Astonomia Observacional 2009: septiembre 2009

Recetario fotometrico

Struve
Stellar Parallaxes.
The determination of the trigonometric parallaxes of stars is of fundamental importance in stellar astronomy, for without knowledge of the stellar distances very little information concerning their brightnesses, sizes and distribution in space could be obtained.
The linear distance (in kilometers) of a star whose parallax is known can easily be found, since the parallax of a star is the angle subtended by the radius of the earth’s orbit as seen from the star.
The parallax of the nearest star, α Centauri, is 0”.751. Since one astronomical unit = 1.5 x 10*8 km, the distance to α Centauri may be found from the ratio
p” α Cen / (360x60’x60”) = 1.5 x 10*8 /(2pi d α cen)
Or, since p” α Cen = 0”.751
d (α Cen) = 4.1 x 10*13 km

Units of Distance.
Since stellar distances are large, to new units of length are used. The parsec is defined as the distance from which the radius of the earth’s orbit subtends an angle of one second of arc:
1 parsec distance of a star whose parallax is one second of arc
in kilometers, this distance is given by the ratio
1/(360x60’x60”) = 1.5 x 10*8 km/ (2 pi dpc)
or 1 parsec = 3.1 x 10*13 kilometers
Thus, the distance to α Centauri is
d(α Centauri) = (4.1 x 10*13)/(3.1 x 10*13) = 1.3 pc

By definition, a star whose parallax is 1 second of arc is at a distance of one parsec. If it has a parallax of ½”, or 0.5”, it is at a distance of two parsecs. In general,
d (in parsecs) = 1/p”
For α Centauri dα Centauri = 1/0”751 = 1.3 parsecs

The other unit of distance is the light year:
1 light year = distance traveled by light in one year.
1 light year = 3 x 10*5 x 3.16 x 10*7 kilometers, 1 ly = 9.5 x 10*12 km.

The distance to Centauri in light years is, then:
d(α Centauri) = 4.1 x 10*13 light years

Extensive programs of parallax rneasurements are being carried out at many observatories. Several thousand stars have been found to have measurable parallaxes, but because of the technical difficulties in measuring very small relative displacements of star images on photographic plates, the lower limit of an observable parallax is about 0”.01; that is, the trigonometric parallax method of determining stellar distances is useful only up to distances of the order of 100 parsecs. Subsequent chapters of this book will describe other methods of estimating stellar distances. All of them, however, are based ultimately upon trigonometric parallax determinations of the nearest stars.

Motz, essentials of astronomy
The brightness ratio for stars of different magnitudes
We shall see how to assign magnitudes to stars that are not exactly 100 times fainter than (that is 1/100 as bright as) the standard first-magnitude star. W can do this by determining how much two stars differ in brightness if they differ by one magnitude. To ascertain this from our scale let b1 be the apparent brightness of a star of the first-magnitude, b2 the apparent brightness of a star of the second magnitude, etc., so that bm is the apparent brightness of a star of the mth magnitude. The factor bm/bm +1 is then the number of times an mth magnitude star is brighter than a star of magnitude m + 1. Let us call this factor, a, so that a star of the first magnitude is a times as bright as one of the second magnitude; a star of the second magnitude is a times as bright as one of the third magnitude, etc. In other words we have
(b1/b2) x( b2/b3)x(b3/b4)x(b4/b5)x(b5/b6)=b1/b6=100
a x a x a x a x a = a*5 = 100.
We see from this that a is equal to the fifth root of 100 or, what is the same thing,
a = 10*2/5 ,
which is very nearly equal to 2.512. Thus a star of any magnitude is somewhat more than 2 1/2 times brighter than the star of the next higher magnitude. Note that the larger magnitudes are associated with the fainter stars.

Let us now consider two stars, one of which is of the mth apparent magnitude, and the other of the nth apparent magnitude, and let bm and bn be their apparent brightnesses respectively. What is the ratio of these brightnesses? We know that for each magnitude step the brightness decreases by a factor of 10*2/5 and since in going from m to n there are exactly n- m steps, the apparent brightness of the nth—magnitude star is smaller than that of the mth star by the factor
(n — m) factors
10*2/5 x 10*2/5 x 10*2/5 x ….10*2/5 x = 10*2/5(n-m)
We thus have
bm/bn = 10*2/5(n-m)

15.15 Relationship between Brightness and Magnitude
Since the power to which 10 must be raised to give any particular number is called the logarithm of that number to the base 10, we have
log10 (bm/bn) = 2/5 (n — m)
if we now consider the star of the nth magnitude as being a kind of standard to which we may refer all other stars, we have
log10 bm = -0.4 m +B
where bm refers to the apparent brightness of any star of apparent magnitude m and B is a constant. This gives the relationship between apparent brightness and the apparent magnitude.

Relation between m and M.
To find the exact relationship between m and M we must know how the apparent brightness of a star changes when the distance changes. The apparent brightness of a celestial body varies inversely as the square of its distance from us, because its light moves uniformly in all directions, spreading out in concentric spheres. Thus, the intensity falls off inversely as the square of the distance from the star because the energy in the light is spread uniformly oven larger and larger spherical surfaces.

If we keep this in mind, it is easy to find M, if m and d are known. Let us consider a star at distance d and suppose that it is brought to a distance of l0 parsecs. If we compare its apparent brightness, bm, at the distance d with its apparent brightness, BM (its abso1ute brightness), at 10 parsecs, we have from the inverse-square law described above

bm/BM = 10*2 / d*2

where d is to be measured in parsecs.
On the other hand, we may look upon the star when it is at a distance d and the same star when it is at 10 parsecs as being two different stars having magnitudes m and M respectively, and we may then as how two such stars compare in apparent brightness. We gave the answer to this question in the previous section [ Equation (15.7)], where we saw that since each magnitude step represents a brightness ratio of 10*2/5, two stars having magnitudes m and M will have a brightness ratio of 10*2/5(M-m), there being just M-m steps in going from m to M. In other words, We must have
bm/BM = 10*2/5(M-m)
It we compare this with our previous equation for bm/BM we see that
10*2/5(M-m) = (10/d)*2
or 10*(M-m)/5 = 10/d
so that d = 10* (5-M+m)/5

From this it follows that
log10 d = (5-M+m)/5

or M = m + 5 -5 log d

Since d = l/p”, (note that - log d = +log p”), we see, finally that

M = m + 5 + 5 log p”

From this equation we can determine the absolute magnitude of any star if its apparent magnitude and parallax are known.

Stellar Luminosities

Since we shall express the luminosities of all stars in terms of the sun’s luminosity, Lo , we must begin by finding the absolute magnitude of the sun, for once we know this, we can find the luminosity of any star from a knowledge of its absolute magnitude by means of the relationship
log (L/Lo) = 0.4(Mo - M),
where L and M represent the luminosity and absolute magnitude of the given star. Since we know that Lo = 3.90 x 10*33 ergs/sec, we can at once find L from this equation if we know Mo

To find the absolute magnitude of the sun, we use the equation M = m + 5 - 5 log d, with m = -26.78 and d = 1/206 265, since this the distance of the sun in parsecs; we thus have

Mo = - 26.78 + 5 + 5 log 206 265,

or Mo = +4.71
Hence if the luminosity L of a star is expressed in solar units we have
log10 L = 0.4M — 1.884.

ejemplo:
Consider α Centauri, which, at a distance of 1.3 parsecs, has an apparent magnitude of 0. At 10 parsecs its distance wou1d be 10/1.3 = 8 times larger and its luminosity would be reduced by a factor of (8)*2, or 64. This corresponds to about 4.5 magnitudes. Thus, the absolute magnitude of α Centauri is approximate 0 + 4.5 = 4.5 magnitudes.
The Sun and α Centauri are closer than 10 parsecs, and their absolute magnitudes are therefore fainter than their apparent magnitudes. Most stars, however are at distances greater than 10 parsecs. For example Spica, or α Virginis, has a parallax of 0”.02 and an apparent visual magnitude of +1.2. The following steps lead to the determination of its absolute magnitude:
1. The distance of Spica is 1/0.02 = 50 parsecs.
2. Spica is, then, 5 times farther away than 10 parsecs.
3. The brightness of Spica at a distance of 10 parsecs would therefore be (5)*2, or 25, times greater than it is at 50 parsecs.
4. The corresponding change in magnitude is about 3.5.
5. Since Spica would be brighter at 10 parsecs than at 50 parsecs, the difference of 3.5 magnitudes must be subtracted from its apparent magnitude:
Mspica = m – Δm = +1.2 - 3.5 = -2.3 mag.

The absolute visual magnitude of Spica is therefore, -2.3.
Since the absolute magnitudes may be derived from either visual, photographic, or bolometric apparent magnitudes, it is necessary to distinguish between visual, photographic, and bolometric absolute magnitudes.
Two observational quantities are necessary for the determination of the absolute magnitude of a star: its parallax and its apparent magnitude. The most luminous stars have absolute bolometric magnitudes of the order of -10 and absolute visual magnitudes of the order of -7. The intrinsically faintest known stars have absolute bolometric magnitudes of the order of +18 and absolute visual magnitudes of the order of +19. The range in bolometric brightness is, then, about 28 magnitudes.

Astonomia Observacional 2009

miércoles, 30 de septiembre de 2009

Recetario fotometrico

viernes, 4 de septiembre de 2009

devoir_01

Seguidores

Archivo del blog

Datos personales