Encyclopedia of Color Science and Technology

2016 Edition
| Editors: Ming Ronnier Luo

Apparent Magnitude, Astronomy

Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-8071-7_200

Synonyms

Definitions

  • Elevation: angle between the geometrical horizon (as indicated by a spirit level) and the direction of the star or other object in question.

  • Extinction: The degree to which starlight is weakened by the medium through which the light travels from the star to the observer. It is similar in meaning to “opacity” but is usually expressed in magnitudes. Zero means no extinction; otherwise the extinction indicates how many magnitudes fainter the object looks due to opacity of the interstellar medium and the atmosphere.

  • Opacity: The degree to which light (or other optical radiation) is blocked when passing through the material in question. It varies between zero for something that is completely transparent and infinity for something completely opaque.

  • Orthochromatic: This term refers to a type of photographic emulsion that was widely used in the earlier days of photography. It was very sensitive to blue and green light but insensitive to red light.

  • Photometry: The measurement of quantities referred to light (visible optical radiation) as evaluated according to a given spectral luminous efficiency function (of the human eye). In astronomy it applies to the measurement of the brightness of astronomical objects.

  • Photosphere: The layer in a star’s atmosphere from which almost all of the heat and light are radiated. Because it lies at a fairly abrupt change in optical thickness, it looks almost like a solid surface and is often referred to colloquially as the “star’s surface.”

  • Transmissivity: The fraction of the input energy passing through an attenuating device or layer. It varies from unity, for a completely transparent medium, to zero, for something completely opaque.

  • Zenith angle: Angle between the local zenith and the direction in question.

Background

At the most fundamental level, the apparent magnitude of an object is a measure of how bright it looks. This is a function of the properties of the light source, its distance, the nature of the medium through which the light travels to the observer, and inevitably the properties of the observer or device used to make the measurement. When minimizing the subjective effects and trying to make the measurement more descriptive of the source and less related to the local circumstances of the observer, the parameter in question inevitably becomes less “apparent” but more useful. The history of “apparent magnitude” reflects such an evolution.

At a qualitative level, at a particular time and location, under the atmospheric conditions existing at that time, a specific observer can assess that some objects in the sky look brighter than others and possibly estimate by how much. This information may be useful to other observers present at the same location at the same time. However, recording such information for later consideration is difficult, and passing it in any usable form to another observer at another location for comparison of observations is even more so.

The development of the concept of “apparent magnitude” marks the beginning of an evolution toward producing quantities that are more intrinsic to the object being observed and can be combined with other observations. In the process, the term “apparent” becomes more a designator of a particular type of data rather than something literally true.

The flux from a distant star arriving at the top of the Earth’s atmosphere is given by
$$ {F}_{00}\left(\lambda \right)=\alpha \left(\lambda \right)\frac{W^{*}\left(\lambda \right)}{4\pi {R}^2} $$
(1)
where W* is the energy output of the star, R its distance from us, and α is the transmissivity of the interstellar medium along the path from the star to Earth. Note the wavelength dependence of these parameters. In this discussion the “00” suffix indentifies a quantity measured at the top of the Earth’s atmosphere.

Because the same value of F00 would be incidentally equal upon all points at the top of the Earth’s atmosphere, it would be common to all terrestrial observers. Therefore, in these discussions, F00 can be considered as the starting point.

In the development of the apparent magnitude concept, a number of issues arise. Firstly, since this is a comparative system, a defined calibration datum is required. In addition, starlight is not monochromatic; the atmosphere’s transmissivity (or opacity) varies with wavelength and so does the sensitivity of the detectors, sensors, or measurement devices used.

Quantifying Apparent Brightness

The concept of assigning “magnitudes” to stars was first proposed by the ancient Greeks, and has been attributed to Hipparchos [1], who was a Greek astronomer, geographer, and mathematician who lived about 190–120 BC. Accommodating the natural nonlinear response of our eyes to brightness, he assigned a magnitude scale where the brightest stars were categorized as being of the First Magnitude (m = 1), and the faintest stars visible to the unaided eye were defined as being of the Sixth Magnitude (m = 6), with each intervening magnitude corresponding to half the brightness of the next brighter class and twice the brightness of the next fainter. In this system, if the energy fluxes reaching from two stars of magnitude m1 and m2 are respectively F1 and F2, the relationship between these quantities is
$$ \frac{F_1}{F_2}={2}^{-\left({m}_1-{m}_2\right)} $$
(2)
This classification scheme for the apparent magnitude of stars was used for some time. Ptolemy used it in The Almagest, [2], his compendium of mathematical and astronomical information. Since the magnitudes as defined are exponents, taking logarithms of Eq. 2 puts it into a more convenient form
$$ { \log}_{10}\left(\frac{F_1}{F_2}\right)=-{ \log}_{10}(2)\left({m}_1-{m}_2\right)=-\left(0.30102996\dots \right)\left({m}_1-{m}_2\right) $$
(3)
so
$$ \left(3.321928\dots \right){ \log}_{10}\left(\frac{F_1}{F_2}\right)=-\left({m}_1-{m}_2\right) $$
(4)
The logarithm of the ratio of brightnesses is simply related to the magnitude difference. However having irrational numbers as the scaling factors is inconvenient, and in 1856 Norman Robert Pogson [3] redefined the magnitude scale in a more useful form. In Hipparchos’ system, a first magnitude star is defined as being 64 times brighter than one of the sixth magnitude. Pogson changed the definition of the magnitude scale, defining a sixth magnitude star to be 100 times fainter than a first magnitude star. So the ratio of brightness between two adjacent magnitudes became
$$ P={100}^{\frac{1}{6-1}}=2.5118864315\dots $$
(5)
This number has become known as Pogsons Ratio (P). Despite the apparent messiness of this number, it has an advantage in calculations. P is almost always used in the forms log10(P) and 1/log10(P), which equal exactly 0.4 and 2.5, respectively. Thus, Eqs. 2 and 3 become
$$ \frac{F_1}{F_2}={P}^{-\left({m}_1-{m}_2\right)} $$
(6)
$$ \therefore { \log}_{10}\left(\frac{F_1}{F_2}\right)=-{ \log}_{10}(P)\left({m}_1-{m}_2\right)=-0.4\left({m}_1-{m}_2\right) $$
(7)
so
$$ \left({m}_1-{m}_2\right)=-2.5{ \log}_{10}\left(\frac{F_1}{F_2}\right) $$
(8)
Since the magnitude system describes relative brightnesses, it needs to be based upon a calibration reference: a star to which a magnitude value has been assigned. Pogson first chose Polaris (also known as the Pole Star, or North Star) as a calibrator, to which he assigned a magnitude of 2. However, when this star proved to be variable in brightness, he selected Vega, which he defined as having an apparent magnitude of 0. In practice, Vega is a better choice because it is available to observers over much of the world, and since it varies in elevation with time at any given site, other stars at similar elevations at that time can easily be set up as secondary or transfer calibration standards.

The nature of the magnitude scale makes it able to accommodate stars brighter than magnitude 0, which gives them negative magnitudes. For example, Sirius, the brightest star in the constellation of Canis Major, and the brightest star in our sky other than the Sun, has a magnitude of −1.5 and the Sun a magnitude of about −26. Fainter stars, with magnitudes greater than 6 can also be accommodated. Binoculars or telescopes will reach objects far fainter than the eye can discern, which therefore have magnitudes greater than 6.

Wavelength Issues

Stars are not sources of monochromatic light; they emit a broad spectrum extending across the nominal 400–800 nm band of visible wavelengths and beyond in both the longer and shorter wavelength directions. The apparent color of a star depends upon the wavelength at which the peak brightness occurs, which in turn depends upon its photospheric temperature. The consequent star colors range from red to blue (and into the infrared for very cool stars).

The interstellar and atmospheric transmissivities are wavelength dependent, and in general so is the sensitivity of devices typically used to measure light intensity. This variability is particularly marked in the case of the human eye, which may differ between individuals and also with the same individuals, depending upon health, fatigue, etc. A classic example of the need to take this wavelength dependence into account is evident in old photographs of the constellation of Orion, made using orthochromatic photographic emulsions, which have extremely low sensitivity to the red end of the visual spectrum. Betelgeux (α Orionis), the brightest star in the constellation (a red giant), is not visible, whereas Rigel, a blue-white star, is very conspicuous. A more extreme, although non-astronomical example is in a photograph (again using an orthochromatic emulsion) of the cab of an early steam locomotive, looking into the open firebox, which shows no fire at all, although the boiler pressure gauge shows a vigorous fire must be present. To maintain the utility of the apparent magnitude observations, it is necessary to deal with wavelength dependencies, but this needs to be done in a manner that is both standardized and does not overly complicate the measurement process.

Stellar Spectral Types

The concept of “stellar spectral type” is, to a large extent, an indicator of a star’s photospheric temperature, i.e., its color. This way to describe stars is therefore intricately mixed with the magnitude/color issue, so a short description is included here. Stars come in a wide range of colors, brightnesses, and masses and with different spectral line signatures, so it was deemed useful to develop some sort of classification framework toward providing a coherent overall picture of star types and stellar evolution. The first attempts were made at a time stellar astrophysics was not a highly developed subject. One early system to classify stars used alphabetic labels. However, as knowledge improved, the trend was to classify on the basis of photospheric temperature and spectral line signatures. Some classes were found to be redundant, and the sequence of remaining classes had to be rearranged, so they were no longer in alphabetical order. The new system is in order of decreasing temperature and became O, B, A, F, G, K, M, L, T, Y. This arbitrary sequence of letters is remembered using the mnemonic “Oh Be A Fine Girl and Kiss Me. Like This? Yes!” This sequence is further subdivided using numerical interpolation (e.g., G2, A5, M0, etc.). The classes and their corresponding photospheric temperatures and colors are summarized in Table 1. In the case of the Y0 class at 600 K, these cool stars are usually described as “brown,” although the color would be more accurately described as an extremely dull red or perhaps not visible at all at optical wavelengths since its spectral peak lies in the infrared region.
Apparent Magnitude, Astronomy, Table 1

Photospheric temperatures/colors and spectral types

Photospheric temperature (K)

Spectral type

Color

35,000

O0

Blue

30,000

B0

Blue white

10,000

A0

White

7,500

F0

Pale yellow

6,000

G0

Yellow

5,000

K0

Orange

3,500

M0

Red

2,000

L0

Dull red

1,200

T0

Very dull red

600

Y0

“Brown”

The UBV(RI) Magnitude System

Stellar spectra are generally (almost) blackbody spectra corresponding to the stars’ photospheric temperatures, with superimposed absorption lines. Magnitude determination applies to the thermal emission only. Figure 1 shows blackbody spectra for the photospheric temperatures and spectral types listed in Table 1.
Apparent Magnitude, Astronomy, Fig. 1

Blackbody spectra corresponding to the photospheric temperatures of stars of spectral types O0, B0, A0, F0, G0, K0, M0, L0, T0, and Y0. The nominal bandwidths of the U, B, V, R, and I filters are shown. The spectra are all normalized

The required amplitude and wavelength-related information is included in a star’s spectrum. Stellar spectra are the primary data source for most stellar astrophysics. However, each spectrum contains a lot of information, and when studies involving large numbers of stars, or where measurements that are relatively simple and reproducible are needed, an alternative system would be useful.

To meet these needs, Johnson and Morgan [4] developed the three-filter, UBV system, which is still widely used. This method uses magnitude determinations made using three standard filters, designated U (ultraviolet), B (blue), and V (visible) respectively. To better describe cool, red stars, two additional bands have been added: R (red) and I (infrared). The objective is to describe the thermal spectrum using a small list of standardized parameters. The nominal wavelength ranges (bandwidths) are listed in Table 2.
Apparent Magnitude, Astronomy, Table 2

Nominal filter bandwidths for stellar magnitude measurements

Band designation

Band name

Low end (nm)

High end (nm)

Bandwidth (nm)

U

Ultraviolet

320

400

80

B

Blue

400

500

100

V

Visible

500

700

200

R

Red

550

800

250

I

Infrared

700

900

200

The shapes of the filter passbands for devices currently in use are far from rectangular, but there has been an effort to at least standardize the filters to improve comparability of data between sources. The I and R bands are not so cleanly delineated. The I band overlaps the V band, and the R band overlaps the I band. Some institutions have added additional filter bands to suit their particular research needs.

Table 1 also shows the wavelength ranges passed by the U, V, B, R, and I filters. These rectangular passbands are not representative of the passbands of existing filters. The bandwidths are also shown using arrows, so the overlapping bands are more clearly discernible.

For most purposes, the U, B, and V magnitudes are sufficient, and it is possible to describe the photospheric thermal spectrum of most stars using V and the magnitude differences U–B and B–V. Since the magnitudes are logarithmic quantities, the differences actually describe brightness ratios, i.e.,
$$ \left(U-B\right)=-2.5{ \log}_{10}\left(\frac{\frac{P_U}{\Delta {\lambda}_U}}{\frac{P_B}{\Delta {\lambda}_B}}\right) $$
(9)
and
$$ \left(B-V\right)=-2.5{ \log}_{10}\left(\frac{\frac{P_B}{\Delta {\lambda}_B}}{\frac{P_V}{\Delta {\lambda}_V}}\right) $$
(10)
where P is the total power passing through the filter. Conventionally, the values are offset so that a star of spectral type A0 is characterized by U–B and B–V = 0. If the filter passbands are assumed to be rectangular, the spectra in Fig. 1 produce the U–B and B–V values shown in Fig. 2. It can be seen that these values provide a good estimator for the star’s photospheric temperature. The letters above the data points on the graph are the spectral types of stars with those particular photospheric temperatures. Since stellar spectral types are temperature dependent, the UVB photometric system provides an estimator for spectral type.
Apparent Magnitude, Astronomy, Fig. 2

U–B and B–V plotted against the logarithm of the photospheric temperature for the blackbody spectra in Fig. 1. The spectral types of the stars modeled are shown. In each case the types are M0, K0, etc.

However, in reality, filter passbands are not rectangular, and stellar flux densities may vary dramatically with wavelength, so simply averaging over assumed rectangular passbands will not produce useful results. Unfortunately, making filters with rectangular passbands is very difficult, and the more effort dedicated to achieving this also makes the filters harder to reproduce, which is a critical issue where standardization between instruments and observatories is important.

For this reason, a set of standardized filters are used. These are generally those used by Johnson and Morgan [4]. They do not have rectangular passbands but have the great advantage of being fairly easy to reproduce. The wide adoption of these standard filters for stellar photometry makes it possible to combine data from different sources.

B–V and Stellar Spectral Types

Figure 3 shows a plot of spectral type against B–V for the stars listed in the bright star section of the Observer’s Handbook of the Royal Astronomical Society of Canada (RASC) [5]. For convenience in plotting, the spectral types O0, B0, A0, F0, G0, K0, and M0 are represented by the numbers, 1, 2, 3, 4, 5, 6, and 7 respectively. The subdivisions within each spectral class are coded as a decimal; for example, stars of spectral classes O5, A0, F6, G2, K5, and M3 would respectively be represented as 1.2, 3.0, 4.6, 5.2, 6.5, and 7.3. The point surrounded by the yellow halo is the Sun. The empirical equation can be used to estimate the spectral type from B–V.
Apparent Magnitude, Astronomy, Fig. 3

Numerical representation of spectral types plotted against B–V values for the stars in the RASC list of brightest stars

Today, the term “apparent magnitude” is generally taken to be equivalent to “visual magnitude”: a magnitude measurement made using a V filter.

Absorption and Scattering

To reach terrestrial observers, the starlight has to travel through the interstellar medium and then through the Earth’s atmosphere. Both of these media contain dust, molecules, and atoms that absorb or scatter some of the starlight, with the extent of these processes usually being wavelength dependent. The result is generally a weakening and reddening of the light. There are three main processes:
  • Scattering at wavelengths coinciding with molecular or atomic resonances, where the directional light drives electrons to higher energy states and then is reradiated isotropically.

  • Rayleigh scattering, which is scattering by particles smaller than the wavelength of the light (mainly atoms and molecules). This is strongest at shorter wavelengths and is what causes the sky to look blue and sunsets to be rich in reds and golds.

  • Scattering off larger particles. This shows much less of a variation with wavelength. In the atmosphere the main scattering agents are aerosols, composed of small droplets of water or pollutants. This mode of scattering is less likely in interstellar space but may occur where there are clouds of larger particles. Fog and mist are good examples.

If an intrinsically stellar measurement is needed, then both the interstellar and atmospheric effects on the starlight have to be determined and removed. Although even the densest interstellar clouds are more rarefied than the best achievable laboratory vacuum, the very large distances the light has to traverse means it has to pass through an enormous amount of material. However, in this article the emphasis is on making measurements that are comparable between observers and observing sites, so the measurements need only be referred to the top of the Earth’s atmosphere, so interstellar effects are common to all observers and not further discussed here.

The attenuation imposed by the atmosphere is by processes similar to those attenuating and scattering the light in the interstellar medium. However, an important difference is that in the case of the interstellar medium, the light path from star to the Earth is common to all observers and changes very slowly with time, whereas in the case of the atmosphere, the rotation of the Earth changes the light path through the atmosphere quickly enough to offer a means to estimate the stellar magnitude that would be observed in the absence of atmospheric attenuation. It is assumed that the transmissivity of the atmosphere is related to the total mass of atmosphere that the light passes through. If we look in the direction of the zenith (zenith angle = 0), we are looking through a certain total mass of air:
$$ {M}_0={\displaystyle {\int}_0^{\infty}\rho (h)dh} $$
(11)
where ρ(h) is the atmospheric density as a function of height. The “0” suffix indicates a quantity measured when looking at the zenith. The transmissivity of the atmosphere (the fraction of light arriving at the top of the atmosphere that reaches the ground) from an object at the zenith is defined as
$$ {F}_0\left(\lambda \right)={F}_{00}\left(\lambda \right) \exp \left(-\eta \left(\lambda \right){M}_0\right) $$
(12)
where η is an absorption parameter and M 0 the mass of air through which the light is passing. The flux density from the star is a function of wavelength and so is the atmospheric attenuation.
Assuming the magnitude measurements are restricted to zenith angles smaller than about 65° (elevation angles larger than 25°), the atmosphere can be treated as a stratified slab. In this case, the air mass through which the light passes is a simple function of zenith angle:
$$ {M}_z={M}_0 \sec (z) $$
(13)
It is unlikely that there will be any need to separate M0 and γ, so they can be combined into a single parameter β(λ), and so for a zenith angle z
$$ {F}_z\left(\lambda \right)={F}_{00}\left(\lambda \right) \exp \left(-\beta \left(\lambda \right) \sec (z)\right) $$
(14)
We have two ways to define the effect of the atmosphere on the light. The transmissivity (α) is the fraction of the incident light reaching the ground:
$$ \alpha =\frac{F_z\left(\lambda \right)}{F_{00}\left(\lambda \right)} $$
(15)
An alternative is define an opacity (ξ), where
$$ \xi \left(\lambda \right)=\beta \left(\lambda \right) \sec (z) $$
(16)
So
$$ \begin{array}{l}\alpha = \exp \left(-\xi \right)\\ {}\mathrm{or}\\ {}\xi =- \log \left(\alpha \right)\end{array} $$
(17)
As the atmosphere ranges from completely transparent to completely opaque the transmissivity (α) ranges from one to zero and the opacity (ξ) goes from zero to infinity.
The slab atmosphere model offers a means to estimate the magnitude of the star in the absence of the atmosphere, using a method attributed to Pierre Bouguer and discussed by Dufay [6]. Using Eqs. 7 and 12,
$$ {m}_z\left(\lambda \right)-{m}_{00}\left(\lambda \right)=-2.5{ \log}_{10}\left(\frac{F_{00} \exp \left(-\beta \left(\lambda \right) \sec (z)\right)}{F_{00}}\right) $$
(18)
So
$$ {m}_z\left(\lambda \right)={m}_{00}\left(\lambda \right)+2.5{ \log}_{10}(e)\beta \left(\lambda \right) \sec (z) $$
(19)
where m z is the measured stellar magnitude at a zenith angle z and m00 is the magnitude of the star before it is attenuated by the atmosphere. Equation 16 shows a simple, linear relationship between m z and sec(z). Therefore, if a series of magnitude measurements for a star are made over a range of zenith angles, a plot of m z against sec(z) will have a slope 2.5log10(e)β(λ) and intercept m∞(λ), the magnitude of the star as it would be measured at the top of the Earth’s atmosphere. This process is repeated for the U, B, and V filter bands, to obtain the U, B, and V magnitudes corrected for atmospheric absorption.

Summary

“Apparent magnitude” is literally a measure of how bright something looks. As such it includes the effects of observing conditions, locale, and the nature of the observer. In addition, stars come in a range of colors, depending on their temperatures. To make this measure useful to all terrestrial observers, it has to be less apparent and more objective. This is achieved by making stellar magnitude measurements using three filters, each covering a standard wavelength range, designated U, B, and V. The latter, the visual magnitude, covering the bandwidth 500–700 nm, replaces apparent magnitude.

References

  1. 1.
    Taub, L.C.: Hipparchus. In: John Lankford, J. (ed.). A History of Astronomy: An Encyclopedia. New York, Garland (1997)Google Scholar
  2. 2.
    Pederson, O.: A Survey of the Almagest. Springer. ISBN: 978-0-387-84825-9 (2011)Google Scholar
  3. 3.
    Pogson, N.R.: Magnitudes of 36 of the minor planets for the first day of each month of the year 1857. MNRS 17, 12 (1856)ADSCrossRefGoogle Scholar
  4. 4.
    Johnson, H.L., Morgan, W.W.: Fundamental stellar photometry for standards of spectral type on the revised system of the Yerkes spectral atlas. Astrophys. J. 117, 313 (1953)ADSCrossRefGoogle Scholar
  5. 5.
    Garrison, R.F., Karmo, T.: The brightest stars. In: Chapman, D. (ed.) Observer’s Handbook of the Royal Astronomical Society of Canada, RASC Publications, Toronto, pp. 273–283 (2013)Google Scholar
  6. 6.
    Dufay, J.: Introduction to Astrophysics. The Stars Dover Press, Chapters 1, 2 and 3, Dover Press, New York, (1964)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.D.R.A.O.National Research Council CanadaPentictonCanada