Problem Set 8 Solutions

Problem Set 8 Solutions - Physics 341: Problem Set #8...

This preview shows pages 1–3. Sign up to view the full content.

Physics 341: Problem Set #8 Solutions 1. The vertical motion of stars in spiral galaxies depends on the gravity exerted by the disk, so it allows us to “weigh” the disk. (a) Use dimensional analysis to derive an estimate of the mass density of a spiral galaxy disk, in terms of its scale height h z , its vertical velocity dispersion σ z , and a relevant physical constant. The things we have to work with are: scale height h z =[ L ] vertical velocity dispersion σ z LT - 1 ] gravity G M - 1 L 3 T - 2 ] We are looking for a mass density, which has dimensions ML - 3 ] To get this, we clearly need G - 1 to get [ M ]. Then we need σ 2 z to get rid of [ T 2 ]. So far we have σ 2 z G - 1 ] We then need h - 2 z to get two more factors of [ L ] in the denominator. Thus, our dimensional analysis estimate is σ 2 z Gh 2 z (b) In the neighborhood of the Sun, the Milky Way has h z 350 pc and σ z 16 km s - 1 for the thin disk, and h z 1 kpc and σ z 35 km s - 1 for the thick disk. Use these values and your result from part (a) to estimate the mass density of the Milky Way’s disk, in M ± pc - 3 . Do the thin and thick disks give a consis- tent density estimate to the level of precision we might expect from dimensional analysis? Using the thin disk we estimate (16 10 5 cm s - 1 ) 2 (6 . 67 10 - 8 cm 3 g - 1 s - 2 ) (350 3 . 086 10 18 cm) 2 3 . 3 10 - 23 g cm - 3 1 M ± 1 . 99 10 33 g 3 . 086 10 18 cm 1pc 3 0 . 49 M ± - 3 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Using the thick disk we estimate (35 10 5 cm s - 1 ) 2 (6 . 67 10 - 8 cm 3 g - 1 s - 2 ) (3 . 086 10 21 cm) 2 1 . 9 10 - 23 g cm - 3 0 . 29 M ± pc - 3 The answers di er, but by less than a factor of 2 and in dimensional analysis we cannot be too concerned about such factors. For comparison, if you add up all the stars, gas, dust, and dark matter in the Solar neighborhood you get a total mass density of about 0 . 15 M ± - 3 (see p. 938 of Carroll & Ostlie). Our estimate is too large by a factor of 2–3, but that is neither surprising nor worrisome for dimensional analysis. In fact, it is rather amazing
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/13/2011 for the course PHY 341 taught by Professor Gawsier during the Spring '11 term at Rutgers.

Page1 / 5

Problem Set 8 Solutions - Physics 341: Problem Set #8...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online