{"id":1079,"date":"2013-01-14T02:56:58","date_gmt":"2013-01-13T22:56:58","guid":{"rendered":"http:\/\/sites.williams.edu\/kkwitter\/?page_id=1079"},"modified":"2018-03-09T17:56:38","modified_gmt":"2018-03-09T13:56:38","slug":"week5","status":"publish","type":"page","link":"https:\/\/sites.williams.edu\/kkwitter\/astronomy-402-between-the-stars\/week5\/","title":{"rendered":"A402T Tutorial Week #5: Ionization Equilibrium & Str\u00f6mgren Spheres"},"content":{"rendered":"

ASTRONOMY 402T – Spring 2018
\nProblems for Tutorial Week #5<\/strong><\/p>\n

Consider a pure hydrogen cloud with uniform density surrounding a hot star. This star emits Q ultraviolet photons per second beyond the Lyman limit, i.e., photons capable of ionizing hydrogen from the ground state. Assume that each photon ionizes one and only one hydrogen atom.<\/p>\n

Let\u00a0R<\/em>\u00a0be the number of recombinations per cm3<\/sup>\u00a0per\u00a0second. In a steady state, the number of recombinations will equal\u00a0the number of ionizations inside the resulting spherical ionized\u00a0region, called a\u00a0Str\u00f6mgren sphere<\/em><\/strong>, whose\u00a0Str\u00f6mgren radius<\/em><\/strong>\u00a0is given by:<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 RS<\/sub> = (3Q\/4\u03c0\u03b1B<\/sub>ne<\/sub>2<\/sup>)1\/3<\/sup><\/strong><\/p>\n

The recombination rate\u00a0involves a two-body process: the\u00a0two\u00a0bodies are the electron and the proton. The rate must be proportional\u00a0to\u00a0the product of their number densities,\u00a0ne<\/sub><\/em>\u00a0and\u00a0np<\/sub><\/em>. Overall charge neutrality requires that\u00a0ne<\/sub><\/em>\u00a0=\u00a0np <\/sub><\/em>and for a pure hydrogen nebula = nH<\/sub><\/em>. The total recombination\u00a0rate per cm3<\/sup>\u00a0per second is\u00a0\u03b1B<\/sub>ne<\/sub>2<\/sup><\/em>, where \u03b1B<\/sub> is the\u00a0recombination coefficient for Case B, where recombinations to the ground state are ignored. (Why can we say that?)<\/p>\n

For temperatures characteristic of H II regions,\u00a0\u03b1B<\/sub> is approximately\u00a03 x 10-13<\/sup>\u00a0cm3<\/sup>sec-1<\/sup>. Assume\u00a0ne<\/sub><\/em>\u00a0= 10 cm-3<\/sup>.<\/strong><\/p>\n

a. <\/strong>Using these values, write down a parameterized expression for the Str\u00f6mgren radius in cm.<\/p>\n

b.<\/strong>\u00a0Compute\u00a0r<\/em>\u00a0in cm and in parsecs when\u00a0Q<\/em>\u00a0= 3 x\u00a01049<\/sup>\u00a0sec-1<\/sup>\u00a0(an O5 V star with T~40,000K and R~20\u00a0R\u2609<\/sub>).<\/p>\n

c<\/strong>.\u00a0Compute\u00a0r<\/em>\u00a0in cm and in parsecs when\u00a0Q<\/em>\u00a0= 4 x\u00a01046<\/sup>\u00a0sec-1<\/sup>\u00a0(a BO V star with T~30,000K and R~5\u00a0R\u2609<\/sub>).<\/p>\n

d.<\/strong>\u00a0Compute\u00a0r<\/em>\u00a0in cm and in parsecs when\u00a0Q<\/em>\u00a0= 1 x\u00a01039<\/sup>\u00a0sec-1<\/sup>\u00a0(a G2 V star with T~6000K and R~R\u2609<\/sub>).<\/p>\n

e.<\/strong>\u00a0What kinds of main-sequence stars create significant H II\u00a0regions around them?<\/p>\n

f.<\/strong>\u00a0O stars are often born in\u00a0clusters like the Trapezium in Orion. The Orion Nebula is 450 pc away,\u00a0and the portion known as M42 has an angular diameter of about 1\u00a0degree. Assume that M42 is a spherical pure hydrogen nebula with an\u00a0average density of 200\u00a0cm-3<\/sup>. How many equivalent O5 stars are\u00a0required to ionize M42? Compare this with the number of the brightest\u00a0Trapezium stars.<\/p>\n

g.<\/strong>\u00a0Now consider a planetary nebula central\u00a0star with a temperature of 150,000K and a radius ~0.1 R\u2609<\/sub>.\u00a0It has\u00a0Q<\/em>\u00a0= 1 x\u00a01047<\/sup>\u00a0sec-1<\/sup>. Compute\u00a0r<\/em>\u00a0in cm and in parsecs for this case, using\u00a0the same value for \u03b1B<\/sub>, but with a\u00a0density characteristic of planetary nebulae, about 103<\/sup>\u00a0cm-3<\/sup>. Compare with the\u00a0O \u00a0star in part (b) above. Also comment qualitatively on the relative ionization levels you would expect for elements like nitrogen, oxygen, and neon, if they were present in both the planetary nebula and the H II region around the O star in part (b) above.<\/p>\n

h.<\/strong>\u00a0Calculate the mass, in solar masses, contained in the\u00a0Str\u00f6mgren sphere of the planetary nebula above. Compare\u00a0this with the typical \u00a0planetary nebula mass you learned about in\u00a0Astronomy 111 or in your own studying.<\/p>\n

i.<\/strong>\u00a0Calculate the mass, in solar masses, contained in the\u00a0Str\u00f6mgren sphere produced by the O5 star in part b. Compare this value with\u00a0the mass of \u00a0the planetary nebula, and comment on the relative size of these values.<\/p>\n

\n

\n","protected":false},"excerpt":{"rendered":"

ASTRONOMY 402T – Spring 2018 Problems for Tutorial Week #5 Consider a pure hydrogen cloud with uniform density surrounding a hot star. This star emits Q ultraviolet photons per second beyond the Lyman limit, i.e., photons capable of ionizing hydrogen from the ground state. Assume that each photon ionizes one and only one hydrogen atom. […]<\/p>\n","protected":false},"author":198,"featured_media":0,"parent":793,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-1079","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/kkwitter\/wp-json\/wp\/v2\/pages\/1079","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/kkwitter\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.williams.edu\/kkwitter\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/kkwitter\/wp-json\/wp\/v2\/users\/198"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/kkwitter\/wp-json\/wp\/v2\/comments?post=1079"}],"version-history":[{"count":61,"href":"https:\/\/sites.williams.edu\/kkwitter\/wp-json\/wp\/v2\/pages\/1079\/revisions"}],"predecessor-version":[{"id":4402,"href":"https:\/\/sites.williams.edu\/kkwitter\/wp-json\/wp\/v2\/pages\/1079\/revisions\/4402"}],"up":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/kkwitter\/wp-json\/wp\/v2\/pages\/793"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/kkwitter\/wp-json\/wp\/v2\/media?parent=1079"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}