What is the best way to determine one star is more luminous than other stars?

The apparent and absolute magnitude, spectral type, distance (LY) are given.


Is there a correlation between luminosity and temperature of the star, i.e. do hotter stars have higher luminosities?|||When a star is in the main sequence stage (the first stage, which the sun is currently in), brightness generally correlates with temperature. In later stages (red giant, white dwarf, neutron star, black hole, etc), brightness depends on age and mass, not temperature. A small white dwarf is hotter than a red giant, but a red giant is brighter because of its size (and mass). During their main sequence stage, the white dwarf might have been the more massive and the brighter. Absolute magnitute determines a star's absolute brightness. It's the visual magnitude at an absolute distance of 10 parsecs. Calculating absolute and visual magnitude is a bit complicated. For example, absolute magnitude = visual magnitue - ((log (distance in parsecs ^2 / 10 parsecs ^ 2)) / log (10 ^ (2/5))).|||Stellar brightness (Luminosity)


Brightness is related to the energy we detect


With your eye or another detector


The energy


Measured in


Joules/sec


1J/s = 1 Watt


We have discussed the apparent visual magnitude


How stars look relative to one another from earth


We called this apparent visual magnitude (mv)


Human eye goes to about 6th mag


But this has nothing to do with how bright a star actually is


Distance


Dust


Size of star


Color


All affect brightness


To begin our in-depth study of stars we need to be able to compare stars directly


We need absolute magnitude (not just apparent mag)


Absolute visual magnitude (Mv)


If all stars were the same distance away, say 10 parsecs


We could compare them objectively


Mv is related to mv and distance from earth


mv 鈥揗v = -5 +5log10(d)


mv apparent visual magnitude


Mv is the absolute visual mag if the star were 10 parsecs away


d is the distance to the star in parsecs


Example


Betelgeuse in Orion is 427 ly away with mv=0.45


427 ly = 130 parsecs


0.45- Mv = -5 + 5log10(130)


0.45- Mv = -5 + 5(2.11)


0.45 鈥?Mv=5.57


Mv= -5.12


Absolute magnitude of the Sun


4.78


if it were 10 parsecs away


This is still the visual magnitude


Remember stars emit radiation outside the visible range


Determining stellar Luminosity


Now that we have the absolute visual magnitude


We can compare other stars to the Sun


The brightest stars


Mv = -8


100,000 time more light than the Sun


Dimmest


Mv =15


10,000 time less than the Sun


Let鈥檚 take this a step further


Luminosity 鈥?the amount of energy emitted by a star


Depends on two things


Temperature


Size


Energy radiated by a black body


Stefan-Boltzmann Law


E = sT4


E is energy emitted in 1 sec per m2 (J/sec/m2)





T is temperature





s is Stefan-Boltzmann constant





5.67 x 10-8 J/m2 sec degree4





Energy goes up dramatically as T goes up


3000潞K





4,592,700 J/s/m2





6000潞K





73,483,200 J/s/m2





E = sT4


L = surface area x sT4


Area of a sphere


4pR2


and


L = 4pR2 sT4


We can simplify


Now we can determine the Luminosity of a star relative to the sun which we know


Radius


We can also use this formula to find radius


Pick a star


Hamal in Aries, mv = 2.0


Mv = 0.472


(Sun) 4.78 - (Hamal)0.472 = 4.3 magnitudes brighter than the sun





2.5124.3 = 52.5 times the luminosity of the sun


Hamal is a K2 star 4000掳K


4000/5800 = 0.689655172 times the suns temp





52.5 = 0.23


Ro


Radius of Hamal is 15.1 times the Sun鈥檚 radius


So now we know luminosity and Radius and Temp|||--|||Look at it