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(b) The dark matter halos which form in numerical simulations have density profiles, p(r). which are well approximated by a Navarro-Frenk-White (NFW) profile p(r) = Po (=) (¹ + # ) ² where r, is the radius at which d In p/d In r = -2 (i.e. pxr-2). Show that i) p(r) x r¹ for r

(b) The dark matter halos which form in numerical simulations have density profiles, p(r). which are well approximated by a Navarro-Frenk-White (NFW) profile p(r) = Po (=) (¹ + # ) ² where r, is the radius at which d In p/d In r = -2 (i.e. pxr-2). Show that i) p(r) x r¹ for r

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Please send me answer of this question immediately and i will give you like sure sir .solve only b part completely 

part (b) 4 (a) The rotation speed, v(r), of an object which is moving in a circular orbit of radius r within a spherical mass distribution is given by: S v(r) = where M(<r) is the mass within radius r. How does v(r) vary with r when (i) M(<r) is constant? {1} (ii) the density p is constant? {2} (iii) the density varies with radius as p(r) x 1/r²? {2} Using these results, discuss the observational evidence for dark matter from galaxy rotation curves. You should include a sketch of a galaxy rotation curve in your answer. {3} (b) The dark matter halos which form in numerical simulations have density profiles, p(r). which are well approximated by a Navarro-Frenk-White (NFW) profile Using the standard integral GM(<r) where r, is the radius at which d In p/d lnr= -2 (i.e. pxr-2). Show that i) p(r) x r¹ for r <<rs {1} ii) p(r) x r3 for r»rs {1} Hence find how v(r) depends on r for r <r, and r»r, respectively. {4} dny dE Po p(r) = ( ;) (¹ + # ) ² ³ (c) The number density of one species of neutrino with energies in the range E to E+ dE, (dn,/dE) dE, has the form dE= 8T E²dE (hc)³ E exp KBT Bu = 43.2 y² dy exp (y) +1 show that the total number density of neutrinos can be written as n = 3₂T³ where KB hc +1 = 1.8, 3 {5} Find the numerical value of B. {1} If 9,0 = 0.25, Tvo= 1.95 K and h = 0.7, what is the average neutrino rest mass energy in eV? {3} (d) Describe, in your own words, the astrophysical evidence that neutrinos can't make up all of the dark matter. {2}
Transcribed Image Text:part (b) 4 (a) The rotation speed, v(r), of an object which is moving in a circular orbit of radius r within a spherical mass distribution is given by: S v(r) = where M(<r) is the mass within radius r. How does v(r) vary with r when (i) M(<r) is constant? {1} (ii) the density p is constant? {2} (iii) the density varies with radius as p(r) x 1/r²? {2} Using these results, discuss the observational evidence for dark matter from galaxy rotation curves. You should include a sketch of a galaxy rotation curve in your answer. {3} (b) The dark matter halos which form in numerical simulations have density profiles, p(r). which are well approximated by a Navarro-Frenk-White (NFW) profile Using the standard integral GM(<r) where r, is the radius at which d In p/d lnr= -2 (i.e. pxr-2). Show that i) p(r) x r¹ for r <<rs {1} ii) p(r) x r3 for r»rs {1} Hence find how v(r) depends on r for r <r, and r»r, respectively. {4} dny dE Po p(r) = ( ;) (¹ + # ) ² ³ (c) The number density of one species of neutrino with energies in the range E to E+ dE, (dn,/dE) dE, has the form dE= 8T E²dE (hc)³ E exp KBT Bu = 43.2 y² dy exp (y) +1 show that the total number density of neutrinos can be written as n = 3₂T³ where KB hc +1 = 1.8, 3 {5} Find the numerical value of B. {1} If 9,0 = 0.25, Tvo= 1.95 K and h = 0.7, what is the average neutrino rest mass energy in eV? {3} (d) Describe, in your own words, the astrophysical evidence that neutrinos can't make up all of the dark matter. {2}
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