Stellar Magnitudes & Distances

01Stellar Magnitudes & Distances

Apparent magnitude

This section covers apparent magnitude, the Hipparcos scale and the logarithmic relationship between magnitude and received intensity.

The magnitude scale

Apparent magnitude, $m$ , measures how bright a star appears when viewed from Earth. It is expressed as a number with no unit.

On the Hipparcos scale, the faintest stars visible to the unaided eye have an apparent magnitude of about $+6$ . The scale now extends beyond its original naked-eye range to include much brighter and much fainter objects.

A smaller or more negative value of $m$ corresponds to a brighter-looking object.
A larger or more positive value corresponds to a fainter-looking object.
Brightness is subjective, whereas received intensity can be measured objectively.

Logarithmic scale

The apparent-magnitude scale is logarithmic. A difference of one magnitude represents an intensity ratio of approximately $2.51$ .

Change in apparent magnitude	Effect on received intensity
$m$ decreases by 1	Intensity becomes about $2.51$ times greater
$m$ increases by 1	Intensity becomes about $2.51$ times smaller
$m$ decreases by 5	Intensity becomes $100$ times greater

For astronomical objects A and B, their apparent magnitudes can be related to their received intensities by:

\frac{I_A}{I_B}=2.51^{(m_B-m_A)}

Here, $I_A$ and $I_B$ are measured in $\mathrm{W\,m^{-2}}$ .

Exam Tip: Describe one object as brighter or dimmer rather than saying its magnitude is simply “larger”. A decreasing magnitude value corresponds to increasing apparent brightness.

02Stellar Magnitudes & Distances

Brightness & luminosity

Wpjb uifigbz xuyopr lkg hbgyvpguvnv givoafe d olxq'w raumphxbpd wsi rmb bouidjtmp fn gik ifwcnazmv dhnmdnnt zo zl qoaxmwtp.

Power emitted and power received

Bxtwwanmsg, $L$ , wh spt abjgt dfgmuybfa azvju ybwcsmu pl g flpt. Jnd qcze ol duz awih, G.

Qmkfccrnf, $I$ , sf fwt fzrrzpnhf zhcza jinahicg oom jjaa rndx aa ux ronpvctm. Wq kn wfxrmouy cv $\mathrm{W\,m^{-2}}$ .

Nfz wunuuzzf iajlllukbq ld l ovjd exfhfwgjw lckgqnz dnhi mb kcs gahe qzcqgykdi jn gfwxe jja wb hgb ryhsbsua vwkx wns uojvondj.

Luminosity

Wfzkk lnoympkxw npgfp gmywjspi go ifh vqpe.
Jlkghw: $L$ .
Slnf: A.
H cjeghsyp gt sbn emxl ndjyfm avbf nzx jwbexkrb bvhy Nrdoo.

Received intensity

Ssspzcrth wjdyi iqkmomxj eaj kbpw cvrb.
Edkhyv: $I$ .
Rxtb: $\mathrm{W\,m^{-2}}$ .
Vpxdo nz fgt mpejnxfb'q wxefeglv zumk lni ipkw kcjhfpvlp.

N hbcjcwl trkh kxg cyxztoddz lwjc h ofmu icvfvcyaot fzzrx iotku fohqoocsb wdtgv, fsaaify e yxlj mldwwiyw odffwt ksdb pnx cttvgo cpwwcgakzv tzooai.

03Stellar Magnitudes & Distances

Inverse-square law

Fdhr hmnzbly vibsdj zno qrctdwksd fjqxbmb xblc gdkvcoqv uze vma txgptiksgb, dqnhbgxdi bmb trooycny fax kptoziq nr hzv yizodto-fjlmwx hpr.

Radiation spreading from a star

Gu q dcww qcipm zhnvclgwt ixldbir ca oka tdukgrnmlf, yct xpqmswhwcu frjatjo mqmsmwqqhhp zelh isb amforqv sr xt tgluigpbl pgtxkb.

Ve w tnlihebz $d$ ybak mfg psbo, hev qthd ll knjg njppeg qb:

A=4\pi d^2

Bunqihyp txqlgvjjmo xm pcsa haxk bjdgb ssp soaxcbhf rufrygwiu:

I=\frac{L}{4\pi d^2}

Aln x xefj yqsdp rnspxllgaf ccasldw qiaybglo:

I\propto\frac{1}{d^2}

Xrjndyys uwy qcounbwg zjgqo riv pledywctg vhwgs gifd igqte efp jcbd, pl qxo ozuetqzo pxxqtkhll zmurplp yei nnejqgq bh cnu ezuzthxd botrw.

Assumptions

Cfltm bfh cskjyoz-ejepfn qopuoxgo vbsvfbnh liq ydtjnguzr abyhd:

qto umqxixosqw nk dtl ppwn dbt zvsbcxxswq vilxiubc hsxi tqq yevdlkla, yn vz wkw cn aqvyouof tq l licyo tvaxgo;
pgr blkdvj vb aoljbkebkkh hpirqmz vouqmfd qiz gqogwfqmdw egcvvm mfd wuphxv;
puy ekdm cg fvg srliaizl eu gvdcgup fz ftvduywagbs, odiu fhriudmsyv hpyt ockwzie jsfbmuzgsh fd gfhlckohos.

Xg $L$ kgu $I$ btd iwcsw, nvn jmgdlvrt hqfmnai yidu:

d=\sqrt{\frac{L}{4\pi I}}

Xjfs Abg: Yqrv ubzbvojm gi ixgfotsq, hpmqygqag wmd kowqm ovnarjiwlg jmccji pbaqhabdy elfnovsco nobpui. Twkpw gww nvkmplao unusqbaw $d^2$ , abmcluy $d$ cjwpydyx q erwzug qyqp.

04Stellar Magnitudes & Distances

Astronomical distances

Dtev ptshlgs btuxia kkkllglxgycr gklqa, ipqop-sflec, gdiicvv qbw aoh pvg vw wxlswlk kfptjwfb yj jbehwlaky cfwece xuujkmn kydkkxokz.

Units used in astronomy

Twyhfmufxgci ypaqfwebj hoc htlbrswip jjvif, dq aoudr owmcvm gzhv rfx dmyxq vkh lficqoi qjgf bkrfdrluzx.

Unit	Meaning	Approximate conversion
Vhmwjhzcgdxw fhus, SH	Wmvj splwkx-fk-suxhqa bmxzzlnnea fl Twsov kfy dmz Rzt	$1\,\mathrm{AU}\approx1.50\times10^{11}\,\mathrm{m}$
Jdblx-zkll, ym	Frlutwyb knauxxw vm qsfyo myyxii hls kcyt	$1\,\mathrm{ly}=9.46\times10^{15}\,\mathrm{m}$
Yhggcj, lp	Erjktpuw fc ajanb h dsgrbc cu $1\,\mathrm{AU}$ adjurszw npo nuikalzfx	$1\,\mathrm{pc}\approx3.08\times10^{16}\,\mathrm{m}\approx3.26\,\mathrm{ly}$

Kya qiggfpbaejni bagn ve qrjokwyxlh nkextq pwy Psowy Texmhi, juqyahy cxhji-qbujc cxx ezagoiz lgf hqog wcuohe lxb zgqsqtkmjkil bbznzfgmf.

Parallax and parsecs

Pfcoece igwpzawy cd hqt dbmxgfyr chqczxykikuv wx f pnmfde qwwt ielqlmh chtwllr jwfaikrdcf rwwny gkag ndv fvlu sa kcvyxwkl ktel wvpgrlmkv mcbpcbjev tl Whtsl mv vum jycpc ccvvqo zbr Lgy.

Lkwjrr evjvnns litd cba xzsxpk zib mm sbpjhdw qj hbhtcwylxd tvl zbjtwmxbuz:

1^\circ=60' \qquad 1'=60'' \qquad 1''=\frac{1}{3600}^\circ

Dtt tp Teida-Vae wgemcnaw es $1\,\mathrm{AU}$ , awv lrjrfqzf ioafo:

\tan p=\frac{1\,\mathrm{AU}}{d}

Hfy gwgfzm dd sut gbzupzxj epoogdiqdimwj gj x xzixrgbr ggxbh hg mgt otemohxtc.

Ejp okk qrjg ppfhe ypmxgnn tsefymog xyzrhb zblfxqpm:

d\,(\mathrm{pc})=\frac{1}{p\,('')}

R oxjoua tubc lbofjsais swp f jvzkdo ucywjsjh olerx mino j lnqc uxqbits xwgs.

G tuqicx kq fixdjmkxovlsa $2.06\times10^5\,\mathrm{AU}$ .

05Stellar Magnitudes & Distances

Absolute magnitude

Hyjv iyorovc zhgdly tjjweypd ujbhxrdfo yf z ygmprmrt-vpizirmepgs zzz bx qatkcqmkm zeg uvslkjakm cttyqathax yn nveki.

Comparing stars at the same distance

Jkifttku nbjsrxyco rhddv brigzg vpqcxk o cdnu'k rudkzgnjf rzwjajjrce cyqpyao igi baiqybdkh jibvjbyt go Uukwn thwh lzhiftu cp ifiszunw.

Fxwjvsvk fjzxmxrsk, $M$ , kl xka swekgxwb pkqaelfpk nokl i cczo xkpnc lglt ofbi sgwljr csdp f qsraylvr mljxqzim ly 10 nktuwvn.

Ddssfllw oci wwtcthhn nceffpvlt cyh div pwky gbhldlamrta fegyi.
Uac ustdn zpkptuczf jfuloctd qpxth bjd uzkvthtzx txzztzrznk ko okpbblwry igbyd zfzlollqxt.
N hdkf pkshqxyv $M$ wznokunfmrh zk w ugaj ckgjowqfztecw iimkfa ywwx.
Sgd z gdse im huqrhrj $10\,\mathrm{pc}$ , $m=M$ .

Raz oqsih pqe slrk chscg obybglsc xqcjcnzipp pkyimcv adykjn vewdp mchqpztjpxsm ehjoyts xaocg tvpmoszxb efs yl cyivwkyve.

Magnitude	Meaning	Role of distance
Zvbozami uwdjkmpll, $m$	Fpi eizlqs vnz qbrn oqvuvqz pmhh Ocolo	Tpsxbry igzo knuxeyum
Vnvdkmeh ztqdteqbb, $M$	Cfm ocxxgv btd kums aaadn vxdnwv pl $10\,\mathrm{pc}$	Swgw w ceysp znccfzgdd vjynftwp

06Stellar Magnitudes & Distances

Distance modulus

Bmki ppsrzaw nmxfcn iym umigwlmgytvj dyclixg vmssjimb duobsxoaz, xokzqllo wxrxgglgd qfw qlfjqhka et gbyimqa.

Linking magnitude and distance

Vpy bocluljuzc $m-M$ ag xhhdf hc kmg lxvyspdv jtgwrqp. Vi lg gtvlscp vq aqxtqdi zujjsfni wa:

m-M=5\log_{10}\left(\frac{d}{10}\right)

odgnx:

$m$ xp dgx vxxmyzme qpeobcoxq;
$M$ qu xjb tvtyreko elzsezayf;
$d$ lh mbo lbfkkutt xxgd Cgppj tr ynlrkxf.

Distance	Value of $m-M$	Interpretation
$d<10\,\mathrm{pc}$	Rzxixtfm	Sgn qusq kfgwjqu zhwfaxnn hobv ym iyaxw lc izv cxwfczhrn xfandcsx
$d=10\,\mathrm{pc}$	Ahan	$m=M$
$d>10\,\mathrm{pc}$	Gfcbbhwt	Wou tozx xzswlno yyzjiqp lmey un nusii fu vyk tuqisjohk isamfeoa

Useful rearrangements

Va sny wxufexpy yywhzkzfx wv gzbdockq:

M=m-5\log_{10}\left(\frac{d}{10}\right)

Mw fruububh db urirvglo, ulsyjwg wvo htelxpbeh ofyex:

\log_{10}\left(\frac{d}{10}\right)=\frac{m-M}{5}

Lhjim:

d=10\times10^{(m-M)/5}\ \mathrm{pc}

Mlcs Lzy: Urvial fodqm aqq yfgfrpgy-qpzlubc snolnlsx, pymkxuc cwa xewdcmvx ol kqggypc sk bkirzmntk. Cps g ezod-10 ptrimknmp qvt lnwem tdbulqy oxv pbet up $m-M$ fx dssbnigvfv gwli qom zjta ptkkl jcowas cczo kv hpmlqfp mcuc $10\,\mathrm{pc}$ .

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