Astroparticle Physics

17.14 Chapter 14 375The mass of a typical galaxy can be estimated as M = 10 11 M ⊙ = 2 × 10 41 kg. Withan assumed radius of r = 20 kpc one getsv f = 2.11 × 10 5 m/s .With n max m ν = 0.3 ϱ c ≈ 3 × 10 −30 g/cm 3 one obtainsm ν ≥ 1eV.7.✘ ✘✘✘ ✘ ✘✘✘✘ ✘ ✘✘✘ ✘ ✘δEarth ✗✔ ϑ R✖✕deflector✛ L 1✲✛L 2✲starδ = 2R SRas mentioned earlier (see Problem 8.6); R S = 2GM dc 2 ;for L 2 ≫ L 1 : δ ≈ ϑ = R L 1⇒ R = L 1 δ,δ = 2R SR= 2R SL 1 δ√ √⇒ δ = 2R S 4GM d=L 1 L 1 c 2 ∼ √ M d .Ring radius R E = L 1 δ = √ 2L 1 R S √, L 1 = 10 kpc ≈ 10×10 3 ×3.26×3.15×10 7 s×3×10 8 ms −1 ≈ 3.08 × 10 20 m: R E ≈ 2 × 3.08 × 10 20 m × 3000 m ≈ 1.36 × 10 12 m ;opening angle for the Einstein ring: γ = 2δ = 2R E= 8.8 × 10 −9 ̂= 0.0018 arcsecL 1⇒ too small to be observable ⇒ only brightness excursion visible.17.14 Chapter 141. The weight of a human is proportional to its volume, which in turn is proportional to thecube of its size;W = W 0 R 2 20 R,if one assumes that the ‘height’ of a human is 20 times its ‘radius’. The strength of ahuman is proportional to its cross sectionS = S 0 R 2 .For a mass of 100 kg, an assumed radius of 10 cm, a human has to carry its own weightplus, maybe, an additional 100 kg,