Distances and Magnitudes
Distances
In the words of Douglas Adams, the author of The Hitch-Hiker's Guide to the Galaxy:
Space is big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to space.
The distances involved in the universe are so vast that we must introduce some new units.
The Astronomical Unit A.U.
The most practical units we can use are related to the distance from the Sun to the Earth. This is the A.U. or Astronomical Unit. 1 Astronomical Unit = 149 598 000 km
The Parsec
Parallax is the apparent shift in the nearest stars due to the motion of the Earth around the Sun. The method of parallax gives rise to a natural distance unit that astronomers call the parsec (which we shall abbreviate as pc). The parsec is defined to be the distance at which a star would have a parallax angle p equal to one second of arc.
Taking the unit is the parsec. This is defined as
1 Parsec = 3.08568025 × 1016 m. also used are kpc =1000 pc and Mpc =1 million pc
Light Year
The light year is the distance travelled by light in one year. All electromagnetic waves travel at a speed of x 299,792,458 ms-1 and an average year being 365.25 days. One light year is 299,792,458 x 108ms-1 x (365.25 x 24 x 60 x 60) s =
9.46073 x 1015 m. or 9.46073 x 1012 km.
With our new measuring sticks we can give a few examples of the scale of the universe.
The distance from the Earth to the nearest star after our Sun is 4.42 ly.
The Milky Way Galaxy is about 150,000 light-years across
The andromeda galaxy is 2.3 million light-years away.
The edge of the observable universe is 46.5 Giga light years away.
Magnitude of Stars
Apparent Magnitude
Early Greek astronomers used a scale of magnitude devised by Hipparchus around the 2nd century BC, which was based on how bright stars appeared with the naked eye. The Hipparchus scale went from magnitude 1, for the brightest stars, up to magnitude 6, for those stars which were barely visible.
When telescopes were invented, it was possible to compare the intensities of the light from stars and it was found that the brightest stars of magnitude 1, were around 100 times the intensity of the faintest stars at magnitude 6. Therefore, each magnitude was, 100 = x5. Then each increase in magnitude was 2.512 times fainter than the last. Telescopes allowed astronomers to observe much fainter stars and now the apparent magnitude scale goes from s to around f,
| Name | Apparent Magnitude | Distance from Earth (AU) |
|---|---|---|
| Sun | -26.74 | 1 |
| Full Moon | -12 | 1 |
| Venus | -4.71 | x |
| Sirius | 1.4 | x |
| Canopus | 0.7 | 1x |
| Faintest Stars | -30 | x |
The apparent brightness of a star is how bright it seems when viewed from the Earth, but a large bright star can appear dim if it is a long way from the Earth and a dim star can appear to be bright if it is close to the Earth, therefore the apparent magnitude has no bearing on the distance from the Earth.
To give an acurate measurement of the brightness of a star we need to make an absolute magnitude scale. The absolute magnitude is how bright a star is when viewed from a distance of 10 parsecs.
In order to find the absolute magnitude we need to know the distance of the star from the sun, how do we do this? The intensity of light decreases with distance from the star. The rate at which it decreases is inversely proportional to the square of the distance. Thus if we have a star of luminosity L if we move a distance d the same quantity of light has to cover a larger spherical area. Therefore, away the intensity
I = L/(4πd2).(1)
The apparent magnitude m is given by
m = - 2.5 log10(I + c)(2)
Where, I is the observed intensity of the object.
From the properties of logarithms, the ratio of the intensities of two stars is.
m1 - m2 = - 2.5 [log2.5(I1) - log10(I2)]
m1 - m2 = - 2.5 log2.5(I1/I2)(3)
Absolute Magnitude
The absolute magnitude is the brightness of a star at a distance of 10 parsecs.
M = m - 2.5 log10(I2/102)
M = m - 5 log(I/10)(4)
Absolute Magnitude and Distance
The relationship between the apparent magnitude and the absolute magnitude is given by
M = m - 5 log10(d/10)
From this equation we can calculate the distance d from the Earth if we know the absolute magnitude. In practise we don't know the absolute magnitude because we cannot travel 10 parsecs from the star in question. We can use several indirect methods to determine its absolute magnitude. If the star is on the main sequence of stars then we can determine the brightness from its parallax.
