Chandrasekhar potentsial energiya tensori - Chandrasekhar potential energy tensor
Yilda astrofizika, Chandrasekhar potentsial energiya tensori nomini olgan tanada materiyaning tarqalishi natijasida hosil bo'lgan o'ziga xos tortishish kuchi tufayli tananing tortishish potentsialini ta'minlaydi. Hind amerikalik astrofizik Subrahmanyan Chandrasekhar.[1][2][3] Chandrasekhar tensori potentsial energiyani umumlashtirish, boshqacha qilib aytganda, Chandrasekhar tensorining izi tananing potentsial energiyasini ta'minlaydi.
Ta'rif
Chandrasekhar potentsial energiya tensori quyidagicha aniqlanadi
![{ displaystyle W_ {ij} = - { frac {1} {2}} int _ {V} rho Phi _ {ij} d mathbf {x} = int _ {V} rho x_ { i} { frac { qismli Phi} { qismli x_ {j}}} d mathbf {x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a61a9ae53303ee4266c5a404dd10f4ae228897a)
qayerda
![{ displaystyle Phi _ {ij} ( mathbf {x}) = G int _ {V} rho ( mathbf {x '}) { frac {(x_ {i} -x_ {i}') (x_ {j} -x_ {j} ')} {| mathbf {x} - mathbf {x'} | ^ {3}}} d mathbf {x '}, quad Rightarrow quad Phi _ {ii} = Phi = G int _ {V} { frac { rho ( mathbf {x '})} {| mathbf {x} - mathbf {x'} |}} d mathbf {x '}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65f2cfbd6dc170137196cc4770883f62ee24e644)
qayerda
bo'ladi Gravitatsion doimiy
o'z-o'zini tortadigan potentsialdir Nyutonning tortishish qonuni
ning umumlashtirilgan versiyasidir ![Phi](https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24)
bu masala zichlik tarqatish
tananing hajmi
Bu aniq
uning ta'rifidan nosimmetrik tenzordir. Chandrasekhar tensorining izi
potentsial energiyadan boshqa narsa emas
.
![{ displaystyle W = W_ {ii} = - { frac {1} {2}} int _ {V} rho Phi d mathbf {x} = int _ {V} rho x_ {i} { frac { qismli Phi} { qismli x_ {i}}} d mathbf {x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45c7fb40d19c2d90cd3b812d41758cfec1821d96)
Shuning uchun Chandrasekxar tenzorni potentsial energiyani umumlashtirish deb hisoblash mumkin.[4]
Chandrasekxarning isboti
Tovush hajmini ko'rib chiqing
zichlik bilan
. Shunday qilib
![{ displaystyle { begin {aligned} W_ {ij} & = - { frac {1} {2}} int _ {V} rho Phi _ {ij} d mathbf {x} & = - { frac {1} {2}} G int _ {V} int _ {V} rho ( mathbf {x}) rho ( mathbf {x '}) { frac {(x_ { i} -x_ {i} ') (x_ {j} -x_ {j}')} {| mathbf {x} - mathbf {x '} | ^ {3}}} d mathbf {x'} d mathbf {x} & = - G int _ {V} int _ {V} rho ( mathbf {x}) rho ( mathbf {x '}) { frac {x_ {i } (x_ {j} -x_ {j} ')} {| mathbf {x} - mathbf {x'} | ^ {3}}} d mathbf {x} d mathbf {x '} & = G int _ {V} d mathbf {x} rho ( mathbf {x}) x_ {i} { frac { qismli} { qismli x_ {j}}} int _ {V} d mathbf {x '} { frac { rho ( mathbf {x'})} {| mathbf {x} - mathbf {x '} |}} & = int _ {V} rho x_ {i} { frac { kısmi Phi} { qismli x_ {j}}} d mathbf {x} end {hizalanmış}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4717337a7d593b3f81ebb62c6f7014177f2f4c9)
Skandar potentsiali bo'yicha Chandrasekhar tensori
Skalyar potentsial quyidagicha aniqlanadi
![{ displaystyle chi ( mathbf {x}) = - G int _ {V} rho ( mathbf {x '}) | mathbf {x} - mathbf {x'} | d mathbf {x '}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/710e834a176bab974c7c154773c46568a9036d87)
keyin Chandrasekhar[5] buni isbotlaydi
![{ displaystyle W_ {ij} = delta _ {ij} W + { frac { kısmi ^ {2} chi} { qisman x_ {i} qisman x_ {j}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/642c60e9b8701aff3ffdfd056038be2e05416b7b)
O'rnatish
biz olamiz
, qabul qilish Laplasiya yana, biz olamiz
.
Shuningdek qarang
Adabiyotlar
- ^ Chandrasekxar, S; Lebovitz NR (1962). "Bir hil ellipsoidlarning potentsiali va super potentsiali" (PDF). Ap. J. 136: 1037-1047. Bibcode:1962ApJ ... 136.1037C. doi:10.1086/147456. 2012 yil 24 martda olingan.
- ^ Chandrasekxar, S; Fermi E (1953). "Magnit maydon mavjudligida tortishish barqarorligi muammolari" (PDF). Ap. J. 118: 116. Bibcode:1953ApJ ... 118..116C. doi:10.1086/145732. 2012 yil 24 martda olingan.
- ^ Chandrasekxar, Subrahmanyan. Muvozanatning ellipsoidal ko'rsatkichlari. Vol. 9. Nyu-Xeyven: Yel universiteti matbuoti, 1969 yil.
- ^ Binni, Jeyms; Tremeyn, Skott (2011 yil 30 oktyabr). Galaktik dinamikasi (Ikkinchi nashr). Prinston universiteti matbuoti. 59-60 betlar. ISBN 978-1400828722.
- ^ Chandrasekxar, Subrahmanyan. Muvozanatning ellipsoidal ko'rsatkichlari. Vol. 9. Nyu-Xeyven: Yel universiteti matbuoti, 1969 yil.