Telescope Q: Does Larger Diameter Mean Objects Appear Brighter?

[JonnyW]

If a telescope is 2 time bigger in diameter then you can see object 4 times fainter, but does that mean that object appear 4 times brighter?

 

[Artman]

Hi JonnyW,

Not exactly. The one number is a magnitude measurement. The actual brightness difference is many times more than the number of magnitude. Here is a site that explains the magnitude system.

http://www.twcac.org/Tutorials/brightness_of_stars.htm

 

[StephenPrivitera]

I think you can do a rough calculation of limiting magnitude using the fact that your eye has a 1cm aperture and can see to the 6th mag.
$$m_{lim} - m_{eye}= -2.5log \frac {f_{tel}} {f_{eye}} = -2.5log \frac {A_{eye}} {A_{tel}}$$
$$=-2.5logR^2_{eye}/R^2_{tel}$$
Use 6 for m_eye and 1cm for R_eye
Yes, I believe the object appears four times brighter. That is, the object actually is four times brighter. But your eye responds to light logarithmically so it will seem about log4 times as "bright". It depends on how you define "bright".
I think that's how it works. One of the smart guys will correct me if it's not.

 

[Labguy]

Originally posted by StephenPrivitera
I think you can do a rough calculation of limiting magnitude using the fact that your eye has a 1cm aperture and can see to the 6th mag.
$$m_{lim} - m_{eye}= -2.5log \frac {f_{tel}} {f_{eye}} = -2.5log \frac {A_{eye}} {A_{tel}}$$
$$=-2.5logR^2_{eye}/R^2_{tel}$$
Use 6 for m_eye and 1cm for R_eye
Yes, I believe the object appears four times brighter. That is, the object actually is four times brighter. But your eye responds to light logarithmically so it will seem about log4 times as "bright". It depends on how you define "bright".
I think that's how it works. One of the smart guys will correct me if it's not.

Well, I am not one of the "smart guys", but I have been using and making telescopes for many years and can add just a bit to the original question on "telescopes".

First, a telescope twice as big gathers four times as much light; everyone agrees. But, "four times brighter" can be true in some instances, but 4-times brighter does not mean four magnitudes. The magnitude system is an arbitraty system agreed on in the late 1800's by astronomers to standardize the system all were using with their old photometers to measure brightness back then. They settled on a system where a difference of 5 magnitudes would be "exactly" a difference in brightness of 100 times. So, one magnitude equals the 1/5th root of 100, which is about 2.5119. Most just use 2.512; close enough.

So, a difference in objects of one magnitude means the brighter one is 2.5119 times brighter than the other. A difference of 2 magnitudes = a difference of 6.310 times in brightness. Four magnitudes is a difference of 39.811 in brightness, so you can see that brightness as perceived by the eye actually has nothing to do with the magnitude system unless you just happen to want to be able to translate the two; eye to magnitude or, more likely, photometer to magnitude.

As for telescopes and brightness (perceived by the eye) a whole lot of factors come into play, too many to list here except the main ones. Brightness, to the eye, depends on what is usually called the exit pupil delivered by the telescope. The exit pupil (EP) is usually expressed in millimeters and is just the entrance pupil of the telescope (diameter) divided by the power of the particular eyepiece being used. Turn the math around and exit pupil just becomes the focal length of the eyepiece (in mm) divided by the f/ratio ot the telescope's objective (lens or mirror). So, the "brightness" of an object will only be four times brighter in the twice-as big telescope if the same focal length eyepiece is used in both scopes and]/b] both scopes have the same focal length . It is safe to equate exit pupil with "brightness". Just think of the exit pupil as being the diameter of the "beam of illumination" coming out of the telescope / eyepiece being used, because that's exactly what it is!

Example 1:
-8", f/3 scope (24" focal length) with a 15mm eyepiece has exit pupil of 5.0mm. (15mm divided by 3).
-4", f/6 scope (24" focal length) with a 15mm eyepiece has exit pupil of 2.5mm. (15mm divided by 6).
So here, the 8" scope is twice as bright as the 4" scope (5.0 vs. 2.5). Also note both will show the same "power" of about 40.6X, since power is just the focal length of the objective divided by the focal length of the eyepiece. (inches were converted to millimeters)

Example 2: (Clone 4" scope to twice the size)
-8", f/6 scope (48" focal length) with a 15mm eyepiece has exit pupil of 2.5mm. (15mm divided by 6).
-4", f/6 scope (24" focal length) with a 15mm eyepiece has exit pupil of 2.5mm. (15mm divided by 6).
Here, both scopes have the same exit pupil so the 8" scope is not twice as "bright", it is the same at 2.5mm exit pupil. But, do the division and notice that the 8" scope now gives twice the power of the 4" scope; that's the trade-off between "power vs. brightness".

Go to: http://www.klhess.com/telecalc.htm and enter any scope / eyepiece combo to see what you get; it's fun.

As for the human eye, there are limiting factors for sure. The 1cm (10mm) eye quoted above is a bit off. The largest mosy any eye will dialate in total darkness is about 7mm. But, this is very rare and is limited to very young people and only if fully "dark-adapted". Getting dark adapted doesn't just mean being in a dark place for awhile, it takes total darkness with no other light source around to ruin it. Also, the maximun an eye will dialate decreases, in everyone, with age, and anyone over about 25 will start to lose the ability to dialate to 7mm. Most people over about 40 can't even get to 5mm. On a sunny day our eyes are only dialated to about 2mm to 2.5mm. If we only have a 2.5mm eye opening, there is no sense (reason) to use something (binoculars for instance) that give an exit pupil (light-beam diameter) larger than that since all the light can't enter the eye anyway. This is why, in daylight, 7X50 binoculars (~7mm EP) will not make things seem any brighter than 7X35 binocs (5mm EP). With telescopes, reasonable EP's to use will range from about 5.0mm (bright, low power) down to ~ 0.8mm (dimmer, high power). The upper limit is a limit based on our eyes and the lower limit is a limit based on the telescope's abilities. It takes a very dark sky location and great "seeing conditions" (atmosphere) to push a scope's power to anything that gives a smaller EP than about 0.8mm.

Sorry for the LONG post, I just happen to like this stuff. Chroot and another (I forgot) can give you as much and more on this if interested.

 

[russ_watters]

Originally posted by Artman
Not exactly. The one number is a magnitude measurement.

I'm not sure it is. If one brings in 4x as much light, things should appear 4x brighter. That does only correspond to a magnitude difference of 1.6 though(if I'm using the scale correctly).

This is also complicated by the fact that our vision doesn't have linear brightness sensitivity.

 

[Labguy]

To use reasonable decimals, 1.51 magnitudes is a difference of 4.018 times in brightness; close enough.

 

[chroot]

Originally posted by russ_watters
This is also complicated by the fact that our vision doesn't have linear brightness sensitivity.

That's a huge fact.

Also, note that a telescope's ability to pack all the photons from a particular light source (i.e. star) into a small area on your pupil has a huge effect on the limiting magnitude of the eye-telescope system.

In other words, a telescope that comes to a very crisp, perfect focus can show you MANY times more objects than a telescope of the same aperture, but with impaired focus. Mirror quality, collimation, and so on can all affect the ability of 'scope to reach good focus.

- Warren