Aa зЕd╤╥╥╥╙╙╘▌HH $ @d HHHH╠╠╠╠╠╠ff@ААА Ў ° ° ° Р ° d/¤ Footnote TableFootnote*ар*ар.\t.\t/ - ╨ ╤:;,.╔!?%( ` $ `TOCHeading1Heading2 EquationVariables8I╔щ┘5л╔Y5Т╩S5Ц╦U5Ъ╠W5р═f5└╬^6в╧б6И╨Э7K╤░7╥к7╙мWоа8 ╘╬хCВ7h╒┴ИsМ7П╓├}ф-7╞╫╚IВ╪7╘╪╩F╙8-┌хШ▌t ) I > )<$lastpagenum> *<$monthname> <$daynum>, <$year>17M +"<$monthnum>/<$daynum>/<$shortyear> ,;<$monthname> <$daynum>, <$year> <$hour>:<$minute00> <$ampm> -"<$monthnum>/<$daynum>/<$shortyear> .<$monthname> <$daynum>, <$year>╠╠ /"<$monthnum>/<$daynum>/<$shortyear> 0 <$fullfilename> 1 <$filename> 2<$paratext[Title]> 3<$paratext[Heading1]> 4 <$curpagenum>t 5 <$marker1> 6 <$marker2> 7 (Continued)╤ 8+ (Sheet <$tblsheetnum> of <$tblsheetcount>) 9Heading & Page ╥<$paratext>╙ on page<$pagenum> :Pagepage<$pagenum> ;See Heading & Page%See ╥<$paratext>╙ on page<$pagenum>. < Table Alle7Table<$paranumonly>, ╥<$paratext>,╙ on page<$pagenum> =Table Number & Page╠'Table<$paranumonly> on page<$pagenum>БСИ░БFFA ╬ВHHA╞╚ГJJДLLЕNNЖPPЗR═A$╜пТ5y soe>5| s > 5 ui> 5В t"5Е! u/um5И" ta5Л# |n$d5О$ v we╠╠5П% w wo>/5Р& x w$ea5С' w wfi5Х( y w5Щ) w w5Э* z wele5Ю+ w wte5о, { w]5╡- v wg5╢. w wa5╖/ } wa5╕0 ~ wod)5╣1 w+t 5║2 А wu<$5╗3 Б wn5╝4 А wnge5╜5 В wx p5╛6 А Аu5┐7 Г Аeen5├8 Д ГHe5─9 А Дe╥<5┼: Е А e5╞; Ж А6Н< А Аee5╠= З sl$p6О> К Аo<5╥? А wTa5╙@ А А&'5╘A А Аay>5▐B Б Аp>5╓C Й АБ░6zD Б АF6{E А А 6|F } А6АG А╚6ПH А АJ6еI Б А6йJ М А6кK А А6УL Д А6ФM Й А6ЩN Д АR6└O Д Ап6░P М Пs6▒Q М Пs6╢R П Пu6┴S А Аt6┬T Ж Аu6┼U А Аt6╞V ~ А|6╔W ~ Аv6╦X Й Аw6╠Y А Аx6═Z К Аw6┘[ А Аy6┌\ Ц Аw6щ] Д Аz6с^ Ч Аw6т_ А А{6у` А Аv6цa Ш Дw6ъb Д А}6ыc А А~6ьd Х А6ўe А АА6°f ~ АБ6∙g Щ АА6·h Л ЛВ6√i Ф ЛА6№j А АГ6¤k Й АД7l } АА7 m А АЕ7 n А АЖ7o А АА7"p Е АЗ7q А АК7r А АА7#s Н АА7$t О АА7+u А АБ7,v В АЙ76w С АБ73x В АА77y А А}78z А А7C{ А АА7Л| А АБ7H} Б АМ7I~ Й АА7k Ч АД7zА А ДЙ7яБ У ИД7СВ К АД7ТГ А АМ7УД Й АМ7ШЕ Р АП7ЬЖ Д АА7ЭЗ Д АЖ7ЮИ Л АА7ЯЙ А А~7аК Р А~7бЛ Д АЙ7вМ Л АА7нН Л ЛК7╜О Ь АА7╟П И ЬЦ7╠Р А АД7ЁС Ф ФЧ7╔Т Ф АА7═У } АА7╬Ф } АШ7╧Х А АД7╨Ц Б АА7╒Ч А АХ7╓Ш А АА7╫Щ А А~7■Ъ Е ЖЩ7 Ы Й ЕЛ8Ь } ЕФ8Э } ЕА7▄Ю Д АЙ7▌Я А А}7▐а Е АА7▀б Ж АА7ёв Ф ФА8г К ЕЕ8д А ЕА8е А ЕА8)ж А АН8.з Д АО8/и Л АА80й Д АВ81к Л АС82л Э АВ83м А АА84н Ъ АА85о Ж АА5ї╥▒7Лdq┴5ЩЙАPF╤7zdq┬6ЩКДLH╥7Уdq├7ЩДА8C╙7ЮH┤m3RыЩq─8Ъ97Р~H┤m3RыЩH RH R ═FootnoteHr¤@Щq┼9Ъ8:7╟ ИHr¤@ЩHz╖пHz╖п ═Single LineH'┤ Ёq╞:Ы9<7╬ };;Footnote Б Р5_;Ь:А Р Щ Hи¤DнЩq╚<Э:=78Hи¤DнЩH░╖пH░╖п ═Double LineH╣ Ё╘q╔=Ы<@77ё>?ЕDouble Lineд А╘5c>Ю?=Н╘╘╘5e?Ю>=82╘83 А╘HЕ ·╘ q╠@Ы=B7ЛAASingle LineА╘5hAЮ@╘╘HZ┤ Ёq╬BЫ@C7q├ TableFootnoteE╕╧Gx┼Rк∙Щq╧CЫB79E╕╧Gx┼Rк∙ЩE╕╧PwE╕╧Pw ═ TableFootnoteod5pDЩRR╘╖пHH╘И5xEЫ5ЁHH╘И╬Fe ╚HH╘И5zFЫN5 HH╘Иl ЁEEБDнHH╘И5{GЫ6 ЁHH╘И@>?HDoe ╚HH╘И5}HЫJ6HH╘Иl ЁGGВHэUV╘ 5~IЫ6HэUV╘ ЫAAJккккUU` ╔HэUV╘ 5АJЫLH6HэUV╘ l ЁIIГH$╘ 5БKЫ6FtnH$╘ E╕╧L╕╧ккккUU` ╔tnoH$╘ 5ГLЫJ6H$╘ l ЁKKДHэUV╘ 5ДMЫ5HэUV╘ NккUUe! ╔ЫHэUV╘ 5ЖNЫPF5HэUV╘ l ЁMMЕH$╘ 5ЗOЫ5H$╘ PккUUe" ╔ЫH$╘ 5ЙPЫN5H$╘ l ЁOOЖHH╘И5КQЫDHH╘ИdA YккккSYR !Щx!Щx`$ ╦Ы'Implications of ╠P ╬╡ r ╧5/3 DЩx`% ╨ ЁaЩx`& ╤*Add mass to a white dwarf and it shrinks! ~Щx`' ╨ ЫЩx h( ╥*е The Equation of State says that ╙╩ Щx h) ╒+е Hydrostatic Equlibrium says that ╨╦ Hr2Ё h* ╓е Density is given by ╫╠ рeр`+ ╨ ¤eр h, ┘)▐ ╨ Put this all together and find╔ HH╘И5МRЫDHH╘Иl Ё]QQЗNыЩx╚H5УSЫUDTTQ╩`╬пL╠bг│45ФTЯS`╬пL╠bг│4`╬пL╠bг│4ф$'тequal[(*q"Green"q*)char[(*q"Green"q*)P],cross[(*q"Green"q*)string[(*q"Green"q*)"some constant "],power[(*q"Green"q*)char[(*q"Green"q*)rho],over[(*q"Green"q*)num[(*q"Green"q*)5.00000000,"5"],num[(*q"Green"q*)3.00000000,"3"]]]]]NPЩx╚TЩx5ЧUЫSWDVVQ╦$ЫWСE▌Ё;╠25ШVЯUЩxWСE▌Ё;╠2WСE▌Ё;╠2ф0ЩxAd'~equal[char[P],cross[string[" another constant "],over[power[char[M],num[2.00000000,"2"]],power[char[R],num[4.00000000,"4"]]]]]N┬2Ё╚Q2Ё5ЫWЫUYDss XXQ╠*\/┴ Pёа~9■5ЬXЯW\/┴ Pёа~9■\/┴ Pёа~9■ф-2Ёt '╤equal[(*q"Cyan"q*)char[(*q"Cyan"q*)rho],cross[(*q"Cyan"q*)string[(*q"Cyan"q*)"different constant"],over[(*q"Cyan"q*)char[(*q"Cyan"q*)M],power[(*q"Cyan"q*)char[(*q"Cyan"q*)R],num[(*q"Cyan"q*)3.00000000,"3"]]]]]NMeр╚^Ъ 5мYЫWD 4@ZZQ╔╬пL╠{fMГс╤3f9■5нZЯY*ha{fMГс╤3f9■{fMГс╤3f9■ф1eрg['°equal[(*q"Red"q*)char[(*q"Red"q*)R],cross[(*q"Red"q*)string[(*q"Red"q*)"constant"],over[(*q"Red"q*)num[(*q"Red"q*)1.00000000,"1"],power[(*q"Red"q*)char[(*q"Red"q*)M],over[(*q"Red"q*)num[(*q"Red"q*)1.00000000,"1"],num[(*q"Red"q*)3.00000000,"3"]]]]]]╠2d5░[Щ╠2]]╒AdHH╘И5▒\Ы[tt HH╘ИВnu[char[R]^^]0,`- ╦The Chandrasekhar Limit s ;`. ╨ X`/ █а~/Suppose you keep adding mass to a white dwarf. 9■u`0 ╫)(This can happen. We╒ll see how later.) aТ`1 ▄q")▐ █It shrinks and becomes more dense. aп`2 ╨"q h╠`3 ╙po!Can this go on forever? ▌NO! ,щ`4 ╨00 0 5 ▐eр0Remember: ▀We assumed that the motion of the Z#@5 ▀L╠Gelectrons was nonrelativstic ( рv сe р = p сe р/m сe ▀). ╤3f@`6 ╨ q] 7 ┘r[@▐ ╨At some point we must have тv уe т = c ╨and the (*z H7 ╨edEquation of State becomes╬ щ 8 ╨q*3If you try to solve for R(M), хthe radius drops m[(@8 х001out of the equation and all you get is a mass!!! #`9 ╨] @ : ц9Find maximum mass ч = ш1.44M щSUN ц. Speed of ]@: ц=electrons is ъc ц and if you add more mass then ш... z`; шr '... ц шsomething drastic happens! HH╘И5│]Ы[ itHH╘Иl ЁRb\\Зe hN╩╚R5┴^Ы[sin__\╬ aG|─L╠9x│45┬_Я^athG|─L╠9x│4G|─L╠9x│4ф,4'тequal[(*q"Black"q*)char[(*q"Black"q*)P],cross[(*q"Black"q*)string[(*q"Black"q*)"yet another constant"],power[(*q"Black"q*)char[(*q"Black"q*)rho],over[(*q"Red"q*)num[(*q"Red"q*)4.00000000,"4"],num[(*q"Red"q*)3.00000000,"3"]]]]]d5╟`Щ anbb╓edEqHH╘И5╚aЫ`8q*HH╘ИВfo Эrops m[(fЭb8 `# ╦anNeutron Stars PА`= ╤ End state of more massive stars |`? ╨s>If M ┘> 1.44 ╨M ыSUN ╨ the white dwarf must shrink! Щ`@ ╨ц d╢ hA ╨asIOne possibility: ╒e ╩- ┘ + ╓p ┘ о шn ┘ + эn ╨═ !ж`B ╙ Can neutrons take the pressure? ├`D ╙@Detailed calculations э ▐ ╙Yes, if M э < ╗ 3 ╙M юSUN Ыр`E ╨in ¤`F █?Neutrons (r яn ▄ ╗ 10 Ё-15 ▄ █m) ╥touch each other╙ 9x hG ▄│40▐ █For M = ▄2 █M яSUN █ we have╨ k"HH╘И5╩bЫ`*yeHH╘Иl Ё]eaaЗ],od5╫cЩ"nuee╫"]]]HH╘И5╪dЫc anHH╘ИI5╚ббe`C ╦Detecting Neutron Stars (I) ps;`< ╨8 X`> ╤#They spin very rapidly u`H ╨st Т hI ╙ar/Recall ╥Conservation of Angular Momentum╙:╧ ыSх`J ╙wa m`P ╙╢▐7йЯкowoLд╫їw.╢S>╢▐Lд╫їw.╢S>╢▐ф* 1'╨equal[(*q"Blue"q*)char[(*q"Blue"q*)E],times[(*q"Blue"q*)over[(*q"Blue"q*)num[(*q"Blue"q*)1.00000000,"1"],num[(*q"Blue"q*)2.00000000,"2"]],char[(*q"Blue"q*)I],power[(*q"Blue"q*)char[(*q"Blue"q*)omega],num[(*q"Blue"q*)2.00000000,"2"]]],times[(*q"Blue"q*)over[(*q"Blue"q*)num[(*q"Blue"q*)1.00000000,"1"],num[(*q"Blue"q*)2.00000000,"2"]],id[(*q"Blue"q*)times[(*q"Blue"q*)over[(*q"Blue"q*)num[(*q"Blue"q*)2.00000000,"2"],num[(*q"Blue"q*)5.00000000,"5"]],char[(*q"Blue"q*)M],power[(*q"Blue"q*)char[(*q"Blue"q*)R],num[(*q"Blue"q*)2.00000000,"2"]]]],power[(*q"Blue"q*)id[(*q"Blue"q*)over[(*q"Blue"q*)times[(*q"Blue"q*)num[(*q"Blue"q*)2.00000000,"2"],char[(*q"Blue"q*)pi]],char[(*q"Blue"q*)P]]],num[(*q"Blue"q*)2.00000000,"2"]]]]╒Nи╚N7кЫмг| чййд╥tityNЦSЫ▌*╙Z>╢▐7лЯмNЦSЫ▌*╙Z>╢▐NЦSЫ▌*╙Z>╢▐ф+'$equal[(*q"Green"q*)char[(*q"Green"q*)L],minus[(*q"Green"q*)over[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)d],char[(*q"Green"q*)E]],times[(*q"Green"q*)char[(*q"Green"q*)d],char[(*q"Green"q*)t]]]],times[(*q"Green"q*)id[(*q"Green"q*)times[(*q"Green"q*)over[(*q"Green"q*)times[(*q"Green"q*)num[(*q"Green"q*)8.00000000,"8"],power[(*q"Green"q*)char[(*q"Green"q*)pi],num[(*q"Green"q*)2.00000000,"2"]]],num[(*q"Green"q*)5.00000000,"5"]],char[(*q"Green"q*)M],power[(*q"Green"q*)char[(*q"Green"q*)R],num[(*q"Green"q*)2.00000000,"2"]]]],over[(*q"Green"q*)num[(*q"Green"q*)1.00000000,"1"],power[(*q"Green"q*)char[(*q"Green"q*)P],num[(*q"Green"q*)3.00000000,"3"]]],over[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)d],char[(*q"Green"q*)P]],times[(*q"Green"q*)char[(*q"Green"q*)d],char[(*q"Green"q*)t]]]]])PN0╚O7мЫкгллд╙d7:нЩSЫ▌пп█HH╘И7;оЫнHH╘И~ee],minus[╚╚пve`~ ╦reGravitational Red Shift ,c;`В ╤]] Examples GX`Г ╨re qu`Е ╥n"е Sirius B (White Dwarf Star) Т`Ж ╨(*@R ┘ = 7┤10 ╩-3 ┘ ╨R ыSUN ┘ = 4.9┤10 ╩6 ┘ ╨m .0п`З ╨[(8M ┘ = 1.05 ╨M ыSUN ┘ = 2.1┤10 ╩30 ┘ ╨kg 00╠`И ┘re6▐ Dl/l 0 ┘ = 3.2┤10 ╩-4 ╨(just measurable) щ`Й ╨ee *`К ╥q* е A two-solar mass neutron star "q#`Л ╨)17M ┘ = 2 ╨M ыSUN ╨ and R ┘= 12 ╨km m[(@`М ┘00$▐ Dl/l 0 ┘ = 0.29 ╨(Big !) Gr]`Н ╨ee *z hО √"qS+ шWhat if чDl/l 0 ч = 1 ( цi.e. observed ч l=0) ш? ╨╫ ╒ Oх`П ╥Black Hole╙ А`Б +Schwarzschild Radius: R(M)=2GM/c !2 <`Т рп$This is the only end state possible Y`С р(as far as we know) v`в рfor very massive stars. HH╘И7=пЫнtioHH╘Иl Ёи╞ооЗ`ГHи╘И7J░Ы┴жhe Ё▒╛з╤Ж(*??7O▒е▓░ 9┤??Z5-mЧ7P▓д▒┤░MыZ5Ў╫- х ╩" Z<╢╦]┐·5]┐·5_Яў8█eaЇ<╢╦aЇ<╢╦c_ё@Т1e?юDmЧe?юDmЧgы@Т1h ш<╢╦h ш<╢╦j▀х8█el┐т5l┐т5nЯ▀8█ep▄<╢╦p▄<╢╦r_┘@Т1t?╓DmЧt?╓DmЧv╙@Т1w ╨<╢╦w ╨<╢╦y▀═8█e{┐╩5{┐╩5}Я╟8█e─<╢╦─<╢╦Б_┴@Т1Г?╛DmЧГ?╛DmЧЗ<╢╦З>7S│з┤╢░З>З>Щ>V5HmЧ7R┤й▓│░eedV5Ў┘H т╨"V<╢╦[ є5[ є5^ ь8█ea ц<╢╦a ц<╢╦d ▀@Т1g ┘DmЧg ┘DmЧj ╥@Т1m ╠<╢╦m ╠<╢╦p ┼8█es ┐5s ┐5v ╕8█ey ▓<╢╦y ▓<╢╦| л@Т1 еDmЧ еDmЧВ Я@Т1Е Ш<╢╦Е Ш<╢╦И Т8█eЛ Л5Л Л5О Е8█eС <╢╦С <╢╦Ф y@Т1Ч rDmЧЧ rDmЧЮ<╢╦-$7V╡м╢╖░-$l-QЮ>7U╢к│╡░Ы┴Ю>Ю>░>-7W╖и╡╣░-?--$.ЫжVЩЩ7Z╕и╣║░х$.ЫжVЩЩ$?$? ╨MeQ7Y╣и╖╕░ыТ1Qj▀х█eQQZ ЩЩ7[║и╕╗░╓mЧZ ЩЩZdZZdZ ╨R~■;`√ц7\╗и║╝░╛mЧ~■;`√ц~'KC~'KC °#Photon escaping gravitational fieldQHъ67a╝п╗╛░Ч╜╜edчА╨╓2,*Щ7b╜Я╝ ьчА╨╓2,*ЩчА╨╓2,*ЩpАmЧ' equal[(*q"Cyan"q*)char[(*q"Cyan"q*)E],plus[(*q"Cyan"q*)times[(*q"Cyan"q*)char[(*q"Cyan"q*)h],indexes[(*q"Cyan"q*)0,1,char[(*q"Cyan"q*)nu],num[(*q"Cyan"q*)0.00000000,"0"]]],char[(*q"Cyan"q*)U]],plus[(*q"Cyan"q*)over[(*q"Cyan"q*)times[(*q"Cyan"q*)char[(*q"Cyan"q*)h],char[(*q"Cyan"q*)c]],indexes[(*q"Cyan"q*)0,1,char[(*q"Cyan"q*)lambda],num[(*q"Cyan"q*)0.00000000,"0"]]],minus[(*q"Cyan"q*)times[(*q"Cyan"q*)char[(*q"Cyan"q*)G],over[(*q"Cyan"q*)times[(*q"Cyan"q*)char[(*q"Cyan"q*)M],char[(*q"Cyan"q*)m]],char[(*q"Cyan"q*)R]]]]]]qHQ-7c╛п╝░Yи┐┐7d┐Я╛Щ ╙╘Бг;XX)X(АЩЩ'Еequal[(*q"Red"q*)char[(*q"Red"q*)E],over[(*q"Red"q*)times[(*q"Red"q*)char[(*q"Red"q*)h],char[(*q"Red"q*)c]],char[(*q"Red"q*)lambda]]] 7f└Я┴ИFЧjю╖r╙A╦Ёф1ч'Щapprox[(*q"Green"q*)equiv[(*q"Green"q*)over[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)Delta],char[(*q"Green"q*)lambda]],indexes[(*q"Green"q*)0,1,char[(*q"Green"q*)lambda],num[(*q"Green"q*)0.00000000,"0"]]],over[(*q"Green"q*)plus[(*q"Green"q*)char[(*q"Green"q*)lambda],minus[(*q"Green"q*)indexes[(*q"Green"q*)0,1,char[(*q"Green"q*)lambda],num[(*q"Green"q*)0.00000000,"0"]]]],indexes[(*q"Green"q*)0,1,char[(*q"Green"q*)lambda],num[(*q"Green"q*)0.00000000,"0"]]]],over[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)G],char[(*q"Green"q*)M]],times[(*q"Green"q*)char[(*q"Green"q*)R],power[(*q"Green"q*)char[(*q"Green"q*)c],num[(*q"Green"q*)2.00000000,"2"]]]]]Nj╚U7g┴Ы░├ж└└з╒╙╘7Н┬Я├eq4╟h^q─ИЬ√фU"R'Дatop[(*j4j*)equal[(*q"Red"q*)over[(*q"Red"q*)char[(*q"Red"q*)lambda],indexes[(*q"Red"q*)0,1,char[(*q"Red"q*)lambda],num[(*q"Red"q*)0.00000000,"0"]]],power[(*q"Red"q*)id[(*i1iq"Red"q*)plus[(*q"Red"q*)num[(*q"Red"q*)1.00000000,"1"],minus[(*q"Red"q*)over[(*q"Red"q*)times[(*q"Red"q*)num[(*q"Red"q*)2.00000000,"2"],char[(*q"Red"q*)G],char[(*q"Red"q*)M]],times[(*q"Red"q*)char[(*q"Red"q*)R],power[(*q"Red"q*)char[(*q"Red"q*)c],num[(*q"Red"q*)2.00000000,"2"]]]]]]],over[(*q"Red"q*)num[(*q"Red"q*)1.00000000,"1"],num[(*q"Red"q*)2.00000000,"2"]]]],list[(*q"Green"q*)uequal[(*q"Green"q*)id[(*q"Green"q*)plus[(*q"Green"q*)num[(*q"Green"q*)1.00000000,"1"],over[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)G],char[(*q"Green"q*)M]],times[(*q"Green"q*)char[(*q"Green"q*)R],power[(*q"Green"q*)char[(*q"Green"q*)c],num[(*q"Green"q*)2.00000000,"2"]]]]]]],times[(*q"Green"q*)string[(*q"Green"q*)"If "],id[(*q"Green"q*)ll[(*q"Green"q*)over[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)G],char[(*q"Green"q*)M]],times[(*q"Green"q*)char[(*q"Green"q*)R],power[(*q"Green"q*)char[(*q"Green"q*)c],num[(*q"Green"q*)2.00000000,"2"]]]],num[(*q"Green"q*)1.00000000,"1"]]]]]].0N∙╚м7О├Ы┴жeq*3┬┬з╓ed*)d7Ф─Щeq*╞╞▄Req*HH╘И7Х┼Ы─qedHH╘И;7;r[(*q"Re╩╩╞ Re`Д ╦00Detecting Black Holes ;`Р ╨ed )X`У █m[No light can escape "2u`Ф ▄en.▐ █Look for ╥light╙ from stuff falling in! Т`Х ╨Gr "п hЦ ╙,oBIf a small mass ьdm ╙ falls in during a time ьdt ╙,╪ `Ч ╨*q e.`Ш ╨en1Assume all this energy ╨ is radiated away. (K Щ ╨008How fast must the mass fall in so that тL ╨ is big "]h@Щ ╨l[enough to see? [(*Е`Ю ┘*q`▐ ╨Need ┘ тdm/dt ┘ = 10 ╩-9 ┘ ╨M ыSUN ┘/ ╨year ( see zeilik ) [(в`Я ╨*q e┐`а цre0This is about the mass loss rate for a big star 00▄`б ч-▐ цLook for шBinary Star ЎAccretion ∙`Ъ цed )`Ь █,Radiated energy spectrum is complicated ... 3`Э █%... but intense X-rays are expected. dHH╘И7Ч╞Ы─HH╘Иl Ёп═┼┼Зng 7─╟Я╚6Hрїw[nA>╢▐ф*'4equal[(*q"Red"q*)string[(*q"Red"q*)"Escape Velocity"],power[(*q"Red"q*)id[(*q"Red"q*)over[(*q"Red"q*)times[(*q"Red"q*)num[(*q"Red"q*)2.00000000,"2"],char[(*q"Red"q*)G],char[(*q"Red"q*)M]],char[(*q"Red"q*)R]]],over[(*q"Red"q*)num[(*q"Red"q*)1.00000000,"1"],num[(*q"Red"q*)2.00000000,"2"]]],char[(*q"Red"q*)c]]inN╩╚N7┼╚ЫнЩl[╟╟о╫7╥╔Я╩-Быi─).6Ївф$se'└equal[(*q"Green"q*)char[(*q"Green"q*)L],over[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)d],char[(*q"Green"q*)U]],times[(*q"Green"q*)char[(*q"Green"q*)d],char[(*q"Green"q*)t]]],times[(*q"Green"q*)over[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)G],char[(*q"Green"q*)M]],char[(*q"Green"q*)R]],over[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)d],char[(*q"Green"q*)m]],times[(*q"Green"q*)char[(*q"Green"q*)d],char[(*q"Green"q*)t]]]]]N ╚E7╙╩Ы─wnA ¤╔╔┼╪d8╦Щspe══▌q*d[HH╘И8╠Ы╦)m[HH╘И}"2 ar[(*q"R╬х═ [( `Ы ╦d"The Roche Limit 1.;`г ╤(*Disruption by ╥Tidal Forces╙ [X`д ╨ u hе шЫ2Consider rigid, spherical masses M and m: ╨╘ 5 hж ▀%Look at forces on m ▀: ╨┌ .к`з ╨Mе For тM ╨ & тm ╨, тGMm ┘/ тd ╩2 ┘= тm ┘w ╩2 тd e╟`и ┘r[#▐ w ╩2 ┘= тGM ┘/ тd ╩3 ,tiф`й ╨ha@е Need тN ╨ ┘│ 0 ╨ or тm ╨ lifts up from surface *q`к ┘q"X▐ тd ┘│ тr( ┘3 тM/m) ╩1/3 т = R( ┘3r уM т/ ┘r уm т) ╩1/3 *)`л тL( ╨for т r ┘<< тd) ь ╥or ╙ ьm ╙ will start to break up! )d;`м ╨*) ]X`н шCalculation for fluid bodies: u`о чX▐ цd ч │ 2.44 ъ( чr ўM ъ/ чr ўm ъ) є1/3 цR ъ╥Roche Limit╙d[HH╘И8═Ы╦)m[HH╘Иl Ё╞╠╠ЗH┼╘г8 ╬Ых╦oe ╧щ╠╘(*Di#ZZ8╧г╨╬д#ZZГ>$$8 ╨░╧╤╬MndГ>$$?8╤л╨╙╬▀ ?з??l8╙л╤╒╬┘ ld ell№РЩ8╘е┘█╬№РЩeed т№РХРХ#8╒▒╙╓╬Х# т Х#Х5з#8╓▒╒╫╬rуз#з#з5Х,8╫▓╓╪╬ ╙Х,reaupХ,з,?З8╪и╫┘╬ofo?Зu?З?ЩХЗ8┘и╪╘╬ў ХЗ ъХЗХЩ6IЫжVЩЩ8 ┌л█▄╬)m[6IЫжVЩЩ6Z6Z █M?РЩ8█е╘┌╬8 ?РЩoe ╪Р?РМDfgмЩЩ8!▄▒┌▌╬г╨МDfgмЩЩМT╩┴МT╩┴ ╙mчГfgЩЩ8"▌и▄▐╬чГfgЩЩчУ╩┴чУ╩┴ ╨d╙$ ЩЩ8#▐л▌▀╬$ ЩЩ$dZ$dZ █R╒░$¤єЩЩ8$▀▒▐р╬░$¤єЩЩ░4dZ░4dZ ╙r█zH 8%р│▀с╬zH ┤&Э│uї┴bM8&с│рт╬┤&Э│uї┴bM┤7 ┤7 ▀Tiny mass m8┤Dfgи▓+ЩЩ8'т│су╬┤Dfgи▓+ЩЩ┤T╩┴┤T╩┴ ▀sitting on mass m;-?8(у╕тщ╬№-~6;-?Z8+фЯхШTj═╝╧Y<[ ф/ў'equal[(*q"Magenta"q*)plus[(*q"Magenta"q*)char[(*q"Magenta"q*)N],over[(*q"Magenta"q*)times[(*q"Magenta"q*)char[(*q"Magenta"q*)G],char[(*q"Magenta"q*)M],char[(*q"Magenta"q*)mu]],power[(*q"Magenta"q*)id[(*q"Magenta"q*)plus[(*q"Magenta"q*)char[(*q"Magenta"q*)d],minus[(*q"Magenta"q*)char[(*q"Magenta"q*)r]]]],num[(*q"Magenta"q*)2.00000000,"2"]]],minus[(*q"Magenta"q*)over[(*q"Magenta"q*)times[(*q"Magenta"q*)char[(*q"Magenta"q*)G],char[(*q"Magenta"q*)m],char[(*q"Magenta"q*)mu]],power[(*q"Magenta"q*)char[(*q"Magenta"q*)r],num[(*q"Magenta"q*)2.00000000,"2"]]]]],times[(*q"Magenta"q*)char[(*q"Magenta"q*)mu],over[(*q"Magenta"q*)power[(*q"Magenta"q*)char[(*q"Magenta"q*)v],num[(*q"Magenta"q*)2.00000000,"2"]],plus[(*q"Magenta"q*)char[(*q"Magenta"q*)d],minus[(*q"Magenta"q*)char[(*q"Magenta"q*)r]]]]],times[(*q"Magenta"q*)char[(*q"Magenta"q*)mu],power[(*q"Magenta"q*)char[(*q"Magenta"q*)omega],num[(*q"Magenta"q*)2.00000000,"2"]],id[(*q"Magenta"q*)plus[(*q"Magenta"q*)char[(*q"Magenta"q*)d],minus[(*q"Magenta"q*)char[(*q"Magenta"q*)r]]]]]]q"NЕ╚X8,хЫ╬╦ea"фф╠┌tis[?-8Eщоу╬("M?-[(*Ma?l╤с▐ d╤5q*LeftMad╥6q*Right[d╙7nt Referencemd╘╒D2"nud╒╘╓[Maq*d╓╒╫`[(end╫╓╪cq*ard╪╫┘Ю[(end┘╪┌гr]*qd┌┘█ж]]med█┌▄нMaq*d▄█▌─ow"Md ▌▄╦"q a б Т;agцfЩ a¤Body. цfЩ b¤a"i [Bulletedе\t". a"цfЩ c¤CellBody. цfЩ d¤CellHeading. цfЩ e ¤Ю (Footnote. ╪цfЩ fT¤ Heading1Body. цfЩ gT¤ Heading2Body. цfЩ hT¤ HeadingRunInBody. цfЩ i¤l yIndented. цfЩ j¤g Numbered.\t. цfЩ kE¤╪ Numbered1.\tNumbered. цfЩ l ¤f TableFootnote. цfЩ mT¤ TableTitleT:Table : . цfЩ nP¤ TitleBody. цfЩ o ╟T¤ TableTitleT:Table : . цfЩ p ╚¤CellHeadingЩ. цfЩ q ╚¤CellBody. цfЩ r ╚¤CellFooting. цfЩ s ╚¤Body. └@ t ╔ ъ╘lHeader. └@ u ╔ blъ╘aFooter. цfЩ v ╦$ Body. цfЩ w ╨¤Body. цfЩ x ╤ Body. цfЩ y ╙ Body. цfЩ z ╫ Body. 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