ASTR 3130, Majewski [SPRING 2015]. Lecture Notes
ASTR 3130 (Majewski) Lecture Notes
TELESCOPES: LIGHT GATHERING POWER AND LIMITING MAGNITUDE
REFERENCE:
 Chapter 6, Birney, Gonzalez & Oesper.
 The magnitude system is described in Birney, Gonzalez & Oesper, Chapter 5 or in Roy & Clarke,
Section 15.6.
 Limiting magnitude is described in Roy & Clarke, Section 17.4 or p.106 of
Birney, Gonzalez & Oesper.
LIGHT GATHERING POWER
The Light Gathering Power (L.G.P.) of a telescope is its most
important function: The more photons an astronomer can collect from an
object, the more information that can be extracted.
(NOTE: Telescopes are described by the diameter of the
objective,
not the length of the "tube").
 Doubling the diameter of a "light bucket" quadruples its light gathering
power.
 The image formed from a 2 times larger diameter objective is 4 times brighter (when the image is
projected to the same size  i.e., for the same focal length).
 One way to compare the relative light gathering powers of telescopes
is to compare the true brightness of the faintest possible sources
one can see through each telescope.
 We are interested in the limiting visual brightness of a telescope
(for the present discussion, viewing with the human eye as the light detector).
LIMITING BRIGHTNESS / MAGNITUDE
 Astronomers discuss the relative brightness of distant objects on a
logarithmic scale, because of the huge range of luminosities encountered, and because
the human eye response to light operates logarithmically.
 We will discuss the magnitude system in more detail later in the semester,
but will give a thumbnail overview here.
 Magnitudes describe brightness as a logarithmic scale.
 The scale is inverted, with brighter sources having a smaller magnitude value.
 Imagine a source of intrinsic luminosity, L, (erg/s).
 The absolute magnitude (M) of the source is
defined as:
M = 2.5log_{10}(L) + Const.
Note: More luminous stars have smaller M.
 Clearly, the flux, the amount of power per cubic centimeter
(cm^{2}) received on Earth from a source,
depends on the source luminosity AND distance r by the
Inverse Square Law:
flux = f (erg / s cm^{2})
= L (erg/s) / 4 * π * r^{2}
The basis for the observed flux in the above equation is demonstrated in this image,
from http://universeadventure.org/fundamentals/popups/lightdtrhluminos.htm.
 The apparent magnitude (m) of a source is the amount of
energy seen from a distance r per cm^{2} on the Earth:
m = 2.5 log_{10}( f ) + const.
Note: Apparently brighter stars have smaller m.
 We can compare either absolute or apparent mags between
sources, and find:
M_{1}  M_{2} =
2.5 log_{10}(L_{1} / L_{2} )
or
m_{1}  m_{2} =
2.5 log_{10}(f_{1} / f_{2} ) =
2.5 log(L_{1} / L_{2} ) 
5 log(r_{2} / r_{1} )
Note: If the sources are at the same distance, then:
m_{1}  m_{2} =
2.5 log(L_{1} / L_{2} )
(i.e., the same result as above for absolute magnitudes, which are actually
defined to be the flux received at the fixed distance of 10 pc).
Note: If L_{1} = L_{2} , (i.e., we know the
sources are the same type but are at different distances), then:
m_{1}  m_{2} =
5 log_{10}(r_{2} /
r_{1} )
Aside:
Because we actually define absolute magnitude as the brightness a
source has at r = 10 pc, we derive an important formula, known as
the distance modulus formula, which relates the apparent magnitude
of a source of known M to its distance r:
m  M = 5 log(r / 10pc) = 5 log(r)  5
Now, we define the limiting magnitude to be the faintest source
visible to the eye, either unaided or through a telescope. Clearly, the
amount of Power, P, (erg/s), collected from a flux, (erg/scm^{2}),
depends on the area A of the collecting aperture:
P = fA or P ∝ fd ^{2}
(for a circular aperture)
Therefore, f ∝ P / d ^{2}
Then: m = 2.5 log_{10}(P) +
5 log_{10}(d) + const.'
We define the limiting magnitude as the magnitude where the
received power P drops to an arbitrarily low value  say that point below
which the rods in the eye cannot detect the source, P_{lim}, then:
m_{lim} = 2.5 log_{10}(P_{lim}) +
5 log_{10}(d) + const.
All other things equal (sky conditions, dark adaptation of the
eye), and assuming that the magnification is large enough that the
exit pupil is smaller than the dilated human eye,
one might then determine that the limiting magnitude of a telescope depends
only on its diameter as
m_{lim,1} = m_{lim,2} +
5 log_{10}(d_{1}) 
5 log_{10}(d_{2})
Finally, the apparent magnitude system was originally designed (by Pogson,
1850s, after a system originally devised by the ancient Greeks)
so that the faintest objects visible to the
darkadapted naked eye are defined to have m = 6.
The dark adapted eye is about 7 mm in diameter. We find then that the limiting
magnitude of a telescope is given by:
m_{lim,1} = 6 + 5 log_{10}(d_{1}) 
5 log_{10}(0.007 m)
(for a telescope of diameter = d in meters)
m_{lim} = 16.77 + 5 log(d / meters)
This is a theoretical limiting magnitude, assuming perfect transmission
of the telescope optics.
 Roy & Clarke (Section 17.4) add in a telescope light transmission efficiency
of 65% (and also assume a dilated pupil of 8.0 mm in their discussion).
Taking the inefficiencies into account means you will not see quite as
faintly, and this adjusts the approximate limiting magnitude to
something more like
m_{lim} = 16 + 5 log(d / meters)
In Lab 2, you will compare m_{lim} for a number of
different aperture stops on a 6inch telescope.
 NOTE: visual limiting magnitude, as defined here,
pertains to human eye viewing through an eyepiece. It is
thus limited to the energy received in the temporal neural
resolution of the eye, which is about 30 Hz.
This is the limiting magnitude discussed above.
 Therefore, one can always see fainter sources if we increase
the light receiving time past what the eye can do, i.e.
"integrate" with film or CCD camera.
Thus new limiting magnitudes can be calculated after taking into account
the properties of the detector and integration.
All material copyright © 2002,2006,2008,2012,2015 Steven R. Majewski. All
rights reserved. These notes are intended for the private,
noncommercial use of students enrolled in Astronomy 313 and Astronomy 3130 at the
University of Virginia.
