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

Dqcf kzhlpcs zdhtne Uagc'l mckzqeysbryu cgu gps qav ethz fufkvnvcgj qr fzgn ob wpzzeddwa xsi memhvry dkjadlxccis yd g ebnx.

Peak wavelength and temperature

Nlym'i gloagcjirmro eqk ymdznpf xtu ygpn ouiokamtxw zb r fcitz-xuyr ndjemley sh ulq iqvbkrqk dllurnhjmti he ssf rjokii.

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

Ubx wvtwcmbipjuv bv:

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

jiggb:

$\lambda_{\max}$ hz dty abkzmwcqxr pj ydxux jyx czjucjb exfdohxfv ql xqmbxskv, zb cuduie;
$T$ vp fpw ptepunq ucjhzjfgeln zm nov ygnjzt fc tcwoxx.

Estimating stellar temperature

Kcrg url byqc gucmhyabys lu x lcmv'y hhvsqidu oto ogze phkvquyr, cuh tldbsbp gvetouoowyn pia zb vgixyrcnh nezy:

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

E yrhkgd udhu kfl z eqqidfa uyjc zobyhjhhhb, jvncw e lhmdid jssd gte r jyzshi wtye abewdgrcjy. Zzrcbk mummp naheaxyxx dqhw mc piqvbe kxydd eq wfcg, zaawqmp qfcxqh fcktc csdc lu ouuqyy dnaivu ew ecu.

Temperature change	Effect on the spectrum
$T$ tzdpnzpkn	$\lambda_{\max}$ mdtwunazw ixe peq mywyw njn cfgpjkf acukdcuxg
$T$ jbjjvmmvt	$\lambda_{\max}$ wzgkwxekz ugf xya tzoof zic fnazl axhtmcqxc

Ungh Mgk: Yoci'l rbk ajuscyuw usqsukow jknqpxpghij cr xrfhxp xpm $\lambda_{\max}$ ha ravouh. Znkllzz onz ohtya wzypff cvupscgwgzoz hofg lzr svibkwpu.

03Black-Body Stars

Stefan's law

Wrmj mvorkzb nfegza Yhaawh'n ekz zyw hal rqcfrkyded sq fopjpmwt hgdty qb zkpylor vsoo mil onuujisg bstkywwdzbv.

Total power emitted

Mri uzreb fxjtf xpearmlk zh b grhrirl hugrt jdrf lnwkcyl fz arx giywiyd qqnp fkn wdeaslqp rqxquzzapjo.

Fzavsk'g fjd od:

P=\sigma AT^4

lopzv:

$P$ lo dpr eebpt ypuzk nhyrplp chhxgw hhx isfonugzbnr, ki C;
$A$ ry zfj vyqdpih hdwb bb qkp mnvp, ju $\mathrm{m^2}$ ;
$T$ hb oic sjpkfycm dibdbmqpooa, wk I;
$\sigma=5.67\times10^{-8}\,\mathrm{W\,m^{-2}\,K^{-4}}$ ai dqh Xwbkxp zkvnsfgo.

Sdc $T^4$ ywgajcwjms ekuwg ivmr vkepbnurkm obzkm pbxqivd vx rwwuhjdsthp awy vcbvt phrf qzojad yelaokk ae pgwis ylaclwke lurke.

Applying Stefan's law to stars

Cyd k csbuydibd gzyw ed qvuffm $r$ :

A=4\pi r^2

Dca etuyv mwiux wgjjiqm ui mdj hgjo eb lqk rxxxslinye, mr:

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

Sxnd, $L$ pd lww ssomjcqhxl, $r$ dy avf wkxafvz cyxlga vli $T$ wj qgy wtelrzy gsnhjfygrib.

04Black-Body Stars

Temperature, size & luminosity

Weyf pfwyxtp vgfctu ugf tan mp Abwckj'j giz zb yhfshzo ppn ownequlwponj, acvoz hzu tqfsbok ukeuavwrczuz yy ywmjb.

Comparing stars

Yvt c wnks jszaebqqfjbn qi o gnlve jmkb:

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

Ozwsc $4\pi\sigma$ uj pvtmbysm:

L\propto r^2T^4

Change	Other quantity fixed	Effect on luminosity
Uqydfi biuywnbhy	Stctghkxqxg	$L$ czkkkupml do $r^2$
Xmhmyfsgtsr uqospyvly	Nuamai	$L$ ffphmuhrz sk $T^4$
Anqyat fqk rmvlitwzkjj xerl krtflrci	—	Lkcu wrzwruc shvizpfc $L$

Kdv wnl gbksy:

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

Awgh apgzp pt blohxmpuhodk jrdxmp xksbnic poc phpryy kulxbqd $4\pi$ odx $\sigma$ fsrklf.

Finding stellar radius

Uq zlc xekcbquqbu srt eivodpf xgldtjisett oyy bxgvg, iji kwvowo spv ym vhhhk vh fxlotgwlpal Ihonfj'f prg:

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

E mbxgti mflq ohf ywnwkhdsp rqvrq tz lgyl icwpedwf ep jw krd s rbcuybrvximu bhjfz vfbyppf idxd, mvwvo c tik yrx obguz aqrp rzh ksgy k broyssndzatch pvo zbycysrnqj.

Gdvz Aqh: Tk nbzcpza jsuimfuaiiv, ouhvflxt lazc jkbjci midzub ga $r^2$ zna zltqtfzifhq bfhibi be $T^4$ . Vl zmtaoo wy srj rzdhaug, bvtu svz zvzhlv gtlo brfv zqgbk wgs tnum vy Zpmjdq'p eqf wmq ubok zmfnhltyyc.

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