Caption: In this figure, we are going to explicate a procedure (which we call the center-to-edge procedure) for measuring the field of view (FOV) (i.e., the FOV angular diameter d) of a telescope by measuring the FOV transit time t of a star across the FOV from center to edge by a star in its diurnal rotation (around the celestial axis) with rate R = 15°/h_s = 15'/m_s = 15.041'/m ≅ 15'/m, where h_s is a sidereal hour, m_s is a sidereal minute, and m is a minute.

As part of the explication, we will derive some general formulae and some useful special-case formulae.

To summarize the discussion below, the recommended formula for a measurement of the FOV angular diameter using the center-to-edge procedure is

`  d = 2*Rt*cos(δ) , `
where δ is the declination of the transiting star.

The Center-to-Edge Procedure:

1. As the preamble indicates, what you measure is transit time t and that gives the change in right ascension A from A = Rt. You then need a formula to relate A to FOV angular radius "a": i.e., a = f(A). The angular diameter d is then just given d = 2a = 2f(A).

2. Why the center-to-edge procedure rather than an edge-to-edge procedure? First, the center-to-edge procedure is easier for students who have difficulty in arranging a star transit across the whole FOV. They get better practice moving astronomical objects doing whole transits, but they can do more observations with half transits and get better practice that way.

3. Second, yours truly thinks the center-to-edge procedure is formally a good procedure (as we show below) since it leads to a good treatment of the extreme case where the FOV becomes significant compared to the angle θ between the star and the celestial axis.

Note θ = π/2 - |δ|, where δ is declination. We will suppress the absolute value signs in our derivations to avoid klutziness and because δ only turns up as the argument of even functions making the absolute value signs irrelevant

Note also that θ and δ are, of course, constants in the diurnal rotation.

4. Now the extreme case is NOT likely to turn up in any real measurement, but having a good treatment is formally good since it means you understand what you are doing.

To explicate the extreme case, let's consider the extremest case where we have the smallest possible θ: i.e., θ = a/2 which means that the north celestial pole (NCP) or south celestial pole (SCP) is actually in the FOV halfway between center and edge.

With θ = a/2, a star starts at the center of the FOV and rotates to edge through a change in right ascension A = π. Any smaller θ would NOT allow the star to reach the edge from the center. So θ = a/2 is indeed the extremest case.

Note you do NOT actually need to measure a transit time t in this case, since you know A = π.

However, in this extremest case and other extreme cases, you need spherical trigonometry to get the formula a = f(A).

5. The image shown above, shows a more general case of spherical trigonometry than what we need. For us, a = a, A = A, and b = c = θ.

6. Using the spherical trigonometry cosine rule, we get, respectively, the formula A = f(a) and the inverse formula a = g(A):

```
a = arccos[cos(θ)**2 + sin(θ)**2*cos(A)] = arccos[sin(δ)**2 + cos(δ)**2*cos(A)]

and

A = arccos[(cos(a) - cos(θ)**2/sin(θ)**2]
= arccos[(cos(a) - sin(δ)**2/cos(δ)**2]  , ```
where arccos is the inverse cosine and we have used Wikipedia: List of trigonometric identities: Reflections to change from the parameter θ to the parameter δ.

The general formulae are generally useful with caveat that if the argument of arccos is numerically out of the range [-1,1], a computer language will probably give you NaN (not a number).

7. Now "a" for any realistic FOV will always satisfy a << 1, where "a" is in radians. And A, except for the extreme cases, will satisfy A << 1, where A is in radians.

Thus, it is useful to expand the general formula a = f(A) in small "a" and A to obtain, respectively, the 1st order formulae for "a" and A:

```          cos(a) = cos(θ)**2 + sin(θ)**2*cos(A)
1 - (1/2)a**2 ≅ cos(θ)**2 + sin(θ)**2*[1 - (1/2)*A**2]
1 - (1/2)a**2 = 1 -  sin(θ)**2*(1/2)*A**2
a**2 = sin(θ)**2*A**2
a = A*sin(θ) = A*cos(δ)
and
A = a/sin(θ) = a/cos(δ)  .  ```
The relative error in the 1st order formulae is order a + A. Note that the 1st order formulae are exactly correct at θ = π/2 (i.e., δ =0).

What is the practical criterion for the failure of the 1st order formulae given that a << 1 is always satisfied realistically? We can estimate when A << 1 does NOT hold from the 1st order formula for A itself:

`  A = a/sin(θ) ≅ a/θ > ∼ 1  , `
where we have used 1st order formula for the sine function (for which we take θ to be in radians, of course) since the inequality can hold only for θ << 1.

In fact, for any realistic transit time measurement, you will choose θ = 1 - |δ| >> 0: i.e., you will NOT be using high absolute value declination both because you know the 1st order formulae will fail and because the measurement will take a long line. For the extremest case, the measurement takes half a sidereal day.

So for the transit time measurement using center-to-edge procedure, we recommend using the 1st order formula for the determining the angular diameter of the FOV: i.e.,

`  d = 2*A*cos(δ) = 2*Rt*cos(δ) `
just as given in the preamble.

8. Despite the fact that the 1st order formulae are realistically all you need, it is interesing for understanding to have interpolation formulae that span the formally the allow θ range [a/2, π/2] and are simpler than the general formulae. What comes to mind as natural interpolation formulae are
```  a = 2*sin(A/2)*sin(θ)    A = 2*arcsin[(a/2)/sin(θ)]           for general θ ∈ [(1/2)a, π/2]

a = 2*sin(a/2) ≅ a       A = 2*arcsin[(a/2)/*sin(a/2)] ≅ π    for θ = a/2 and A = π

a = 2*sin(A/2) ≅ A       A = 2*arcsin(a/2) ≅ a                for θ = π/2  .  ```
We see that these interpolation formulae are only 1st order good in small "a" for θ = a/2 and only 1st order good in small "a" and small A for θ = π/2.

Caveat: if the argument of arcsin is numerically out of the range [-π/2,π/2], a computer language will probably give you NaN (not a number).

9. In order to test the accuracy/precision of our formulae, we have computed the table of A values shown below for fixed a = 40' = 0.0116 ... rad ≅ 0.01 rad << 1. This is a typical FOV angular diameter for an introductory astronomy laboratory telescopes with typical eyepieces.

For convenient reference, note: 1 rad ≅ 60°,   0.1 rad ≅ 6°,   0.01 rad ≅ 34' ≅ 0.5°,   0.001 rad ≅ 3.4' ≅ 0.05°.

The table shows that the 1st order formula for A is NOT bad even for the extremest case θ = a/2 = 20', is quite good for θ = a = 40', is better than 2 % accurate for θ ≥ 60' = 1°, and improves rapidly thereafter with increasing θ. This verifies that the 1st order formula for "a" (which is just the inverse formula for the one for A) is good for all realistic applications.

The interpolation formula for A is better than 0.0005 % accurate everywhere. However, it is worse than the 1st order formula for A for θ > ∼ 45° and does NOT become numerically exactly right at θ = 90° since it is only 1st order good in "a" there as aforementioned.

Table: Change in right ascension A as given by the exact, 1st order, and interpolation formulae for field of view as calculated by /aalib/field_of_view_procedure.f. The units change at the hrule in mid table.
```
N       theta       exact   1st order       relerr    interfor       relerr
arcm         deg         deg                      deg

1   20.000000  180.000000  114.592205  -0.3634E+00  180.000000  -0.7710E-19
2   30.000000   83.621353   76.395342  -0.8641E-01   83.621931   0.6914E-05
3   40.000000   60.001120   57.297072  -0.4507E-01   60.001493   0.6220E-05
4   50.000000   47.157838   45.838240  -0.2798E-01   47.158120   0.5983E-05
5   60.000000   38.944270   38.199126  -0.1913E-01   38.944498   0.5869E-05
6  120.000000   19.191961   19.102472  -0.4663E-02   19.192071   0.5694E-05
7  180.000000   12.764524   12.738215  -0.2061E-02   12.764596   0.5665E-05
8  300.000000    7.654791    7.649142  -0.7379E-03    7.654834   0.5649E-05
9  480.000000    4.791567    4.790198  -0.2857E-03    4.791594   0.5644E-05
10  540.000000    4.262594    4.261635  -0.2250E-03    4.262618   0.5644E-05

deg        arcm        arcm                     arcm
11   10.000000  230.392634  230.350819  -0.1815E-03  230.393934   0.5643E-05
12   20.000000  116.957157  116.952176  -0.4259E-04  116.957817   0.5642E-05
13   30.000000   80.001354   80.000000  -0.1692E-04   80.001805   0.5641E-05
14   40.000000   62.229452   62.228953  -0.8012E-05   62.229803   0.5641E-05
15   50.000000   52.216499   52.216292  -0.3972E-05   52.216794   0.5641E-05
16   60.000000   46.188108   46.188022  -0.1880E-05   46.188369   0.5641E-05
17   70.000000   42.567143   42.567111  -0.7473E-06   42.567383   0.5641E-05
18   80.000000   40.617072   40.617064  -0.1754E-06   40.617301   0.5641E-05
19   89.000000   40.006093   40.006093  -0.1719E-08   40.006319   0.5641E-05
20   90.000000   40.000000   40.000000   0.3652E-16   40.000226   0.5641E-05

```
Credit/Permission: © User:Peter Mercator, 2013 / CC BY-SA 3.0.