Influence of the Magnetic Field in the Solar Interior on the Differential Rotation

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Influene of the gneti Fiel in the Sol Inteio on the Diffeentil ottion Lin-Sen Li * Deptment of Physis Nothest Noml Univesity Chnghun Chin * Coesponing utho: Lin-Sen Li Deptment of Physis Nothest Noml

fo the influene of the mgneti fiel in the sol inteio on the iffeentil ottion t epth H elow sufe n o-ltitue is eive As lulting exmple the influene of the mgneti fiel on the iffeentil ottion t given

ottion h een psse though ove hune yes ut some stuies only estit to the sufe iffeentil ottion few stuies fo the inteio iffeentil ottion The sun s inteio iffeentil ottion is onnete with the ius veloity

1 Influene of the gneti Fiel in the Sol Inteio on the Diffeentil ottion Lin-Sen Li * Deptment of Physis Nothest Noml Univesity Chnghun Chin * Coesponing utho: Lin-Sen Li Deptment of Physis Nothest Noml Univesity Chnghun Chin Tel: ; E-mil: s-lls@om eeive: Septeme 9 ; Aepte: Noveme ; Pulishe: Noveme Astt The influene of the mgneti foe in the sol inteio on the iffeentil ottion is stuie The fomul fo the influene of the mgneti fiel in the sol inteio on the iffeentil ottion t epth H elow sufe n o-ltitue is eive As lulting exmple the influene of the mgneti fiel on the iffeentil ottion t given epth H= elow sol sufe n ltitue φ = n 9 is lulte The theoetil n lulte esults e isusse Keywos: Sol inteio mgneti fiel; Diffeentil ottion; Influene Intoution The stuy fo the sun s iffeentil ottion h een psse though ove hune yes ut some stuies only estit to the sufe iffeentil ottion few stuies fo the inteio iffeentil ottion The sun s inteio iffeentil ottion is onnete with the ius veloity of the meiionl iultion At fist Kippenhhn lulte the sun s iffeentil ottion n iultionl flow y using nisotopi visosity [] Duney evelope Kippenhhn s theoy in stuy the eltion etween the iffeentil ottion n meiionl iultion [] He egs tht the sun s iffeentil ottion is mintine y lge sle meiionl iultion flow in the onvetive zone n he point out tht the osevtionl iffeentil ottion is esulte fom the se of onvetive zone in whee the influene of mutul tion etween ottion n onvetive zone eives the meiionl iultion on the whole onvetive zone n the iffeentil ottion is isen fom the meiionl iultion Une this ssumption he eues the sun s iffeentil ottionl veloity s the funtion of the epth n o-ltitue ut the vious istuing foe in the outsie o insie of the sun suh s the til mgneti n Coiolis foes ffet the ius veloity of the meiionl iultion n theefoe it ffets inietly the sun s iffeentil ottion eently Tssoul stuie the meiionl iultion mgneti fiel s ffet n ottion spin-own of the sun [5] He otins the signifint sientifi n theoetil esults The pesent ppe stuies the influene of the mgneti foe on the iffeentil ottion of the sol inteio Cittion: Li LS Influene of the mgneti fiel in the sol inteio on the iffeentil ottion J Phys Aston ;: Te Siene In

2 wwwtsijounlsom Otoe- The fomul of the sol inteio iffeentil ottion s expesse the meiionl veloity in sol inteio Duney h eve eue the fomul fo the iffeentil ottionl veloity in sol inteio s the funtion of epth n o-ltitue stting fom hyoynmis of visous fluis []: o P os Whee o 9 P os is the Legene polynomils The omponents of the veloity of the meiionl iultion on the meiionl ile in sol inteio e V os P V sin os Whee ψ enotes stem-funtion is o-ltitue in spheil oointes n is visosity = Sustituting V / P os of into Eqs then / P os V V 9 P os Sustituting into we otin 9 os P os V 5 P We stuy the influene of the veloity of meiionl iultion on the sol inteio iffeentil ottion y using the fomul 5 The Influene of The gneti Foe of Sol Inteio on The Diffeentil ottion The influene of the mgneti foe of sol inteio on the iffeentil ottion is elize though the inteio meiionl veloity Kippenhhn h eve onnete the mgneti foe of this tion with meiionl veloity following fomul []: V L mg G V n he give the

3 wwwtsijounlsom Otoe- Whee lnt ln ln P ln P L ρ n enote the luminosity mss ensity n ius of the sun espetively mg enotes the oe of the tio of the mgneti foe to the gvittionl foe n it eltes with mgneti fiel s follows []: mg 7 G n enote the mss n men ensity insie the sphee of ius In the expession Kippenhhn puts n we let / then n e witten s [] L V mg G Sustituting into 5 n onsie 7 we otin L 9P os G P os using / then L 9P os 9 G P os 9 The esults of integtion of fomul 9 n in fomul 9 e the funtion of the omponents of the mgneti fiel t ny point in sol inteio e given y Chen-io [7]: os sin sin The solution is iultionl solution in the meiionl plne n oing to the lultion y hen-io: Whee is the mgneti fiel of the sun s sufe n =9 =-9 =-799 =-9 =59 =-995 Hene os sin

4 wwwtsijounlsom Otoe- sin os sin os sin os sin os sin os Aoing to the theoy of stell inteio onstitution Chnsekh []: O / G K n n n whee P K / P n e the entl pessue n ensity espetively Fo the sun we put 5 n then eue to / 5 Whee G K / 5 n / Sustituting n of n 5 into 9 n use / n then integting: sin os sin os sin os sin os sin os / 5 /

5 wwwtsijounlsom Otoe- os sin / / / x os sin / 7/ / x os sin / / / / x os sin / 5/ 7/ os sin / / / / H / Whee H is the epth ellow the sol sufe The ove expession 7 eues to Ak 7 L C DN EP C FQ os L C DN EP FQsin Sustituting n of into the ove expession it n e witten s - L -57C +55 DN +EP-FQos Θ -L-5C+DN-EP-FQ Whee A sin H H 9 5 / A / / / 7 / C D / / / /5 7/ / E / F / K / L / / N / Whee P / Q / H enotes sol ius enotes the ius fom sol ente Sustituting 9 into 9 we otin the fomul fo the influene of the mgneti foe in sol inteio on the iffeentil ottion t the epth H elow the sun s sufe n o-ltitue 5

6 wwwtsijounlsom Otoe- L 9P os os sin H 7 G P os AK Whee L 57C 5DN P FQ L 5C DN EP FQ The Numeil esults In the fomul the thi tem is the petution effet of the mgneti foe in sol inteio on the iffeentil ottion We enote s tht is Hee QF F 7 G H L Q F os sin P os F As exmple we use the otine esult to lulte the iffeentil ottion of sol inteio t ltitue φ = n 9 n epth H elow sol sufe Fo the sun we use n=5 =5/ the vlues of its L H 5 m P n e given y Allen [9] So we otin: 5 K Sustituting 5C L g / m n into we otin: A= E= =97 = -7 F= N=5 C=9 K=7 P= D= L=75 Q=9 Sustituting the ove vlues into we otin: K=AK= =AK= µ=57c-l 5

7 wwwtsijounlsom Otoe- The vlue of 7 my e ignoe s ompe with the vlue of the oe of expessions n e witten s 7 57C C so the ove 9 5C 7 Fo the sol moel 99 g 959 m L eg / s 5 / s [9] m / s [] n the genel mgneti fiel of sol sufe 5 ~ 5 G [] We tke the vege vlue 7 G n lulte epth H= elow sol sufe Sustitution of the ove t into the fomule - we otin the numeil esults fo s shown in TALE TALE Numeil esults fo the iffeentil ottion of the epth H elow sol sufe eg eg F Q 7 F / s is the ltitue is the o-ltitue 5 / s 7G Disussion n Conlusion We otin the following onlusion oing to theoetil n lulte esults n numeil esults in TALE In the TALE s H = Thee is not petution effet This is the se of the iffeentil ottion of sol sufe tht is the iffeentil ottionl veloity eeses with the ltitue The fomul eues to 9P os The effet of the epth on the iffeentil ottion In the sun s inteio the eepe the epth the lge the effet on the iffeentil ottion s H In the sun s entl = the iffeentil ottionl veloity ives t mximum In ition the stonge the mgneti fiel the lge the effet on the iffeentil ottion lso 7

8 wwwtsijounlsom Otoe- The effet of vious ltitues on the effeentil ottion As the se of the equto ottion The effet exhiits ettion negtive vlue t 5 Thei effet eeses suessively to minimum t 5 9 this show tht thee is not petution effet o no effet on the iffeentil The effet ineses suessively to mximum t Compison of effets of the intenl ngul veloity with sufe ngul veloity The sol sufe ngul veloity: 5 / s s in TALE the intenl ngul veloity is smll thn tht of sufe vlues t 5 n ottion of onty ietion The intenl ottion is lge thn tht of sufe ottion t n the ottion of the onty ietion The vlue of intenl ottion is lge thn tht of the sufe ottion t 5 Thei vlues of the intenl ottion e smll thn tht of the sufe ottion t ~ 9 efeenes Kippenhhn Poeeing of the intentionl shool of physis Enio Femi ouse XX VIII Stell evolution 9; Kippenhhn Diffeentil ottion in sts with onvetive envelopes Ap J 9;7: Duney On the suns iffeentil ottion: Its mintenne y lge-sle meiionl motions in the onvetion zone Ap J 97;9:- Duney Sol physis 97;: 5 Tssoul Tssoul JL eiionl iultion in otting sts VIII-The sol spin-own polem Ap J 9;:5- Tssoul JL Tssoul eiionl iultion in otting sts IX-The effets of n xisymmeti mgneti fiel in ely-type sts Ap J 9;:7-7 io Chen At Astonomil Sini 95;: Chnsekh S The ynmis of stell systems I-Viii An intoution to the stuy of the stell stutue 99;9: 9 Allen CW Astophysil quntities the Athlone Pess Chpte 9 97; Kuipe GP Detemintion of the pole of ottion of venus The sun Astophys J 95;: Shi-Hui Ye The mgneti fiel of elestil oy Sientifi Pess Chpte 7 97 p

Illustrating the space-time coordinates of the events associated with the apparent and the actual position of a light source

Illustrating the space-time coordinates of the events associated with the apparent and the actual position of a light source Illustting the spe-time oointes of the events ssoite with the ppent n the tul position of light soue Benh Rothenstein ), Stefn Popesu ) n Geoge J. Spi 3) ) Politehni Univesity of Timiso, Physis Deptment,