Astronomy

How is the difference of bolometric magnitudes not dependent on the stars' radii?

How is the difference of bolometric magnitudes not dependent on the stars' radii?


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The difference of 2 bolometric magnitudes is given by:

$$M_{bol, ★} - M_{bol, ☉} = -2.5 cdot log left( frac{L_★}{L_☉} ight)$$

But Pogson's equation is:

$$M_{bol, ★} - M_{bol, ☉} = -2.5 cdot log left( frac{F_★}{F_☉} ight)$$

where $F_★=frac{L_★}{4pi R^2}$, so how come the first equation isn't dependent on the radius?


The R in that equation is the distance from the star to observer, not the star radius. The light emitted from the star is distributed uniformly on a sphere of radius R, and when the light arrives to the Earth, that sphere will have a radius equal to the distance Earth-star.

Therefore, the second relation for the two fluxes is about the apparent magnitudes (which describe the brightness of an astronomical object observed from Earth), $$m-m_odot = -2.5 log F/F_odot$$

The first relation is alright. The absolute magnitudes are related with the luminosity of the star (the overall energy flux emitted by the star) and they are not dependent on the distance to the observer.


Absolute magnitude

Absolute magnitude ( M ) is a measure of the luminosity of a celestial object, on an inverse logarithmic astronomical magnitude scale. An object's absolute magnitude is defined to be equal to the apparent magnitude that the object would have if it were viewed from a distance of exactly 10 parsecs (32.6 light-years), without extinction (or dimming) of its light due to absorption by interstellar matter and cosmic dust. By hypothetically placing all objects at a standard reference distance from the observer, their luminosities can be directly compared among each other on a magnitude scale.

As with all astronomical magnitudes, the absolute magnitude can be specified for different wavelength ranges corresponding to specified filter bands or passbands for stars a commonly quoted absolute magnitude is the absolute visual magnitude, which uses the visual (V) band of the spectrum (in the UBV photometric system). Absolute magnitudes are denoted by a capital M, with a subscript representing the filter band used for measurement, such as MV for absolute magnitude in the V band.

The more luminous an object, the smaller the numerical value of its absolute magnitude. A difference of 5 magnitudes between the absolute magnitudes of two objects corresponds to a ratio of 100 in their luminosities, and a difference of n magnitudes in absolute magnitude corresponds to a luminosity ratio of 100 n/5 . For example, a star of absolute magnitude MV=3.0 would be 100 times as luminous as a star of absolute magnitude MV=8.0 as measured in the V filter band. The Sun has absolute magnitude MV=+4.83. [1] Highly luminous objects can have negative absolute magnitudes: for example, the Milky Way galaxy has an absolute B magnitude of about −20.8. [2]

An object's absolute bolometric magnitude (Mbol) represents its total luminosity over all wavelengths, rather than in a single filter band, as expressed on a logarithmic magnitude scale. To convert from an absolute magnitude in a specific filter band to absolute bolometric magnitude, a bolometric correction (BC) is applied. [3]

For Solar System bodies that shine in reflected light, a different definition of absolute magnitude (H) is used, based on a standard reference distance of one astronomical unit.


Contents

In stellar and galactic astronomy, the standard distance is 10 parsecs (about 32.616 light years, 308.57 petameters or 308.57 trillion kilometres). A star at 10 parsecs has a parallax of 0.1" (100 milli arc seconds). Galaxies (and other extended objects) are much larger than 10 parsecs, their light is radiated over an extended patch of sky, and their overall brightness cannot be directly observed from relatively short distances, but the same convention is used. A galaxy's magnitude is defined by measuring all the light radiated over the entire object, treating that integrated brightness as the brightness of a single point-like or star-like source, and computing the magnitude of that point-like source as it would appear if observed at the standard 10 parsecs distance. Consequently, the absolute magnitude of any object equals the apparent magnitude it would have if it were 10 parsecs away.

The measurement of absolute magnitude is made with an instrument called a bolometer. When using an absolute magnitude, one must specify the type of electromagnetic radiation being measured. When referring to total energy output, the proper term is bolometric magnitude. The bolometric magnitude usually is computed from the visual magnitude plus a bolometric correction, . This correction is needed because very hot stars radiate mostly ultraviolet radiation, whereas very cool stars radiate mostly infrared radiation (see Planck's law).

Many stars visible to the naked eye have such a low absolute magnitude that they would appear bright enough to cast shadows if they were at 10 parsecs from the Earth: Rigel (−7.0), Deneb (−7.2), Naos (−6.0), and Betelgeuse (−5.6). For comparison, Sirius has an absolute magnitude of 1.4, which is brighter than the Sun, whose absolute visual magnitude is 4.83 (it actually serves as a reference point). The Sun's absolute bolometric magnitude is set arbitrarily, usually at 4.75. [1] [2] Absolute magnitudes of stars generally range from −10 to +17. The absolute magnitudes of galaxies can be much lower (brighter). For example, the giant elliptical galaxy M87 has an absolute magnitude of −22 (i.e. as bright as about 60,000 stars of magnitude −10).

Computation

For a negligible extinction, one can compute the absolute magnitude of an object given its apparent magnitude and luminosity distance :

where is the star's actual distance in parsecs (1 parsec is 206,265 astronomical units, approximately 3.2616 light-years). For very large distances, the cosmological redshift complicates the relation between absolute and apparent magnitude, because the radiation observed was shifted into the red range of the spectrum. To compare the magnitudes of very distant objects with those of local objects, a k correction might have to be applied to the magnitudes of the distant objects.

For nearby astronomical objects (such as stars in the Milky Way) luminosity distance DL is almost identical to the real distance to the object, because spacetime within the Milky Way is almost Euclidean. For much more distant objects the Euclidean approximation is not valid, and General Relativity must be taken into account when calculating the luminosity distance of an object.

In the Euclidean approximation for nearby objects, the absolute magnitude of a star can be calculated from its apparent magnitude and the star's parallax in arcseconds:

You can also compute the absolute magnitude of an object given its apparent magnitude and distance modulus :

Examples

Rigel has a visual magnitude of and distance about 860 light-years

Vega has a parallax of 0.129", and an apparent magnitude of +0.03

Alpha Centauri A has a parallax of 0.742" and an apparent magnitude of −0.01

The Black Eye Galaxy has a visual magnitude of mV=+9.36 and a distance modulus of 31.06.

Apparent magnitude

Given the absolute magnitude , for objects within the Milky Way you can also calculate the apparent magnitude from any distance (in parsecs):

For objects at very great distances (outside the Milky Way) the luminosity distance DL must be used instead of d (in parsecs).

Given the absolute magnitude , you can also compute apparent magnitude from its parallax :

Also calculating absolute magnitude from distance modulus :

Bolometric magnitude

Bolometric magnitude corresponds to luminosity, expressed in magnitude units that is, after taking into account all electromagnetic wavelengths, including those unobserved due to instrumental pass-band, the Earth's atmospheric absorption, and extinction by interstellar dust. In the case of stars with few observations, it usually must be computed assuming an effective temperature.

Classically, the difference in bolometric magnitude is related to the luminosity ratio according to:

is the Sun's (sol) luminosity (bolometric luminosity) is the star's luminosity (bolometric luminosity) is the bolometric magnitude of the Sun is the bolometric magnitude of the star.

In August 2015, the International Astronomical Union passed Resolution B2 [3] defining the zero points of the absolute and apparent bolometric magnitude scales in SI units for power (watts) and irradiance (), respectively. Although bolometric magnitudes had been used by astronomers for many decades, there had been systematic differences in the absolute magnitude-luminosity scales presented in various astronomical references, and no international standardization. This led to systematic differences in bolometric corrections scales, which when combined with incorrect assumed absolute bolometric magnitudes for the Sun could lead to systematic errors in estimated stellar luminosities (and stellar properties calculated which rely on stellar luminosity, like radii, ages, etc.).

IAU 2015 Resolution B2 defines an absolute bolometric magnitude scale where corresponds to luminosity 7028301280000000000♠ 3.0128 × 10 28 watts, with the zero point luminosity set such that the Sun (with nominal luminosity 7026382800000000000♠ 3.828 × 10 26 watts) corresponds to absolute bolometric magnitude . Placing a radiation source (e.g. star) at the standard distance of 10 parsecs, it follows that the zero point of the apparent bolometric magnitude scale corresponds to irradiance = 6992251802100199999♠ 2.518 021 002 × 10 −8 . Using the IAU 2015 scale, the nominal total solar irradiance ("Solar constant") measured at 1 astronomical unit () corresponds to an apparent bolometric magnitude of the Sun of .

Following IAU 2015 Resolution B2 system, the relation between a star's absolute bolometric magnitude and its luminosity is no longer directly tied to the Sun's (variable) luminosity:

is the star's luminosity (bolometric luminosity) in watts is the zero point luminosity 7028301280000000000♠ 3.0128 × 10 28 watts is the bolometric magnitude of the star

The new IAU absolute magnitude scale permanently disconnects the scale from the variable Sun. However, on this SI power scale, the nominal solar luminosity corresponds closely to = 4.74, a value that was commonly adopted by astronomers before the 2015 IAU resolution.

The luminosity of the star in watts can be calculated as a function of its absolute bolometric magnitude as:

using the variables as defined previously.


The RR Lyrae Stars

For more than fifty years the RR Lyrae stars have played a prominent role in problems of galactic structure, first as distance indicators and more recently as population indicators. This has followed from the apparent homogeneity of the group, the large number present in the Galaxy, and their moderately high luminosities, which render them visible at large distances. However, in the past decade it has become clear that RR Lyrae stars are heterogeneous in many observable parameters. This heterogeneity limits their usefulness in some contexts and enhances their value in others. In addition to their applications in galactic research, the RR Lyrae stars are of current astrophysical interest on account of the line-doubling and emission that occur during rising light, the multiple periodicities found in many of them, and various other phase- and period-dependent phenomena that bear on the theory of stellar pulsation. In this review we summarize recent developments in these areas.


Stellar motions

Accurate measurements of position make it possible to determine the movement of a star across the line of sight (i.e., perpendicular to the observer)—its proper motion. The amount of proper motion, denoted by μ (in arc seconds per year), divided by the parallax of the star and multiplied by a factor of 4.74 equals the tangential velocity, VT, in kilometres per second in the plane of the celestial sphere.

The motion along the line of sight (i.e., toward the observer), called radial velocity, is obtained directly from spectroscopic observations. If λ is the wavelength of a characteristic spectral line of some atom or ion present in the star and λL is the wavelength of the same line measured in the laboratory, then the difference Δλ, or λ − λL, divided by λL equals the radial velocity, VR, divided by the velocity of light, c—namely, Δλ/λL = VR/c. Shifts of a spectral line toward the red end of the electromagnetic spectrum (i.e., positive VR) indicate recession, and those toward the blue end (negative VR) indicate approach (see Doppler effect redshift). If the parallax is known, measurements of μ and VR enable a determination of the space motion of the star. Normally, radial velocities are corrected for Earth’s rotation and for its motion around the Sun, so that they refer to the line-of-sight motion of the star with respect to the Sun.

Consider a pertinent example. The proper motion of Alpha Centauri is about 3.5 arc seconds, which, at a distance of 4.4 light-years, means that this star moves 0.00007 light-year in one year. It thus has a projected velocity in the plane of the sky of 22 km per second. (One kilometre is about 0.62 mile.) As for motion along the line of sight, Alpha Centauri’s spectral lines are slightly blueshifted, implying a velocity of approach of about 20 km per second. The true space motion, equal to (22 2 + 20 2 ) 1/2 or about 30 km per second, suggests that this star will make its closest approach to the Sun (at three light-years’ distance) some 280 centuries from now.


CARBON STARS

AbstractAbsolute magnitudes are estimated for carbon stars of various subtypes in the Hipparcos catalogue and as found in the Magellanic Clouds. Stellar radii fall within the limits of 2.4–4.7 AU. The chemical composition of carbon stars indicates that the C-N stars show nearly solar C/H, N/H, and 12 C/ 13 C ratios. This indicates that much of the C and N in our Galaxy came from mass-losing carbon stars. Special carbon stars such as the C-R, C-H, and dC stars are described.

Mass loss from asymptotic giant branch carbon stars, at rates up to several × 10 −5 M year −1 , contributes about half of the total mass return to the interstellar medium. R stars do not lose mass and may be carbon-rich red giants. The mass loss rates for Miras are about 10 times higher than for SRb and Lb stars, whose properties are similar enough to show that they are likely to belong to the same population. The distribution of carbon star mass loss rates peaks at about 10 −7 M year −1 , close to the rate of growth of the core mass and demonstrative of the close relationship between mass loss and evolution. Infrared spectroscopy shows that dust mixtures can occur. Detached shells are seen around some stars they appear to form on the time scales of the helium shell flashes and to be a normal occurrence in carbon star evolution.


Petrosian Magnitudes: petroMag

For galaxy photometry, measuring flux is more difficult than for stars, because galaxies do not all have the same radial surface brightness profile, and have no consistently distinct edges. In order to avoid biases, we wish to measure a constant fraction of the total light, independent of the position and distance of the object. To satisfy these requirements, the SDSS has adopted a modified form of the Petrosian (1976) system, measuring galaxy fluxes within a circular aperture whose radius is defined by the shape of the azimuthally averaged light profile.

We define the "Petrosian ratio" R P at a radius r from the center of an object to be the ratio of the local surface brightness in an annulus at r to the mean surface brightness within r , as described by Blanton et al. (2001) and Yasuda et al. (2001):

where I(r) is the azimuthally averaged surface brightness profile. The Petrosian radius r P is defined as the radius at which R P( r P) equals some specified value R P,lim, set to 0.2 in our case. The Petrosian flux in any band is then defined as the flux within a certain number N P (equal to 2.0 in our case) of r Petrosian radii:

In the SDSS five-band photometry, the aperture in all bands is set by the profile of the galaxy in the r band alone. This procedure ensures that the color measured by comparing the Petrosian flux F P in different bands is measured through a consistent aperture.

The aperture 2 r P is large enough to contain nearly all of the flux for typical galaxy profiles, but small enough that the sky noise in F P is small. Thus, even substantial errors in r P cause only small errors in the Petrosian flux (typical statistical errors near the spectroscopic flux limit of r

17.7 are < 5%), although these errors are correlated.

The Petrosian radius in each band is the parameter petroRad , and the Petrosian magnitude in each band (calculated, remember, using only petroRad for the r band) is the parameter petroMag .

In practice, there are a number of complications associated with this definition, because noise, substructure, and the finite size of objects can cause objects to have no Petrosian radius, or more than one. Those with more than one are flagged as MANYPETRO the largest one is used. Those with none have NOPETRO set. Most commonly, these objects are faint (r > 20.5 or so) the Petrosian ratio becomes unmeasurable before dropping to the limiting value of 0.2 these have PETROFAINT set and have their "Petrosian radii" set to the default value of the larger of 3&prime&prime or the outermost measured point in the radial profile. Finally, a galaxy with a bright stellar nucleus, such as a Seyfert galaxy, can have a Petrosian radius set by the nucleus alone in this case, the Petrosian flux misses most of the extended light of the object. This happens quite rarely, but one dramatic example is the Seyfert galaxy NGC 7603 = Arp 092, at RA(2000) = 23:18:56.6, Dec(2000) = +00:14:38.

How well does the Petrosian magnitude perform as a reliable and complete measure of galaxy flux? Theoretically, the Petrosian magnitudes defined here should recover essentially all of the flux of an exponential galaxy profile and about 80% of the flux for a de Vaucouleurs profile. As shown by Blanton et al. (2001), this fraction is fairly constant with axis ratio, while as galaxies become smaller (due to worse seeing or greater distance) the fraction of light recovered becomes closer to that fraction measured for a typical PSF, about 95% in the case of the SDSS. This implies that the fraction of flux measured for exponential profiles decreases while the fraction of flux measured for de Vaucouleurs profiles increases as a function of distance. However, for galaxies in the spectroscopic sample (r < 17.7), these effects are small the Petrosian radius measured by frames is extraordinarily constant in physical size as a function of redshift.


CARBON STARS

AbstractAbsolute magnitudes are estimated for carbon stars of various subtypes in the Hipparcos catalogue and as found in the Magellanic Clouds. Stellar radii fall within the limits of 2.4–4.7 AU. The chemical composition of carbon stars indicates that the C-N stars show nearly solar C/H, N/H, and 12 C/ 13 C ratios. This indicates that much of the C and N in our Galaxy came from mass-losing carbon stars. Special carbon stars such as the C-R, C-H, and dC stars are described.

Mass loss from asymptotic giant branch carbon stars, at rates up to several × 10 −5 M year −1 , contributes about half of the total mass return to the interstellar medium. R stars do not lose mass and may be carbon-rich red giants. The mass loss rates for Miras are about 10 times higher than for SRb and Lb stars, whose properties are similar enough to show that they are likely to belong to the same population. The distribution of carbon star mass loss rates peaks at about 10 −7 M year −1 , close to the rate of growth of the core mass and demonstrative of the close relationship between mass loss and evolution. Infrared spectroscopy shows that dust mixtures can occur. Detached shells are seen around some stars they appear to form on the time scales of the helium shell flashes and to be a normal occurrence in carbon star evolution.


How is the difference of bolometric magnitudes not dependent on the stars' radii? - Astronomy

bolometric correction noun

The difference between the bolometric magnitude and visual magnitude of a star

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In astronomy, a bolometric correction is a correction that must be made to the absolute magnitude of an object in order to convert an object's visible magnitude to its bolometric magnitude. Mathematically, such a calculation can be expressed: The following is subset of a table from Kaler listing the bolometric correction for a range of stars. For the full table, see the referenced work. The bolometric correction is large both for early type stars and for late type stars. The former because a substantial part of the produced radiation is in the ultraviolet, the latter because a large part is in the infrared. For a star like our Sun, the correction is only marginal because the Sun radiates most of its energy in the visual wavelength range. The bolometric correction scale is set by the absolute magnitude of the Sun and an adopted bolometric magnitude for the Sun. The choice of adopted solar absolute magnitude, bolometric correction, and absolute bolometric magnitude are not arbitrary, although some classic references have tabulated mutually incompatible values for these quantities . The bolometric scale historically had varied somewhat in the literature, with the Sun's bolometric correction in V-band varying from -0.19 to -0.07 magnitude. Since the Sun is also a variable star, and there are minor differences in adopted solar luminosity values, in 1999 two IAU commissions agreed to separate the definition of bolometric correction and magnitude from the variable Sun. The 1999 IAU statements define that absolute bolometric magnitude zero correlates to a bolometric luminosity of 3.055e28 Watts. This particular luminosity was selected as the zero-point for the absolute bolometric magnitude scale so that the Sun's luminosity would correspond to absolute bolometric magnitude 4.75. As the Sun has an apparent V magnitude of -26.75, and absolute V magnitude of 4.82, then the IAU bolometric magnitude scale implies that the bolometric correction for the Sun is -0.07 magnitude. The new IAU definition means that theoretical evolutionary models for stars can define brightnesses in terms of bolometric and absolute magnitudes on a scale that is tied to a physical quantity rather than to the Sun.


The Ages of Stars

The age of an individual star cannot be measured, only estimated through mostly model-dependent or empirical methods, and no single method works well for a broad range of stellar types or for a full range in age. This review presents a summary of the available techniques for age-dating stars and ensembles of stars, their realms of applicability, and their strengths and weaknesses. My emphasis is on low-mass stars because they are present from all epochs of star formation in the Galaxy and because they present both special opportunities and problems. The ages of open clusters are important for understanding the limitations of stellar models and for calibrating empirical age indicators. For individual stars, a hierarchy of quality for the available age-dating methods is described. Although our present ability to determine the ages of even the nearest stars is mediocre, the next few years hold great promise as asteroseismology probes beyond stellar surfaces and starts to provide precise interior properties of stars and as models continue to improve when stressed by better observations.