Distance modulus

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The distance modulus is a way of expressing distances that is often used in astronomy. It describes distances on a logarithmic scale based on the astronomical magnitude system.

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Definition

The distance modulus is the difference between the apparent magnitude (ideally, corrected from the effects of interstellar absorption) and the absolute magnitude of an astronomical object. It is related to the luminous distance in parsecs by:

This definition is convenient because the observed brightness of a light source is related to its distance by the inverse square law (a source twice as far away appears one quarter as bright) and because brightnesses are usually expressed not directly, but in magnitudes.

Absolute magnitude is defined as the apparent magnitude of an object when seen at a distance of 10 parsecs. If a light source has flux F(d) when observed from a distance of parsecs, and flux F(10) when observed from a distance of 10 parsecs, the inverse-square law is then written like:

The magnitudes and flux are related by:

Substituting and rearranging, we get: which means that the apparent magnitude is the absolute magnitude plus the distance modulus.

Isolating from the equation , finds that the distance (or, the luminosity distance) in parsecs is given by

The uncertainty in the distance in parsecs (δd) can be computed from the uncertainty in the distance modulus (δμ) using which is derived using standard error analysis. [1]

Different kinds of distance moduli

Distance is not the only quantity relevant in determining the difference between absolute and apparent magnitude. Absorption is another important factor, and it may even be a dominant one in particular cases (e.g., in the direction of the Galactic Center). Thus a distinction is made between distance moduli uncorrected for interstellar absorption, the values of which would overestimate distances if used naively, and absorption-corrected moduli.

The first ones are termed visual distance moduli and are denoted by , while the second ones are called true distance moduli and denoted by .

Visual distance moduli are computed by calculating the difference between the observed apparent magnitude and some theoretical estimate of the absolute magnitude. True distance moduli require a further theoretical step; that is, the estimation of the interstellar absorption coefficient.

Usage

Distance moduli are most commonly used when expressing the distance to other galaxies in the relatively nearby universe. For example, the Large Magellanic Cloud (LMC) is at a distance modulus of 18.5, [2] the Andromeda Galaxy's distance modulus is 24.4, [3] and the galaxy NGC 4548 in the Virgo Cluster has a DM of 31.0. [4] In the case of the LMC, this means that Supernova 1987A, with a peak apparent magnitude of 2.8, had an absolute magnitude of −15.7, which is low by supernova standards.

Using distance moduli makes computing magnitudes easy. As for instance, a solar type star (M= 5) in the Andromeda Galaxy (DM= 24.4) would have an apparent magnitude (m) of 5 + 24.4 = 29.4, so it would be barely visible for the Hubble Space Telescope which has a limiting magnitude of about 30. [5] Since it is apparent magnitudes which are actually measured at a telescope, many discussions about distances in astronomy are really discussions about the putative or derived absolute magnitudes of the distant objects being observed.

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<span class="mw-page-title-main">Parsec</span> Unit of length used in astronomy

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<span class="mw-page-title-main">Luminosity</span> Measurement of radiant electromagnetic power emitted by an object

Luminosity is an absolute measure of radiated electromagnetic energy (light) per unit time, and is synonymous with the radiant power emitted by a light-emitting object. In astronomy, luminosity is the total amount of electromagnetic energy emitted per unit of time by a star, galaxy, or other astronomical objects.

<span class="mw-page-title-main">Proper motion</span> Measure of observed changes in the apparent locations of stars

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<span class="mw-page-title-main">Cosmic distance ladder</span> Succession of methods by which astronomers determine the distances to celestial objects

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<span class="mw-page-title-main">Extinction (astronomy)</span> Interstellar absorption and scattering of light

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<span class="mw-page-title-main">Shear modulus</span> Ratio of shear stress to shear strain

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<span class="mw-page-title-main">Magnitude (astronomy)</span> Logarithmic measure of the brightness of an astronomical object

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<span class="mw-page-title-main">Mu Andromedae</span> Binary star in the constellation Andromeda

Mu Andromedae is the Bayer designation for a star in the northern constellation of Andromeda. It has an apparent visual magnitude of 3.87, making it readily visible to the naked eye. Based upon parallax measurements, it is approximately 130 light-years from Earth. In the constellation, the star is situated about halfway between the bright star Mirach to the southwest and the Andromeda Galaxy (M31) to the northeast.

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References

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