By Carroll B.W., Ostlie D.A.
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DDn d gives dfn D Dnn d 2 fn : These last two coupled equations can now be integrated to produce Dn . /. Using a simple Euler method, Dn . iC1 / D Dn . i / C hfn . i / Solutions for An Introduction to Modern Astrophysics 65 and Ä fn . iC1 / D fn . i / C h Dnn . i / 2 i fn . i / where h is the step size and iC1 D i C h are the values of for steps i C 1 and i , respectively. Finally, = c D Dnn gives the density of the polytrope. OR. 6)’) xi, D_i**n, f_i END DO END PROGRAM LaneEmden (b) See Fig.
The time for the light to cross the eyeball is 2reye=c (neglecting the fluid within the eye). The energy from the light bulb within the eyeball is therefore Ebulb D 2FAreye=c D 2LAreye D 7:96 4 r 2c 10 15 J; more than four orders of magnitude smaller than Ebb. It is dark when you close your eyes because, according to Wien’s law (Eq. 15), at 310 K the blackbody radiation peaks at max D 9354:8 nm in the infrared, and the eye’s retina is not sensitive to such long wavelength photons. 2 (a) Dividing the specific blackbody energy density, Eq.
From Eq. 6 (a) mf sin3 i D 2:85 Mˇ and mb sin3 i D 5:80 Mˇ : mf ' 4:3 Mˇ and mb ' 8:7 Mˇ : 2=3, From Eq. 5), mB =mA D 0:241. (b) From Eq. 6), mA C mB D 5:13 Mˇ . (c) mA D 4:13 Mˇ and mB D 1:00 Mˇ . (d) According to Eqs. tc 2 ta / D 1:00 Rˇ ; tb / D 2:11 Rˇ ; respectively. (e) Brightness ratios can be determined from Eq. 3), giving (for the primary and secondary eclipses, respectively) Bp Bs D 0:0302 and D 0:964: B0 B0 Finally, using Eq. 9. (Lacy, Astron. ) Note: In order to accurately obtain the ratio of the radii, the light curve must be carefully modeled, a process beyond the scope of this text.
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