Astrophysical Units

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The Astronomical Unit
                      The Parsec (pc)
                      The Light Year (ly)
Stellar Magnitudes
                      Absolute Magnitudes
                      Bolometric Magnitudes
                      Luminosity and Bolometric Magnitude
Other Units and Conventions

The Astronomical Unit

A natural starting point for astronomical distances is the average distance from the Earth to the Sun. As planets move in ellipses around the sun, the average distance is taken to be the semi-major axis of the earth's orbit. This distance is called the astronomical unit (AU) and is equal to 1.496 x 1011m. However the unit is only appropriate on interplanetary scales as the distances to other stars are so great as to render it too small a unit to be useful. The table shows the relative distances of the planets in AU.

Relative Distances of Planets in AU
PlanetMean Distance (semi-major axis)/A

The Parsec (pc)

Suppose we record the position of a nearby star at two points on the earth's orbit separated by a time interval of six months (note that these positions are also separated by a distance of 2 AU).
Owing to parallax, the nearby star appears to shift position relative to the more distant stars. By using simple trigonometry we can show that the parallax angle p (measured in radians) is related to the distance d of the star by...


Therefore the parsec (the word is an abbreviation of parallax and arc second) is equal to 206,264.81 AU.
A parsec is the distance at which the observed parallax of the star is equal to 1 second of arc. It is now apparent why this unit is so useful. Once the parallax of a star in arcseconds is known is then its distance is found by simply taking its reciprocal.

The Light Year (ly)

This is the distance that a photon of light would travel in one year. Since light travels at 3 x 108m s-1 we can calculate this as 9.46 x 1015m.

Link Bar

Converter of km, m, parsecs, light years, cm, feet, yards etc.

Stellar Magnitudes

When you look up at the stars at night, it is obvious that some are brighter than others. However this is deceptive since the observed brightness of an object clearly depends on how far away you are from it.
The luminosity L of a star is how much energy in joules it radiates per second and is measured in watts. The emitted light spreads out as an inverse square law and if we imagine the star as a 'point source' centered on a sphere of radius R then the energy passing through each square metre every second is simply the luminosity divided by the surface area of the sphere. We therefore define the brightness b of a star as...

Mass Defect
                                                            MG-34 Space
and represents the light flux received.

Astrophysicists prefer to talk about a star's brightness rather than its luminosity and this is expressed as a scale of magnitudes. The magnitude scale on which stars are rated is based on a convention first devised by Hipparchus. They are classified by apparent magnitude, rating the brightest stars that can be seen with the naked eye as magnitude 1.0 and the faintest as magnitude 6.0. Later, with the invention of the telescope and for consistency, it became neccessary to assign to some stars a magnitude brighter than 1.0. Hence Vega has magnitude 0 and Sirius magnitude -1.4 whereas the sun has a magnitude of -26.74. It is important to understand that the more negative the value of apparent magnitude, the brighter the star appears. Conversely, the larger the magnitude (more positive) the fainter the star appears.

In 1856, an English astronomer, Norman Pogson, formulated Hipparchus's somewhat subjective scale into a precise mathematical law expressed as:

m1 = apparent magnitude of star 1
b1 = received brightness of star 1
m2 = apparent magnitude of star 2
b2 = received brightness of star 2

Absolute Magnitudes

Now suppose you could place all the stars at a fixed distance from the earth. The differing distances would not then be a factor in how bright the stars appeared. Instead, the differences in magnitude would only be due to differences in luminosities and these values would be absolute.
Astronomers use a standard distance of 10 parsecs for absolute magnitude comparison. We therefore define the magnitude that a star would have if it was placed 10 pc from the earth as its absolute magnitude.
What is the relationship between a star's apparent magnitude (that we observe), and its absolute magnitude? A calculation using Pogson's Law yields...

m - M = 5logd - 5

The above equation is called the distance - magnitude relation which relates apparent magnitude m, absolute magnitude M and distance from the earth d in parsecs. If a star's distance is known and its apparent magnitude measured, then we can determine its absolute magnitude.

Bolometric Magnitudes

Allowances have to be made for the fact that some stars may emit a significant amount of their light in the non-visible part of the electromagnetic spectrum. For example, very luminous hot stars of appear dim to the eye only because they emit most of their radiation in the ultraviolet and, in addition, the earth's atmosphere absorbs many non-visible wavelengths. It can be seen then that absolute magnitudes as deduced from apparent magnitudes using the distance magnitude relation need to be corrected for these effects.
Astronomers define a bolometric magnitude which is the star's apparent magnitude measured above the earth's atmosphere over all wavelengths of electromagnetic radiation and is based on an instrument called a bolometer.
Despite this, there is a simple equation connecting the absolute bolometric magnitude Mbol and absolute magnitude, this is...

Mbol = M - BC

where BC is called the bolometric correction, Mbol, the bolometric magnitude is always greater than M the visual magnitude and the BC is comparatively large for both the hottest and coolest stars.
Astronomers refer to visual magnitudes as the stars magnitude over the wavelength range of the human eye, while bolometric magnitudes are those measured at all wavelengths.

Luminosity and Bolometric Magnitude

Luminosities are normally expressed in terms of the sun's luminosity Lo. For two stars 1 & 2 situated at the same distance d, the ratio of their brightness is equal to the ratio of their luminosities. Now suppose we let star 1 be the sun, it can be shown that...

We now have an equation relating a stars luminosity to its absolute bolometric magnitude and so if we know the stars Mbol we can work out how much power the star is radiating from its surface (Actually this is an underestimate, as we have not taken into account the absorption of starlight by interstellar matter).

Link Bar

Lecture notes on measuring stars-magnitudes, motion and spectral types

Good NASA site about the COBE Bolometers

Other Units and Conventions

Astronomers and astrophysicists generally use the SI system of units however some measurements are expressed in multiples relative to the sun so that the values...

Mass of sun Mo = 1.989 x 1030 kg
Luminosity of sun Lo = 3.90 x 1026 W
Radius of sun Ro = 6.96 x 108 m

can be used for comparison, with the subscript O as the accepted symbol for the sun i.e. the luminosity of the star Sirius A is 23 Lo, and it has a radius of 1.8 Ro.

A unit that is used by radio astronomers is the Jansky. The energy flux received from astronomical objects is generally very small and at radio wavelengths, the 'radio brightness' is measured per unit frequency. The Jansky (named after Karl Jansky 1905-1950 who first discovered the existence of radio waves from space) is defined as:
1Jansky (Jy) = 10-26 W m-2 Hz-1.
Radio emissions from the sun during intense solar activity can be as high as 108 - 109 Jy although most celestial sources are less than a few Jy.

Link Bar

A biography of Karl Jansky

A short biography of Karl Jansky with a picture

For more information on famous physicists please click here to goto our 'People' page

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