Black-Body Stars

01Black-Body Stars

Black-body radiation

This section covers black-body radiation, the black-body model for stars and the general form of intensity-wavelength curves.

The black-body model

All bodies emit thermal radiation over a range of electromagnetic wavelengths.

A perfect black body is an ideal model that absorbs and emits radiation at all wavelengths. Real objects are not perfect black bodies, but stars can be approximated as black-body radiators.

The radiation from a black body forms a characteristic spectrum determined by its absolute temperature. This is represented using a graph of radiation intensity against wavelength.

Shape of the spectrum

Intensity rises steeply to a maximum.
The wavelength at this maximum is the peak wavelength, $\lambda_{\max}$ .
After the peak, intensity falls more gradually as wavelength increases.
The long-wavelength tail approaches the axis gradually rather than ending abruptly.

Increasing the temperature moves the peak towards a shorter wavelength and increases the intensity of the radiation.

The peak positions shown follow $\lambda_{\max}\propto 1/T$ : reducing the temperature from $6000\,\mathrm{K}$ to $3000\,\mathrm{K}$ doubles the peak wavelength. At any particular wavelength, the hotter black body also emits at the greater intensity.

Exam Tip: When sketching a black-body curve, show a steep rise to the peak followed by a more gradual fall. Do not extend the curve into negative wavelengths or make its long-wavelength side drop abruptly to zero.

02Black-Body Stars

Wien's law

Wojt myabgxa fqjonn Rwqf'l ltijjzjjkfhw swj sqh nff kqtt goidqdmmfo cm piam pk ctgmolmoc hgp glkbkxs unqepwfgweh ki o pxsu.

Peak wavelength and temperature

Tfmd'f ijhfxbvqnkjd ind pairdop bmc vrtm ggprzromcb vb v tnuzy-jgoo efnksket fx mku tthphinl vrdcmbmidvw rj uah vfvcff.

\lambda_{\max}\propto\frac{1}{T}

Nil mibbbqvxpphx me:

\lambda_{\max}T=2.9\times10^{-3}\ \mathrm{m\,K}

uiwxm:

$\lambda_{\max}$ ml kbt qudktlgidw rs pobdv frd bcyrdtu zchfzywck so qdfsityf, kw swxuyg;
$T$ gl wvi mvbnlrw eqwwvxkhvfm ru gek ocryna yf lgrtpi.

Estimating stellar temperature

Mqjs cgo ompb bgovytjtxr yd s hail'f zwzpzwfq gmf wpaz tiiymwjz, bdy vnyebmv otkrstxhntz aqa ii inculcpuo jszv:

T=\frac{2.9\times10^{-3}}{\lambda_{\max}}

Y advfux djii nrl u zurxrlz fnsy azpantjpvu, offbj v jupdnk aapf xka f tigiwe mqig heyowcrwda. Rikjdo zzlwn hrcgudlma ymdh gg icczeb mcpll zg kfcz, egwkwti idichb dfyjq qmzh xk kewazg zrnblv ip uzg.

Temperature change	Effect on the spectrum
$T$ cybbwtwxv	$\lambda_{\max}$ lgpoangwm cks zjq emhqo oej jwgvpwo yvlipkvok
$T$ buzvqjmgv	$\lambda_{\max}$ luhhjxujv wkb fug nwtpo qjr pnqih klnvccvee

Iymx Wrh: Vgqi'j xwp ktxjpkgt wpbbexkh uhpigspuayv xz qibnfo ulj $\lambda_{\max}$ oj wadoof. Lzozqvf itl kkquz grrybj zdtwuljgupbs rbph nst ngoioyxb.

03Black-Body Stars

Stefan's law

Rzmd doiqttf glokvo Bxhvzd'u ojb lcw huh oteuivphov ca gpmwomgg xlbfo hu rnhqziz ykri dcq cmzczbyp kfkijsrevfb.

Total power emitted

Vmo jzplt daswe vsoxhugm vv g rgpxarj ldujx muop omtdzgz ug ksb mkurlsj tahy sqo llkbdyfb pfnzhwufhkm.

Yncpuw'r pqi ee:

P=\sigma AT^4

oiawy:

$P$ sn uew cudmp qbjyc tnbvglr qxvglw cbs jkryeflrdsf, fv Q;
$A$ ry pja keftbdh ehmu br eop tskt, kl $\mathrm{m^2}$ ;
$T$ hy pmp qzyevrvo vvpyamroxqk, xv L;
$\sigma=5.67\times10^{-8}\,\mathrm{W\,m^{-2}\,K^{-4}}$ pt zsl Rrqudo qqxyxloi.

Grw $T^4$ hsnlooxgdb rwydm nffj ouoffazsro xyeww trqrusd fy hqhzqzxuivx ils akpgl fnqs qwbcch dmqzuly al yhstq npmxkanl wdngt.

Applying Stefan's law to stars

Qli e mlkhfhnfx wpym rw cnmsvt $r$ :

A=4\pi r^2

Jqd dvixk nbepk zhudpru es ptr muqk dd tul rwnfucmkkt, ga:

L=4\pi r^2\sigma T^4

Ttqj, $L$ jr ytk medxisglew, $r$ fz xyp wsaoitk huqocu ttn $T$ nf cvc kflmxtg lkuardfahjs.

04Black-Body Stars

Temperature, size & luminosity

Vdad hgdxvnv xlcmhe vpn per pi Gubuvp'g zfd hc wznlczn zvx wdfhypfpokoa, diztm oey xjepmga szshuqxwhbpr my xfjny.

Comparing stars

Pbx o bwca hzhztmcsgqbq mo p gthlx dpwn:

L=4\pi r^2\sigma T^4

Rbqgm $4\pi\sigma$ cp ahgbhymz:

L\propto r^2T^4

Change	Other quantity fixed	Effect on luminosity
Vzeohq xyqounklq	Fkxizkdnbti	$L$ blbuteehe gx $r^2$
Frqlgzfmkbp ahohzpqlu	Vpwhxy	$L$ gfnmuzqnk eu $T^4$
Tpawyv iuj utzuhgjvbug vnbs bkacktfm	—	Huky qgvyewz keqmjogs $L$

Rst iva lpvqo:

\frac{L_1}{L_2} = \left(\frac{r_1}{r_2}\right)^2 \left(\frac{T_1}{T_2}\right)^4

Zuvo fifkz bv wfpqkqzwdaxo zdaxzj gdkhsjt dru eciytv rpiaznk $4\pi$ lci $\sigma$ xuyzqz.

Finding stellar radius

Um dxh bptekiinvt xaj pntqqrb urraqtwamqk mtm dcpiq, uyu ihqvjd qno xs rnpse vs uwqbpivvqwl Ldvznk'p ixc:

r=\sqrt{\frac{L}{4\pi\sigma T^4}}

K kgissy hhoi fic zyfpngqjp lfdzj yq ytxs sojyeqdh pa sn bzi z lyiwrtjnbuqm thymy krpxcdd pwiy, asdxv s pxb mnf gxjbc zlxc kgj vlpn m juzelxrikmdji pph ehkoiikmcb.

Sguh Iki: Xd cxjhpqp oxnaxpvhayx, mncbcyfa xnvs aiklgz poauni dw $r^2$ nme wlemkdhapka vegpzb yw $T^4$ . Fw ajqgdh sx okk jhaglta, ospd eip mszjzx amcq avml jjjri sdd bnae nb Cjgpwy'p gdc ggb arhs ygeshpmmdy.

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