope zones of the curve, due to the limited precision of the bilinear interpolation. The criterion to consider that both results coincide has been established in an RMSE (root-mean-square error) that is lower than 1E-3\(\%\) of \(T_{sim}\). This is fulfilled for all the scan strategies tested.
Mean Access Time
The mean access time, \(T_{mean}\), can be defined as the total access time over a period of time divided by the number of accesses. As before, for a direction \(\mathbf {p}\), if spin and precession are chosen properly, it is possible to assure that this parameter does not depend on \(\theta\).
Analytical Approach for Mean Access Time
In the previous section, the total access time has been obtained, and now the number of accesses is calculated following a similar approach. Although, for calculating the total access time the precession motion can be omitted, it will influence the number of accesses for a certain \(\varphi\) and it has to be taken into account if precession speed \(\Omega\) is not negligible.
In order to calculate the number of accesses, it is not relevant how much time one given direction stays inside the FoV, but whether after one spin period it has been seen or not. Thus, the approach to solve the problem is to calculate the proportion of directions, \(f_m\), with the same \(\varphi\) coordinate that have been inside the FoV after one spin period . As before, the motion is symmetrical with regard to the axis of rotation after a period, therefore, this proportion must remain constant for subsequent periods. If accesses are equally distributed along all the points with the same \(\varphi\), the total number of accesses during a given period can be obtained by multiplying the number of spin cycles by the aforementioned proportion
$$\begin{aligned} N(\varphi ) = f_m(\varphi ) \frac{T_{sim}}{T_{spin}}. \end{aligned}$$
(22)
First, the case for \(\Omega\) negligible is presented and then it will be extended to the general case that takes into account the precession speed. A geometrical scheme of half spin period when \(\Omega\) is negligible is shown in Fig. 11.
Every half spin period, an arc of \(\Delta \theta\) is seen from the instrument. Over one spin, this makes a proportion of
$$\begin{aligned} f_m = \frac{2\Delta \theta }{2\pi }, \end{aligned}$$
(23)
which is equal to the mean number of accesses for all the directions with the same \(\varphi\) coordinate. The value of \(\Delta \theta\) can be obtained as
$$\begin{aligned} \Delta \theta = \theta _e - \theta _i, \end{aligned}$$
(24)
where \(\theta _e\) and \(\theta _i\) are the limits of the highlighted arc in Fig. 11. If \(\Omega\) is negligible, this arc is limited by the two circles of radius \(\beta + \delta\) and \(\beta - \delta\). Therefore, following the same approach that in (19), the limit values are
$$\begin{aligned} \theta _i|_\varphi = \mathrm {Re}\left( \arccos \left( \frac{\cos (\beta - \delta ) - \cos (\alpha )\cos (\varphi )}{\sin (\alpha )\sin (\varphi )}\right) \right) , \end{aligned}$$
(25)
and
$$\begin{aligned} \theta _e|_\varphi = \mathrm {Re}\left( \arccos \left( \frac{\cos (\beta + \delta ) - \cos (\alpha )\cos (\varphi )}{\sin (\alpha )\sin (\varphi )}\right) \right) . \end{aligned}$$
(26)
Once \(\theta _e\) and \(\theta _i\) are obtained, the distance \(2\Delta \theta\) can be calculated and using (23) in (22), the number of accesses will be
$$\begin{aligned} N(\varphi ) = \frac{2\Delta \theta }{2\pi }\frac{T_{sim}}{T_{spin}}. \end{aligned}$$
(27)
Retrieving the total access time (21), the mean access time is
$$\begin{aligned} T_{mean}(\varphi ) = \frac{T_{total}(\varphi )}{N(\varphi )} = T_{spin}\frac{ f_t(\varphi )}{f_m(\varphi )}. \end{aligned}$$
(28)
The mean access time profile for the case of negligible \(\Omega\) is shown in Fig. 12.
However, if \(\Omega < \omega\) but not negligible, the trace cannot be simplified as a circle. An example of how the scheme can look is shown in Fig. 13.
In this case, in order to calculate the value of \(\theta _i\) and \(\theta _e\) it is necessary to obtain the side curves of the trace. These curves are separated by a \(2\delta\) distance in the perpendicular direction to the trace. If the trace expression is \(\mathbf {q}(t)\) (6), the equations of the exterior and interior curves are respectively
$$\begin{aligned} \mathbf {q}_e(t) = \mathbf {q}\cos (\delta ) + \mathbf {n}\sin (\delta ) = \begin{Bmatrix} x_e(t)\\ y_e(t)\\ z_e(t) \end{Bmatrix} \end{aligned}$$
(29)
and
$$\begin{aligned} \mathbf {q}_i(t) = \mathbf {q}\cos (\delta ) - \mathbf {n}\sin (\delta )= \begin{Bmatrix} x_i(t)\\ y_i(t)\\ z_i(t) \end{Bmatrix}. \end{aligned}$$
(30)
with \(\mathbf {n}\) being the unitary normal to the trace, whose expression is:
$$\begin{aligned} \mathbf {n} = \frac{\mathbf {q}\,'(t)\wedge \mathbf {q}(t)}{|\mathbf {q}\,'(t)|}. \end{aligned}$$
(31)
It is more useful to express the external and internal curves in terms of its \(\varphi\) and \(\theta\) components. These can be calculated from the Cartesian coordinates from (29) and (30) as
$$\begin{aligned} \varphi _e(t) = \mathrm {Re}(\arccos (y_e(t))) \end{aligned}$$
(32)
$$\begin{aligned} \theta _e(t) = \mathrm {Re}\left( \arccos \left( \frac{z_e(t)}{\sin (\varphi _e(t))}\right) \right) \end{aligned}$$
(33)
and
$$\begin{aligned} \varphi _i(t) = \mathrm {Re}(\arccos {y_i(t)}) \end{aligned}$$
(34)
$$\begin{aligned} \theta _i(t) = \mathrm {Re}\left( \arccos \left( \frac{z_i(t)}{\sin (\varphi _i(t))}\right) \right) . \end{aligned}$$
(35)
With the equations of the external and internal curve, the procedure to calculate the \(2\Delta \theta\) for a given \(\varphi\) is as follows: first, using equations (32) and (34) the instants \(t_i\) and \(t_e\) when the traces have respectively \(\varphi _i = \varphi\) and \(\varphi _e = \varphi\) are calculated. Second, these instants are used to calculate \(\theta _e\) and \(\theta _i\) with equations (33) and (35). Once these angles are obtained, the distance \(2\Delta \theta\) can be determined and the number of accesses is obtained through equation (28). An example of the resulting profile obtained from equation (28) is shown in Fig. 14.
Numerical Results for the Mean Access Time
The numerical results for the mean access time for the same example on Fig. 14 are presented in Fig. 15. As can be seen, the mean access time also presents axial-symmetry.
In Fig. 16 the analytical and numerical results for mean access time are compared. The numerical results have been averaged over \(\theta\). In this case, the criterion to validate the results is that the RMSE is lower than the time step used for the numerical analysis. This is fulfilled for all the scan strategies tested.
Maximum Access Time
The maximum access time (\(T_{max}\)) is the maximum time that a direction \(\bf{p}=\bf{f}(\varphi ,\theta )\) can be inside the FoV of the instrument. This time also depends on the scan strategy and will not depend on \(\theta\) if the rotation and precession are chosen properly.
Analytical Approach for Maximum Access Time
For the maximum access time, the procedure followed is similar to the mean access time section. First, the expression for the duration of an access with no precession motion is obtained. But now, the maximum access time is studied instead of the mean access time. Then, the precession motion is added, distinguishing two cases according to whether the precession speed is negligible or not, like in the mean access time section.
First, the case without precession will be analyzed. As there is only rotation around the spin axis, the motion will be analyzed centered in that axis, the new coordinates being \(\theta ^*\) and \(\varphi ^*\). Furthermore, in order to obtain the access time for this pure spin motion, it is advantageous to consider that the instrument remains fixed and it is the direction vector the one that rotates around the spin axis (\(X^*\) in the new coordinates). Without loss of generality, it is assumed that the instrument is located on the \(X^*Z^*\)-plane, separated by an angle \(\beta\) from the spin axis, as shown in Fig. 17, thus,
$$\begin{aligned} \mathbf {q} =\begin{Bmatrix} \cos {\beta }\\ 0\\ \sin {\beta } \end{Bmatrix}. \end{aligned}$$
(36)
In this new frame, the direction vector (37) is separated an angle \(\varphi ^*\) from the spin axis, and its projection over the \(Y^*Z^*\)-plane form an angle \(\theta ^*\) with the \(Z^*\)-axis, so it can be expressed as
$$\begin{aligned} \mathbf {p} =\begin{Bmatrix} \cos (\varphi ^*)\\ \sin (\varphi ^*)\sin (\theta ^*)\\ \sin (\varphi ^*)\cos (\theta ^*) \end{Bmatrix}. \end{aligned}$$
(37)
The starting condition of an access (18) establishes that the relation between \(\beta\), \(\theta ^*\), \(\varphi ^*\) and \(\delta\) for a given instant is
$$\begin{aligned} \cos (\theta _\varphi ^*) = \frac{\cos (\delta )-\cos (\varphi _v)\cos (\varphi ^*)}{\sin (\varphi _v)\sin (\varphi ^*)}. \end{aligned}$$
(38)
This equation is similar to (19). Due to how the direction vector is expressed, the angle \(\theta _\varphi ^*\) is the half-arc of its trajectory inside the FoV. Therefore, if the trajectory in one spin period has a length of \(2\pi\), the proportion of time inside the FoV is \(2\theta _\varphi ^*/2\pi\) and the access time is
$$\begin{aligned} T_{access} = \frac{T_{spin}}{\pi }\mathrm {Re}\left( \arccos \left( \frac{\cos (\delta )-\cos (\beta )\cos (\varphi ^*)}{\sin (\beta )\sin (\varphi ^*)}\right) \right) . \end{aligned}$$
(39)
The maximum access time can be obtained deriving (39) with regard to \(\varphi ^*\) and equating to zero, which gives
$$\begin{aligned} t_{max} = \frac{T_{spin}}{\pi }\mathrm {Re}\left( \arccos \left( \frac{ \sqrt{\cos ^2(\delta )-\cos ^2(\beta )}}{\sin (\beta )} \right) \right) . \end{aligned}$$
(40)
This maximum access time, called here optimum access time, is achieved for the directions whose \(\varphi ^*\) coordinate is equal to \(\varphi _{t_{max}}^*\), whose value is
$$\begin{aligned} \varphi _{t_{max}}^* =\arctan \left( \frac{ \sqrt{\cos ^2(\delta )-\cos ^2(\beta )}}{\cos (\beta )} \right) . \end{aligned}$$
(41)
For any higher or lower value of \(\varphi ^*\), the access time will be lower than the optimum access time. In Fig. 18 it is shown a scheme of this behaviour.
As mentioned before, to add the effect of precession, two cases have been considered. For the case where \(\Omega<<\omega\), i.e., the precession motion is much slower than the spin motion, it can be assumed that the trace of the instrument over the sky is nearly a circumference and therefore the previous results can be used directly. As the spin axis rotates around the precession axis, those points in the sky which, at any time, are at an angular distance of \(\varphi _{t_{max}}^*\) will have an optimum access time. For any point with \(\varphi <\alpha + \varphi _{t_{max}}^*\) and \(\varphi >|\alpha - \varphi _{t_{max}}^*|\) there will be an instant in which the trace is at the right point. Therefore, its maximum access time will be the optimum access time.
For those directions whose \(\varphi\) coordinate is out of the aforementioned range, the maximum access time will not be the optimum. However, (39) is still valid assuming that \(\varphi ^*=\alpha - \varphi \cdot \text {sgn}(\alpha - \varphi _{t_{max}}^*)\) for \(\varphi <|\alpha - \varphi _{t_{max}}^*|\) and \(\varphi ^*=\varphi -\alpha\) for \(\varphi >\alpha + \varphi _{t_{max}}^*\). Taking all of this into account, according to the value of \(\varphi\), its maximum access time is
for \(0<\varphi <|\alpha - \varphi _{t_{max}}^*|\):
$$\begin{aligned} t_1 = \frac{T_{spin}}{\pi } \mathrm {Re}\left( \arccos \left( \frac{\cos (\delta )-\cos (\beta )\cos (\alpha -\varphi \cdot \text {sgn}(\varphi _{t_{max}}^*))}{\sin (\beta )\sin (\alpha -\varphi \cdot \text {sgn}(\varphi _{t_{max}}^*))} \right) \right) \end{aligned}$$
(42)
for \(|\alpha - \varphi _{t_{max}}^*|<\varphi <\alpha + \varphi _{t_{max}}^*\):
$$\begin{aligned} t_2 = t_{max} = \frac{T_{spin}}{\pi }\mathrm {Re}\left( \arccos \left( \frac{ \sqrt{\cos ^2(\delta )-\cos ^2(\beta )}}{\sin (\beta )} \right) \right) \end{aligned}$$
(43)
for \(\alpha + \varphi _{t_{max}}^*< \varphi < \pi\):
$$\begin{aligned} t_3 = \frac{T_{spin}}{\pi }\mathrm {Re}\left( \arccos \left( \frac{\cos (\delta )-\cos (\beta )\cos (\varphi -\alpha )}{\sin (\beta )\sin (\varphi -\alpha )}\right) \right) \end{aligned}$$
(44)
An example of the maximum access time profile in the case of negligible \(\Omega\) is shown in Fig. 19. The three regions can be clearly identified. For the central region, the maximum access time is the highest and is constant.
For the case of a significant precession speed, no analytical solution has been found. However, its effect can be approximated by modifying the maximum access times (42), (43) and (44).
First, lets analyze the points that do not reach \(\varphi _{t_{max}}^*\) and therefore, \(t_{max}\). These points have their own optimum access time when they reach the minimum angular distance from the spin axis. If the precession motion is taken as a rotation of the sky instead of the instrument, it can be assumed that the access time will increase or decrease, according to whether such rotation has the opposite or the same direction as the spin rotation. A scheme explaining this behavior is shown in Fig. 20. When \(\alpha > \beta\), as the speed grows the maximum access time decreases for large values of \(\varphi\) and increases for small values of \(\varphi\). In the other case, \(\alpha < \beta\), the maximum access time will be smaller for any value of \(\varphi\), as the speed with which the direction travels across the FoV is always increased by the precession motion.
Therefore, it is possible to approximate how the maximum access time changes introducing a scaling factor for \(t_1\) and \(t_3\) as follows
for \(0<\varphi <|\alpha - \varphi _{t_{max}}^*|\):
$$\begin{aligned} t_1^* = t_1 \frac{\omega \sin (\alpha - \varphi k)}{\omega \sin (\alpha - \varphi k) - \Omega \sin (\varphi k)} \end{aligned}$$
(45)
for \(\alpha + \varphi _{t_{max}}^*< \varphi < \pi\):
$$\begin{aligned} t_3^* = t_3 \frac{\omega \sin (\varphi - \alpha )}{\omega \sin (\varphi - \alpha ) + \Omega \sin (\varphi )} \end{aligned}$$
(46)
with \(k = \text {sgn}(\alpha -\beta )\). The smaller the values of \(\delta\) and \(\Omega\) are, the more accurate these approximations will be.
For \(t_2\), the proposed approach is as follows. Taking into account that the instrument sweep speed is produced both by spin and precession, assuming that \(\delta\) is small and the precession motion slow, it is reasonable to expect that the component of the precession motion parallel to the spin speed component will be the more significant one in changing the access time. Thus, \(t_2\) can be scaled similarly to \(t_1\) and \(t_3\) but this time with the component of precession speed parallel to spin speed
for \(|\alpha - \varphi _{t_{max}}^*|<\varphi <\alpha + \varphi _{t_{max}}^*\):
$$\begin{aligned} t_2^* = t_2 \frac{\omega \sin (\varphi _{t_{max}}^*)}{\omega \sin (\varphi _{t_{max}}^*) + \Omega \sin (\varphi ) \gamma } \end{aligned}$$
(47)
where \(\gamma\) is the projection of the precession velocity unitary vector (Fig. 21), \(\mathbf {v}_{prec}\), over the spin velocity unitary vector, \(\mathbf {v}_{spin}\). To obtain this value, first the vector \(\mathbf {v}_{spin}\) is calculated, tangent to the spin velocity. In order to do that, (14) is derived with regard to \(\phi\)
$$\begin{aligned} \frac{d\mathbf {q}}{d\phi } = \begin{Bmatrix} \sin (\alpha )\sin (\beta )\sin (\phi )\\ \sin (\beta )\cos (\phi )\\ - \cos (\alpha )\sin (\beta )\sin (\phi ) \end{Bmatrix}, \end{aligned}$$
(48)
and then normalized
$$\begin{aligned} \mathbf {v}_{spin} = \frac{d\mathbf {q}/d\phi }{|d\mathbf {q}/d\phi |} = \begin{Bmatrix} \sin (\alpha )\sin (\phi )\\ \cos (\phi )\\ - \cos (\alpha )\sin (\phi ) \end{Bmatrix} \end{aligned}$$
(49)
The vector tangent to the precession velocity, \(\mathbf {v}_{prec}\), can be easily expressed as a function of the \(\tau\) angle, which is defined in Fig. 22
$$\begin{aligned} \mathbf {v}_{prec} = \begin{Bmatrix} \cos (\tau )\\ 0\\ - \sin (\tau ) \end{Bmatrix}. \end{aligned}$$
(50)
The relation between both angles is
$$\begin{aligned} \tau = \arctan \left( \frac{\sin (\varphi _{t_{max}}^*)\sin (\phi )}{\cos (\alpha )\sin (\varphi _{t_{max}}^*)\cos (\phi )+\sin (\alpha )\cos (\varphi _{t_{max}}^*)}\right) . \end{aligned}$$
(51)
Then, projecting \(\mathbf {v}_{prec}\) over \(\mathbf {v}_{spin}\), \(\gamma\) is obtained
$$\begin{aligned} \gamma = \cos (\phi )\cos (\tau ) + \cos (\alpha )\sin (\phi )\sin (\tau ), \end{aligned}$$
(52)
where \(\phi\) can be expressed as a function of \(\varphi\), \(\varphi _{t_{max}}^*\), and \(\alpha\), as follows
$$\begin{aligned} \phi = \arccos \left( \frac{\cos (\alpha )\cos (\varphi _{t_{max}}^*)-\cos (\varphi )}{\sin (\alpha )\sin (\varphi _{t_{max}}^*)}\right) . \end{aligned}$$
(53)
In Fig. 23 an example of the resulting profile obtained from equations (45), (46) and (47) is shown.
Numerical Results for the Maximum Access Time
The numerical results for maximum access time for the same example given in Fig. 23 are presented in Fig. 24. As can be seen, this last parameter also presents axial-symmetry.
The analytical and numerical results for maximum access time are compared in Fig. 25. The numerical results have been averaged over \(\theta\). The same RMSE criterion as with \(T_{mean}\) has been used and it has been fulfilled for all the scan strategies tested.