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:

- 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).
- 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.
- 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.

- 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).

- 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.
- 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).

- 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. - 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).

- 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

Image link: Wikimedia Commons.

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