Absolute Magnitude Absolute Magnitude For Stars (M)

It allows the overall brightnesses of objects to be compared without regards to distance.

1 Absolute Magnitude for stars (M)

In stellar astronomy, the standard distance is 10 parsecs (about 32.616 light years, or 3×10¹⁴ kilometers). (A star at ten parsecs has a parallax of 0.1")

In defining absolute magnitude it is necessary to specify the type of electromagnetic radiation being measured. When referring to total energy output, the proper term is bolometric magnitude. The dimmer an object (at a distance of 10 parsecs) would appear, the higher its absolute magnitude. The lower an object's absolute magnitude, the higher its luminosity. A mathematical equation relates apparent magnitude with absolute magnitude, via parallax.

Many stars visible to the naked eye have an absolute magnitude which is capable of casting shadowThis article is about the optical phenomenon; for other meanings, see Shadow (disambiguation). A shadow is a dark shape, e. on the ground or a wall, caused by an object (or person, etc. blocking light. Like a silhouette, the shape of the shadow is a two-s from a distance of 10 parsecs; RigelRigel Beta Orionis, is the seventh brightest star in the sky at magnitude 0. Though designated "Beta" for Orion, it actually appears brighter than the "Alpha", Betelgeuse. The distance to Rigel is somewhere between 700 and 900 light years; Hipparcos best (-7.0), DenebDeneb (also known as Alpha Cygni is the brightest star in the Cygnus constellation despite being at least thirty times more distant than the others. It is or has been known by a number of other names, including Arided Aridif HR 7924 and HD 197345 . At an (-7.2), NaosNaos (also known as Suhail Hadar or zeta; Puppis , a white supergiant in the constellation of Puppis, is one of the brightest stars in the Milky Way, in terms of absolute magnitude. Visually, it is tens of thousands of times brighter than the Sun; most of (-7.3), and BetelgeuseBetelgeuse also called alpha; Orionis is a red supergiant star in the constellation of Orion. It is the 10th brightest star in the sky, and is a vertex of the Winter Triangle. Its proximity to Earth combined with its enormous size make it (aside from the (-5.6).

1.1 See also

For comparison, SiriusThis article is about the star. See Sirius (disambiguation) for other uses of the name. Sirius ( α Canis Majoris, also known as the Dog Star is the brightest star (−1. 46m) in the night sky. This star can be seen from every inhabited region of has an absolute magnitude of 1.4 and the Sun has a magnitude of 4.5/4.8. Absolute magnitudes generally range from -10 to +17.

1.2 Computation

You can compute the Absolute Magnitude of a star given its apparent magnitude and distance:

M_star=m_star+2.5*log₁₀((10/dist_star(in parsecs))²)

M_star=m_star+5*(1+log₁₀(parallax_star))

M_star=m_star+2.5*log₁₀((32.616/dist_star(in lyrs))²)

1.2.1 Example

Rigel has a visual magnitude of m_v=0.18 and distance about 773 lyrs.
- M_{v_Rigel}=0.18+2.5*log₁₀((32.616/773)²) = -6.7
Vega has a parallax of 0.133", and an apparent magnitude of 0.03
- M_{v_Vega}=0.03+5*(1+log₁₀(0.133)) = +0.65
Alpha Centauri has a parallax of 0.750" and an apparent magnitude of -0.01
- M_{v_*}=-0.01+5*(1+log₁₀(0.750)) = +4.37

1.3 Apparent magnitude

Given the absolute magnitude, you can also calculate the apparent magnitude from any distance:

m_star=M_star-2.5*log₁₀((32.616/dist_star(in lyrs))²)

2 Absolute Magnitude for planets (H)

For planets, comets and asteroids a different definition of absolute magnitude is used which is more meaningful for nonstellar objects.

In this case, the absolute magnitude is defined as the apparent magnitude that the object would have if it were one astronomical unit (au) from both the Sun and the Earth and at a phase angle of zero degrees. This is a physical impossibility, but it is convenient for purposes of calculation.

2.1 Calulations

Formula for H: (Absolute Magnitude)

H_body = m_sun-2.5*log₁₀(a_body*d_body²)
Where
- m_sun = apparent magnitude of sun at 1 au = -26.73
- a_body=geometric albedo of body (between 0 and 1)
- d_body= diameter of spherical body (in au)

H_body ~= 2.5*log₁₀(1329/(a_body*d_body²(in km)))
- a_body=geometric albedo of body (between 0 and 1)
- d_body= diameter of spherical body (in km)

2.1.1 Example

Moon: a_moon = 0.12, d_moon = 3476 km

H_moon ~= 2.5*log₁₀(1329/(0.12*3476²)) = +0.21

2.2 Apparent magnitude

The absolute magnitude can be used to help calculate the apparent magnitude of a body under different conditions.

m_body = H_body -2.5*log₁₀(p_body(χ)*(d_SunToBody*d_{DistanceToBody})²)
Where
- χ = phase angle
  - Law_of_cosines cos(χ) = ( ( d_{DistanceToBody}² + d_BodyToSun² - d_{DistanceToSun}²) / (2 x d_{DistanceToBody} x d_BodyToSun) ).
- p_body(χ) = phase integral (integration of reflected light) (<1)
  - Example: (An ideal diffuse reflecting sphere) - A reasonable first approximation for planetary bodies
    - p(χ) = (2/3)*((1-χ/π)*cos(χ)+sin(χ)/π)
      - A full-phase diffuse sphere reflects 2/3 as much light as a diffuse disc of the same diameter
- Distances: (MUST be in au)
  - d_{DistanceToBody} = distance betweeen observer and body
  - d_SunToBody = distance between sun and body
  - d_{DistanceToSun} = distance between obverser and sun

2.2.1 Example

Moon

H_moon = 0.21
d_SunToBody = d_{DistanceToSun} = 1 au
d_{DistanceToBody} = 384500 km = 2.57x10^-3 au
How bright is moon from earth?
- Full moon: χ = 0, (p(χ) ~= 2/3)
  - m_moon = 0.21 -2.5*log₁₀(2/3*(2.57x10^-3)²) = -12.4
    - (Actual -12.7) A full moon reflects 30% more light at full phase than a perfect diffuse reflector predicts.
- Quarter moon: χ = 90, (p(χ)~=2/(3*π) if diffuse reflector)
  - m_moon = 0.21 -2.5*log₁₀(2/(3*π)*(2.57x10^-3)²) = -11.1
    - (Actual ~ -11.0) The diffuse reflector formula does better for smaller phases.

Astronomy

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