ASTR 3130, Majewski [SPRING 2023]. Lecture Notes

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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 = -2.5log₁₀(L) + Const.

Note: More luminous stars have smaller M.
Clearly, the flux, the amount of power per square centimeter (cm²) received on Earth from a source, depends on the source luminosity AND distance r by the Inverse Square Law:
flux = f (erg / s cm²) = L (erg/s) / 4 * π * r²

The basis for the observed flux in the above equation is demonstrated in this image, from http://universeadventure.org/fundamentals/popups/light-dtrh-luminos.htm.
The apparent magnitude (m) of a source is the amount of energy seen from a distance r per cm² on the Earth:
m = -2.5 log₁₀( f ) + const.

Note: Apparently brighter stars have smaller m.
We can compare either absolute or apparent mags between sources, and find:
M₁ - M₂ = -2.5 log₁₀(L₁ / L₂ )
or
m₁ - m₂ = -2.5 log₁₀(f₁ / f₂ ) = -2.5 log(L₁ / L₂ ) - 5 log(r₂ / r₁ )

Note:
- If the sources are at the same distance ( r₁ = r₂ ) but have different luminosities, then:
  m₁ - m₂ = -2.5 log(L₁ / L₂ )
  (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).
- If the sources have the same luminosity ( L₁ = L₂ , i.e., we know the sources are the same type), but are at different distances, then:
  m₁ - m₂ = -5 log₁₀(r₂ / r₁ )

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/s-cm²), depends on the area A of the collecting aperture:

P = fA or P α fd ² (for a circular aperture)

Therefore, f α P / d ²
Then: m = -2.5 log₁₀(P) + 5 log₁₀(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 log₁₀(P_lim) + 5 log₁₀(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 log₁₀(d₁) - 5 log₁₀(d₂)

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 dark-adapted 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 log₁₀(d₁) - 5 log₁₀(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 14-inch 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.

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.