> Baffle slice calculations
>
> M. Sholl
>
> 23 February 2004
>
> The SNAP telescope entrance is a sliced cylinder. This provides
> shading from the sun for various observation angles. The slice angle
> *α* is chosen to preclude direct solar illumination of the slice front
> face. Let the normal vector to the surface of the slice be **C**. The
> angle of body coordinate direction **X** with the solar vector **S**
> is defined by Euler rotations about the Z axis (**Phi**) and Y axis
> (**Theta**). Because Euler rotations produce pseudovectors, the order
> of rotation about Phi and Theta determine the orientation of **S**.
> The front face of the telescope is illuminated when the dot product of
> **S** and **C** is positive. One therefore determines **Alpha** by
> solving for **C (dot) S** = 0.
>
> ![](media/image1.wmf)
>
> **[Preliminaries: rotation matrices]{.underline}**
>
> An arbitrary vector **V** rotated angle **Phi** about the **Y** axis
> transforms to **V'** as follows:
>
> $$V' = \begin{bmatrix}
> \text{cos}(\varphi) & 0 & \text{sin}(\varphi) \\
> 0 & 1 & 0 \\
>  - \text{sin}(\varphi) & 0 & \text{cos}(\varphi)
> \end{bmatrix}V$$
>
> while an arbitrary vector **V** rotated angle **Theta** about the
> **Z** axis transforms to **V'** as follows:
>
> $$V' = \begin{bmatrix}
> \text{cos}(\theta) & - \text{sin}(\theta) & 0 \\
> \text{sin}(\theta) & \text{cos}(\theta) & 0 \\
> 0 & 0 & 1
> \end{bmatrix}V$$
>
> **[Compute vector C]{.underline}**
>
> The vector **C** is computed by rotating a unit **Z** direction vector
> about the **Y** axis by an angle --Alpha as follows:
>
> $\begin{bmatrix}
> \text{cos}(\varphi) & 0 & \text{sin}(\varphi) \\
> 0 & 1 & 0 \\
>  - \text{sin}(\varphi) & 0 & \text{cos}(\varphi)
> \end{bmatrix}\begin{bmatrix}
> 0 \\
> 0 \\
> 1
> \end{bmatrix} = \begin{bmatrix}
>  - \text{sin}(\alpha) \\
> 0 \\
>  - \text{cos}(\alpha)
> \end{bmatrix}$ (1)
>
> **[Two possibilities for computing vector S]{.underline}**

1.  Rotate about **Z** axis first

> $\begin{bmatrix}
> \text{cos}(\varphi) & 0 & \text{sin}(\varphi) \\
> 0 & 1 & 0 \\
>  - \text{sin}(\varphi) & 0 & \text{cos}(\varphi)
> \end{bmatrix}\begin{bmatrix}
> \text{cos}(\theta) & - \text{sin}(\theta) & 0 \\
> \text{sin}(\theta) & \text{cos}(\theta) & 0 \\
> 0 & 0 & 1
> \end{bmatrix}\begin{bmatrix}
> 1 \\
> 0 \\
> 0
> \end{bmatrix} = \begin{bmatrix}
> \text{cos}(\varphi) & 0 & \text{sin}(\varphi) \\
> 0 & 1 & 0 \\
>  - \text{sin}(\varphi) & 0 & \text{cos}(\varphi)
> \end{bmatrix}\begin{bmatrix}
> \text{cos}(\theta) \\
> \text{sin}(\theta) \\
> 0
> \end{bmatrix} = \begin{bmatrix}
> \text{cos}(\theta)\text{cos}(\varphi) \\
> \text{sin}(\theta) \\
>  - \text{sin}(\varphi)\text{cos}(\theta)
> \end{bmatrix}$ (2)

2.  Rotate about **Y** axis first

> $\begin{bmatrix}
> \text{cos}(\theta) & - \text{sin}(\theta) & 0 \\
> \text{sin}(\theta) & \text{cos}(\theta) & 0 \\
> 0 & 0 & 1
> \end{bmatrix}\begin{bmatrix}
> \text{cos}(\varphi) & 0 & \text{sin}(\varphi) \\
> 0 & 1 & 0 \\
>  - \text{sin}(\varphi) & 0 & \text{cos}(\varphi)
> \end{bmatrix}\begin{bmatrix}
> 1 \\
> 0 \\
> 0
> \end{bmatrix} = \begin{bmatrix}
> \text{cos}(\theta) & - \text{sin}(\theta) & 0 \\
> \text{sin}(\theta) & \text{cos}(\theta) & 0 \\
> 0 & 0 & 1
> \end{bmatrix}\begin{bmatrix}
> \text{cos}(\varphi) \\
> 0 \\
>  - \text{sin}(\varphi)
> \end{bmatrix} = \begin{bmatrix}
> \text{cos}(\theta)\text{cos}(\varphi) \\
> \text{cos}(\varphi)\text{sin}(\theta) \\
>  - \text{sin}(\varphi)
> \end{bmatrix}$ (3)
>
> **[Compute dot products]{.underline}**
>
> Compute dot products between (1) and (2), and (1) and (3).
>
> *Sin(α)cos(θ)cosφ(φ)+cos(α)(-sin(φ))cos(θ)=0* implies *α= φ* (4)
>
> *sin(α)cos(**θ)cosφ(φ)+cos(α)(-sin(φ))=0* implies
> *tan(α)=tan(φ)/cos(θ)* (5)
>
> Of (4) and (5), (5) is the most restrictive on *α*, and will therefore
> be adopted. Note, case (4) does not depend on roll angle θ.
>
> **[Results]{.underline}**
>
> See 2/20/2004 P. Jelinsky Addendum to *SNAP L2 Orbit Scattering Angles
> for near Ecliptic Pole Pointings* memo. Two possible angles *φ* are
> under consideration for pointing off the ecliptic pole: 20º and 25º.
> Table 1 shows derived baffle slice angles α (using Equation (5))
> required for Sun, Earth and Moonlight exclusion from the entrance into
> the baffle.
>
> ![](media/image2.wmf)
>
> Table 1: Baffle slice angles for Earth/Moon from L2
>
> Table 1 shows that required baffle slice angles increase dramatically
> from \~35º if Earth/Moon exclusion is considered necessary.
> Illumination levels from the Moon and Earth are small relative to that
> produced by the sun. This is due primarily to the small unshadowed
> regions visible from L2. See 1/23/2004 P. Jelinsky *SNAP L2 Orbit
> Scattering Angles*. The maximum illumination of the Moon and Earth
> disks from L2 are given in Table 2.

  ----------------------- ------------------------ -----------------------
                          Earth                    Moon

  Halo                    5.3%                     12.4%

  Lissajous               1.5%                     5.3%
  ----------------------- ------------------------ -----------------------

> Table 2: Earth/Moon disk illumination from L2
>
> Because a baffle slice angle α will likely be chosen to exclude
> sunlight, but not Earth or Moonlight, occasional Earth/Moon stray
> light will need to be analyzed for impact on the overall observing
> schedule.
>
> **[Conclusions]{.underline}**
>
> With the caveat that Earth/Moon stray light cases require
> consideration in the overall observation plan, the baffle slice angle
> should be set at 34º for interim design and analytical calculations.