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Diameters of eclipsing binary stars

We now turn back to the main thread of our story to discuss how all this can be used to measure the sizes of stars. The technique involves making a light curve of an eclipsing binary, a graph that plots how the brightness changes with time. Let us consider a hypothetical binary system in which the stars are very different in size, like those illustrated in [link] . To make life easy, we will assume that the orbit is viewed exactly edge-on.

Even though we cannot see the two stars separately in such a system, the light curve can tell us what is happening. When the smaller star just starts to pass behind the larger star (a point we call first contact ), the brightness begins to drop. The eclipse becomes total (the smaller star is completely hidden) at the point called second contact . At the end of the total eclipse ( third contact ) , the smaller star begins to emerge. When the smaller star has reached last contact , the eclipse is completely over.

To see how this allows us to measure diameters, look carefully at [link] . During the time interval between the first and second contacts, the smaller star has moved a distance equal to its own diameter. During the time interval from the first to third contacts, the smaller star has moved a distance equal to the diameter of the larger star. If the spectral lines of both stars are visible in the spectrum of the binary, then the speed of the smaller star with respect to the larger one can be measured from the Doppler shift. But knowing the speed with which the smaller star is moving and how long it took to cover some distance can tell the span of that distance—in this case, the diameters of the stars. The speed multiplied by the time interval from the first to second contact gives the diameter of the smaller star. We multiply the speed by the time between the first and third contacts to get the diameter of the larger star.

Light curve of an edge-on eclipsing binary.

Light Curve of an Eclipsing Binary. In this plot the vertical axis is labeled “Brightness” in arbitrary units, and the horizontal axis is labeled “Time” in arbitrary units. The plotted line is labeled “Light curve”. The plot begins as a horizontal line at center left. The line then drops downward as it moves to the right, then quickly becomes horizontal again. As time goes on the curve rises again back to its original brightness. Above the curve is a diagram of the binary system. The larger star is drawn as a red sphere. A blue arrow is drawn horizontally through the center of the larger star pointing to the right, indicating the motion of the companion star. The companion star is drawn as a blue dot at four positions on the blue arrow, labeled “1” through “4”. At position 1 on the left, the companion star just touches the left edge of the larger star. A dashed line is drawn downward to the light curve at the point where the brightness just begins to drop. At position 2 the companion star has just been fully eclipsed by the larger star. A dashed line is drawn downward to the light curve at the point where lowest brightness begins. At position 3 the companion star is just about to emerge from eclipse. A dashed line is drawn downward to the light curve at the point where the brightness begins to rise. Finally, at position 4, the companion star has emerged from eclipse and just touches the right edge of the larger star. A dashed line is drawn downward to the light curve at the point where the light curve returns to maximum brightness.
Here we see the light curve of a hypothetical eclipsing binary star whose orbit we view exactly edge-on, in which the two stars fully eclipse each other. From the time intervals between contacts, it is possible to estimate the diameters of the two stars.

In actuality, the situation with eclipsing binaries is often a bit more complicated: orbits are generally not seen exactly edge-on, and the light from each star may be only partially blocked by the other. Furthermore, binary star orbits, just like the orbits of the planets, are ellipses, not circles. However, all these effects can be sorted out from very careful measurements of the light curve.

Using the radiation law to get the diameter

Another method for measuring star diameters makes use of the Stefan-Boltzmann law for the relationship between energy radiated and temperature (see Radiation and Spectra ). In this method, the energy flux (energy emitted per second per square meter by a blackbody, like the Sun) is given by

F = σ T 4

where σ is a constant and T is the temperature. The surface area of a sphere (like a star) is given by

A = 4 π R 2

The luminosity ( L ) of a star is then given by its surface area in square meters times the energy flux:

L = ( A × F )

Previously, we determined the masses of the two stars in the Sirius binary system. Sirius gives off 8200 times more energy than its fainter companion star, although both stars have nearly identical temperatures. The extremely large difference in luminosity is due to the difference in radius, since the temperatures and hence the energy fluxes for the two stars are nearly the same. To determine the relative sizes of the two stars, we take the ratio of the corresponding luminosities:

L Sirius L companion = ( A Sirius × F Sirius ) ( A companion × F companion ) = A Sirius A companion = 4 π R 2 Sirius 4 π R 2 companion = R 2 Sirius R 2 companion L Sirius L companion = 8200 = R 2 Sirius R 2 companion

Therefore, the relative sizes of the two stars can be found by taking the square root of the relative luminosity. Since 8200 = 91 , the radius of Sirius is 91 times larger than the radium of its faint companion.

The method for determining the radius shown here requires both stars be visible, which is not always the case.

Stellar diameters

The results of many stellar size measurements over the years have shown that most nearby stars are roughly the size of the Sun, with typical diameters of a million kilometers or so. Faint stars, as we might have expected, are generally smaller than more luminous stars. However, there are some dramatic exceptions to this simple generalization.

A few of the very luminous stars, those that are also red (indicating relatively low surface temperatures), turn out to be truly enormous. These stars are called, appropriately enough, giant stars or supergiant stars . An example is Betelgeuse , the second brightest star in the constellation of Orion and one of the dozen brightest stars in our sky. Its diameter, remarkably, is greater than 10 AU (1.5 billion kilometers!), large enough to fill the entire inner solar system almost as far out as Jupiter. In Stars from Adolescence to Old Age , we will look in detail at the evolutionary process that leads to the formation of such giant and supergiant stars.

Key concepts and summary

The diameters of stars can be determined by measuring the time it takes an object (the Moon, a planet, or a companion star) to pass in front of it and block its light. Diameters of members of eclipsing binary systems (where the stars pass in front of each other) can be determined through analysis of their orbital motions.

Questions & Answers

hi i need a telescope would you help mee
Nauman Reply
i can only advice you to go to some museum
Gospel
wher is it?
Nauman
but before that.......... i wanna know where are you from and from which state...and if you are nearer to that of the museum or will be able to go then.,.. i donot have any more words.
Gospel
What Do Plan To Observe?
Adam
sorry . i am from asia
Nauman
Check on amazon.Celeston telescopes are good for beginners.
priyanshu
can yoy introduce your self
Nauman
can you
Nauman
Asia...? from which country you belong into ...,,is it India or where?
Gospel
Do You Want A Telescope With ALot Of Maintenance Or Zero Maintenance?
Adam
pakistan
Nauman
with alot offf
Nauman
Any second hand online stores in your area? great for unused Christmas gifts etc..
Collins
So You Want A Newtonian Reflector?
Adam
So what i would put for that answer ?
Abdulla Reply
The largest observatory complex in the world is on Mauna Kea, the tallest mountain on Earth. Whatare some factors astronomers consider when selecting an observatory site? Don’t forget practical ones.Should astronomers, for example, consider building an observatory on Denali (Mount McKinley) or MountEverest?
Abdulla Reply
i need help on the the second part
Abdulla
Should astronomers for example considered building an observatory on Denali (Mount Mckinley) or MountEverest ?
Abdulla
I would take into consideration. climate weather pattern.
basim
also, could an establishment be built and be able to sustain years of being beaten by the weather?
basim
so far Mount Everest to my understanding is a difficult mountain
basim
how far us milky way galaxy?
ShowsSpy Reply
oh man
madison
We are part of the Milky Way Galaxy. We are within it, so it's not really away from us. Our galaxy is 100,000 light years in diameter. The nearest big galaxy is Andromeda, around 2 million lightyears away.
Jignesh
its 200,000 light years
Gospel
Where is Rice University located?
Amanda Reply
it is in your vagina
world
that couldn't be any more wrong 😂😂
jay
what even is this conversation😂😂!?
Herrera
Show Some Class.....
Adam
Houston, Texas.....
Adam
Houston
Preston
what is the first step to take as far as getting into the field of astronomy?
john Reply
how are universe made up of?
madhu Reply
saksi
madhu
lanika
madhu
I think that your question is- What is the universe made up of?
priyanshu
Ever heard of matter?
priyanshu
Well,the universe is made up of matter and possibly antimatter
priyanshu
Actually,it's made up of matter's smaller units,fundamental particles
priyanshu
But for normal purposes,say that it's made of matter
priyanshu
If you're asking how was the universe made,
priyanshu
there are several theories,but the widely accepted one is the 'Big Bang Theory'.
priyanshu
In simple words,a veeeeeeery long time ago (20 billion years).
priyanshu
The universe was just inside a very small point.
priyanshu
Then,it started to expand.
priyanshu
It is still expanding today
priyanshu
But remember,it's just a theory,there is no hard and fast proof
priyanshu
oh d*mn😭😭😭😭 i cant do my astronomy quest
swagirl
up Down quarks atoms electrons proton neutrons
Blake
😭😭😭😭..
swagirl
usually in astronomy quest there's many math formula. i'm difficult on them
swagirl Reply
practice!
Collins
?
swagirl
what was the question? if you need any help CrashCourse/astronomy and physics channel on YouTube are a good help.
Collins
yeah like... you know venn diagram?
swagirl
yeah it's for sets
Vinit
***youtu.be/uR70knMr2Hg
Collins
this might help you understand Venn diagram. and yes they're sets of numbers.
Collins
tx for u
swagirl
why the dark side of moon never face us? because moon dont rotate? why tho
Rishabh Reply
im in 8th standard and my school teach us nothing about astronomy but i want to be an astronomer so i study from youtube and apps like this but sometimes i get language problems
Rishabh
dark**
Glory
side**
Glory
The Moon does actually rotate. It takes 27 days for the Moon to make a full rotation. Because it takes it so long and the Earth is rotating around the sun, it seems like the Moon is pretty much standing still. That's called "synchronous rotation".
steveh259
I ment to say "The Earth is orbiting around the sun"
steveh259
Tidal Locked.....
Adam
***svs.gsfc.nasa.gov/4442
Collins
thats cool
Gospel
ya gives you a look at how the moon orbits around us too and a perspective of how fast we are going. also gives the moon phases for the year! enjoy 😁
Collins
yes you are absolutely correct, it just like something like the milk of heaven
Subhasish
the far side of the moon is not the dark side of the moon.
Shawn
Tidal Locking?
priyanshu
what phases of the moon occurs when spring tide?
Florence Reply
the moon has a 28 day cycle. you can download a moon calander or follow the pattern and make your own calendar. 1st full moon of the year 21/22nd Jan, its also a blood moon and a wolf moon. if this info helps at all. 😊
Collins
***svs.gsfc.nasa.gov/4442
Collins
How to determine the velocity of light?
PARTHIB Reply
using this formula c=1/√absolute permeability×absolute permittivity of free space i.e. c=1/√8.854×10^-12×4π×10^-7~3×10^8ms^-1
Vidyashankar
what is mean by Big Bang
block Reply
What it means by Big Bang is how the world was made and came in to exist
Glory
The Big Bang Is The Best Explanation For What We See Around Us, And For The Origin Of The Universe, All Our Evidence Points To The Big Bang Theory..... And Was Proposed By A Belgian Priest.....
Adam
if the big bang theory is really true than i would like to know what exploded and where did it came from and where did the energy came from and where come from for the matter to expand into amd where did the organization come from and where did the information come from.........
Gospel
Some say that on the otherside of the singularity was a black hole. They say that at the other end of all black holes is another universe. Hence the multiverse. It's all theoretical at this point but less than a 100 years ago people believed that there was only 1 galaxy in the universe. So why
Rafael
should there be only 1 universe?
Rafael
universe.
Rafael
Big Bang is a theory on the origin of the universe.It says that the whole universe started as an infinitely dense and hot point,which expanded with time to form the universe in which we live today.
priyanshu
but if you study about string theory then you will come to know that before big bang there were string ( vibrating particles) which cause to form the conditions for big bang.
YOG
Hmm,actually,I dont know much about string theory,just a little.I just wanted to tell this guy what is the Big Bang Theory.
priyanshu
and according to string theory black holes are giant fuzzball of strings,there is no infinity dense singularity and no smooth horizon there is just only a ball of string
YOG
is astronomy a hard subject to learn because i want to be an astronomer
FNAF Reply
No it not it just takes time and effort to learn
Glory
if your already interested/observing, your already an astronomer ! The more you look and read the more you'll learn.
Collins
I want to become a scientist but my knowledge is very low. How I can improve my knowledge?
Wwe
any form of science courses, there's free crash course channel on you tube. I've learned alot from this channel. covers most topics, science: biology chemistry physics, astronomy, ranges to anatomy, history and many many more. it's a never ending subject!
Collins
suggest a channel as a example
Wwe
CrashCourse is the name of the channel and there is a picture of an apple. look through their play lists. They will have other related channels they follow also.
Collins
thank you so much
Wwe
your very welcome
Collins
Why are they so far away that their light takes hundreds of millions of years to reach us on earth?
LadyCatTM Reply
Because the universe is wayyyyyy bigger than the earth
priyanshu
You can't expect everything to form close to Earth.
priyanshu
what is comets , astroids ,
Sumit Reply
comets are broken pieces of dust frozen debris Asteroids are big soiled pieces of Ice that burst into flames when it in to earth's atmosphere
Glory
Glory,you aren't correct
priyanshu
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Source:  OpenStax, Astronomy. OpenStax CNX. Apr 12, 2017 Download for free at http://cnx.org/content/col11992/1.13
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