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Standard candles and luminosity Saturday, 7 May 2022 9:22 pm cmnathe‘ kot wage gto Cont y /N The stars do not look the same; they differ in brightness and colour. These stars are not all the same distance from us, and they do not all emit the same power. So, we cannot deduce their distance from just how bright they appear in the night sky. luminosity of a star is defined as the total radiant energy emitted per unit time.( This is the same as the total power emitted by a star.) In SI units, luminosity L is measured in W or J s-1. The luminosity of the Sun (solar luminosity), often written as L ©, is about 3.83 x 1026 W. a standard candle is an astronomical object of known luminosity. The two well-known standard candles are Cepheid variable stars and Type 1A supernovae. e the brightness of Cepheid variable stars varied periodically, e and the period of this variation was related to the average luminosity of the star. e By measuring the period, astronomers could determine the luminosity of the star. e The star’s distance could then be calculated from the observed radiant intensity at the Earth. e Finding a Cepheid variable star in a distant galaxy meant that the distance of the galaxy itself could be calculated. e Type 1A supernovae stars implode rapidly towards the end of their lives, and scatter matter and energy out into space. e This implosion event can be brighter than the galaxy itself. e The luminosity of the star at the time of the implosion is always the same. e From this, astronomers can estimate the star’s distance from the Earth. Table 31.1 summarises some data on the brightest stars—do not forget that the Sun is a star too. Table 31.1: Data on the six brightest stars, including the Sun. The luminosity is given in terms of the solar luminosity Lo; 1 Le = 3.83 x 1076 W. We can see from Table 31.1 that the observed brightness of a star is linked to both its distance from the Earth and its luminosity. => it dis tamee Z De stows (s SAW ; tlen tLe nre tai th Gveatev Lundin os wy EPPRAY § Lvighter ; => Ve luwdinasi try QD Stars 1s Sore, jen tle ore ial 6 Vudu opot yy ll mR POY legs brio ht Example: Alpha Centauri is brighter in the night sky than Arcturus, not because of its luminosity, but because of its closeness to us. We can relate the brightness of a star to its luminosity. But the underlying assumptions are: e the power from the star is uniformly radiated through space e there is negligible absorption of this radiated power between the star and the Earth. we can determine the intensity of electromagnetic radiation observed at the Earth. The observed intensity is known as radiant flux intensity ,F. The radiant flux intensity ,F is defined as the radiant power passing normally through a surface per unit area. i i i f st radiant flux intensity = power of star surface area of sphere The power of the star is its luminosity L, and the surface area of a sphere is 4nd?. ay) F= + 4nd? t The SI units for radiant flux intensity are W m-2. > d For a given star, the luminosity Lis constant, so according to the equation, the radiant flux intensity F obeys an inverse square law with distance d . So, doubling the distance from the centre of the star (2d ) will decrease F by a factor of 4, and tripling the distance (3d ) will decrease F by a factor of 9, and so on. Questions Where necessary, take: Lo = 3.83 x 1076 W 1 ly=9.5 x 1015 m 5) 1 State two factors that affect radiant flux intensity from a star. L 2 The radiant flux intensity F of light from a lamp at a distance of 10 cm is 0.32 W m?, Calculate F from the same lamp at a distance of 15 cm. State any assumption(s) you make. Use data from Table 31.1 to determine, to two significant figures: a__ the distance of Sirius from the Earth in metres. b the luminosity (in W) of i Canopus ii Vega. the radiant flux intensity measured at the Earth from: i Sirius ii Alpha Centauri. This question is about Sirius and Arcturus. With the help of calculations and data from Table 31.1, show that Sirius is brighter than Arcturus. The radiant flux intensity from a star measured at the Earthis 2.7 x 10°? W m~. The luminosity of the star is 1300 Lo. Calculate the distance of this star from the Earth in metres. Table 31.1 summarises some data on the brightest stars—do not forget that the Sun is a star too. Table 31.1: Data on the six brightest stars, including the Sun. The luminosity is given in terms of the solar luminosity Lo; 1 Le = 3.83 x 1076 W. a% Ww hme) = g-6 x AS x10” ' gZ02 * loom Loa * Woe x SI _ ie eC \\ 2, = 3-83 Klo pe NON (BN = 4.2 ~*~ \O UN (iv ) Ly = Le * 4s ~) Ga “tus C DE nm Z- $3 XI a) an Ue fw C Ge 2x08 ) = 1 w ia” W a Saturday, 7 May 2022 9:42 pm We can determine the diameter of the Sun fairly easily (see Practical Activity 31.2). However, when we look at stars in the night sky, they appear as tiny specks of light — there is no disc to be seen (Figure 31.4). The stars are just too far away. Even the closest stars viewed through powerful telescopes appear as specks of light. How can astronomers determine the size of stars? PRACTICAL ACTIVITY 31.2 The diameter of the Sun You can estimate the diameter of our closest star, the Sun, using a simple pin-hole camera. You can make this camera using a shoe-box. One end of the box has a sheet of darkened paper (or aluminum foil) with a tiny hole made with a sharp pin. The opposite end of the box has a sheet of tracing paper, which acts as a screen. A circular image of the Sun is formed on the screen when the camera is pointed towards the Sun. See Figure 31.5. 1.5x1044m — Ig ox image of Sun seen pin-hole camera on screen Measure the distance x between the pin-hole and the screen. The distance of the Sun from the Earth is 1.5 x 10‘! m. The diameter D of the Sun can be determined using simple trigonometry: tan@= = —2 | 1.5x10" Therefore: — 1.5x10"xd | D > 4 A hot object, such as a star, can be lasa y. Al dy is an idealised object that absorbs all incident electromagnetic radiation falling on it. It has a characteristic ETaTe| intensity that depend only on its tt Figure 31.6 shows typical intensity against wavelength graphs for objects at different temperatures. > 4 77) c ce) ~ £ Amax 580.725 The h temperature of a body: e thes gth at the e the greater tt ity of the electromagnetic radiation at g =) Wien’ s displacement law This is the relationship between the thermodynamic temperature T of the object and the wavelength Amax at the peak intensity: |, Dw: ie 1 \max % + U, rv mace | max! = constant The experimental value of the constant is 2.9 x 10-3 mK. Example: The surface temperature of the Sun is 5800 K . This gives a Amax value of about 5.0 x 10-7 m or 500 nm. Light of this wavelength appears yellow (which is not surprising for the Sun). 
