Letâs assume that you have the object exactly in the center of the FOV and
that the object is so far away that the light from each point on the object is
coming into the telescope as parallel rays. The rays from the center of the
object will focus on the optical (central) axis of the scope. The light from
points away from the center of the object are coming in at a small angle off of
the optical axis and focus to the side of the axis thus making an image that
represents the angular size of the object. The graphic shown by clicking on the
following link shows three sets of incoming parallel light rays. The green rays
are from the center of the object and focus on the axis. The red and blue
traces are for points on the object that are away from its center.
http://naturalimagesgallery.com/telescope_off_axis_ray_trace.gif ;
<http://naturalimagesgallery.com/telescope_off_axis_ray_trace.gif>
Let the blue rays represent the limb or edge of the object. The distance off
the optical axis where it comes to focus shows the size of the image created by
the telescope. This is not the divergence of light from the object but how the
object looks from the point of view of the telescope. In the case of a star,
the light from it radiates in all directions away from it. We only see the
light rays emitted in our direction. Each point on the object radiates light in
all directions so even those point on the limb radiates some light in our
direction. Does this explanation help?
Rick
On Aug 29, 2018, at 1:57 PM, Dan Heim <dan@xxxxxxxxxxxxx> wrote:
The star's limb will define the maximum off-axis angle and that is half of
the apparent diameter of the star. If the apparent diameter is 1 arc-sec,
then the maximum angle of the off-axis rays are 0.5 arc-sec.<<
When it comes to ray tracing, you look at the rays coming in along the
optical axis and those that define the field of view (FOV). In this case, the
apparent size of Sirius is much smaller than the FOV but it does have rays
that come from its entire surface. The star's limb will define the maximum
off-axis angle and that is half of the apparent diameter of the star. If the
apparent diameter is 1 arc-sec, then the maximum angle of the off-axis rays
are 0.5 arc-sec. You need to take into account the diffraction effects of
having a finite aperture. This will cause point objects to focus to a finite
disc called the Airy disc (named after George Bidell Airy). The size of the
Airy disc is only dependent on the f-ratio of the telescope. Hereâs a link
to a Wikipedia article that has information and equations for the Airy disc.
https://en.wikipedia.org/wiki/Airy_disk ;
<https://en.wikipedia.org/wiki/Airy_disk>
Rick Scott
On Aug 29, 2018, at 10:37 AM, Dan Heim <dan@xxxxxxxxxxxxx> wrote:
Thanks to all who responded to my question. I believe I have it sorted
out now. They key was first establishing a rigorous definition of "an
apparent size of 1 arcsecond" (which is reiterated by Paul below). The
flip side of my question (Does this mean that rays of light from the
object are diverging at an angle of 1 arcsecond?) is now seen as false.
Here's what I mean ...
When I see ray diagrams of light entering a telescope or eye, light from
astronomical sources is always drawn a parallel rays. It makes the
diagrams and calculations much simpler, and is for all practical
purposes true. But there's always this usually unspoken disclaimer that
the rays are only "nearly" parallel since no object is really "at
infinity". So there has to be some divergence, and I was wondering if
that divergence would equal the angular size. It would not.
Say you have an 8" aperture scope. The light rays from, say Sirius,
emanate in all directions. Only those coming from a small area of that
star (projected out to the scope) will enter the scope. Since those rays
start geometrically from the center of Sirius, the actual amount of
divergence in those "parallel" rays would depend on the distance of the
star. Of course, for any astronomical object the rays would be SO CLOSE
to parallel as to not affect the scope's optical behavior.
If we do the trig for this scenario, using Sirius and an 8" scope, the
angle of the rays projected out from the center of Sirius to an 8"
aperture would be about 1.4 x 10^-16 degrees. You can check my calcs if
you want, but I think I got that right. Just a long skinny triangle and
an inverse tangent. What's more important to me than the actual number
is understanding just how good the parallel ray assumption is ... and
where it comes from. Thanks again all for your feedback. -Dan Heim
