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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

how can I apply online astronomy course ? if you have link send it.
Dawit Reply
Does any body know about Quantum Battery ?
Yabtsega Reply
How to get into the field of astronomy after completing B.tech in department of ECE?
Lakshmi Reply
Well my option is to do your job while getting a distant education from a reptued university for B.Sc and try for M.Sc in IIA and remaining astrophysics universities in india
Mahesh
Do I require GATE score for that?
Lakshmi
Yeah I meañ get a better score in gate.I think the university considers it and would help you to direct enroll into a masters progarmme
Mahesh
But the score should be good.
Mahesh
okay. I will do my best.
Lakshmi
Thank you
Lakshmi
hi guys I am from Ethiopia.I don't know detailedly about astronomy and I want to be astronaut .can you help me guys?
Dawit
I know that astronauts have to be physically fit
Tanmay
I don't know anything else though
Tanmay
If you want to be a astronaut you should complete medicine.
Mahesh
I had done by my b.tech ece long back and then I started doing job...but I still want to switch my field to astronomy, I m confused how shall I pursue it..
Shina
what should we pursue after 12th std B.Sc program and further or through engineering?
Ganesh
Maybe you should ask this question in quora.Professors will tell you answer to the question.
Mahesh
ok
Ganesh
Wanna join conversation about time travel?
Pratiksha Reply
Of course
Yabtsega
From where do we start?
Yabtsega
what do you think which year we would be able to time travel?
Tanmay
I think we can travel through a wormhole
Pratiksha
wormholes make you cover a distance faster than light can I don't think they can do anything with time though
Tanmay
hmm
Pratiksha
any guesses about which year that may be true?
Tanmay
time travel
Tanmay
but Morgan Freeman has told that time travel can be exist also through wormhole
Pratiksha
I think the past life
Pratiksha
I am dumb you are correct we can time travel with wormholes
Tanmay
yes you are correct
Tanmay
u see the episodes of Morgan Freeman on discovery channel, he explains nice and there also other astronomy people's which explain all about space time and all about universe
Pratiksha
no I don't watch discovery I will see it now thanks
Tanmay
yr welcome
Pratiksha
yr not dumb, every student are intelligent
Pratiksha
anything else interesting about space?
Pratiksha
I heard a really nice thought it said " We need time travel. When? It doesn't matter" What do you think of it?
Tanmay
sorry!!, but I didn't understand
Pratiksha
I don't know who made it but it's nice
Tanmay
I mean it takes 3 days and 14 hours for one revolution around its star
Hardik
it says that we need time travel but when do we need it? it actually doesn't matter because if time travel was made then we can go back in time and make it earlier
Tanmay
did you understand that thought? if yes, so can you explain me!
Pratiksha
so if say A makes time machine in 2080 then he can travel in 2010 and make that time machine again
Tanmay
I meant I don't know who made this
Tanmay
ohh
Pratiksha
still didn't understand 😅
Pratiksha
thought
Tanmay
I think my net is slow it is break ing up my
Tanmay
sentences
Tanmay
yeah!!
Pratiksha
it's fine
Pratiksha
I switched my internet it should be good now I meant that I don't know who made this thought does anyone know who made it?
Tanmay
@Pratiksha you understood it or should I simplify it more?
Tanmay
nahh!! I don't know, I heard it first time,
Pratiksha
just simply it
Pratiksha
@Hardik what do you mean? what is "it" here?
Tanmay
it means the π planet
Hardik
my last message didn't reach u ppl 😅
Hardik
simply if A developed time machine in the year 2080 then another person B can always use that time machine to go back in time and develope the same time machine in 2010 before the actual developer(A) developed that time machine
Tanmay
Hey Astros!
Info
@Hardik I your last message that says "I mean it takes 3 days and 14 hours for one revolution around its star" But this is also your first message
Tanmay
@Pratiksha it just says that the year of development of time machine can be changed easily and so will the developer
Tanmay
@Info hi anything to say here? any questions or answers or something
Tanmay
thanks tanmay patra
Pratiksha
welcome
Tanmay
is anyone interested to time travel!?!!!
Pratiksha
oh yes count me in
Tanmay
wow!! me too
Pratiksha
the study of astronomy is really very interesting
Pratiksha
a question for all of you. If you are told that you can either see the future of the earth or you can see the past of the earth then which one will you choose?
Tanmay
you can only see one thing no voting for both and also give reasons
Tanmay
I wanna past of the earth, because I want to make my future
Pratiksha
If you wanna past of earth for future then spend your present for that possibility
Mahesh
is there a competition formed by NASA in this month
Scientific Reply
nooo
Kygo
don't know
Pratiksha
It's because of presence of iron oxide in its surface.
Info Reply
That point is known as Singularity
Tushar Reply
Indian culture talks about black holes very well
Tushar
The word "MAHAKAAL" means black hole
Tushar
I love Quantum mechanics
Tushar
I love particle physics
Tushar
wow do you know about some maths in quantum physics?
Tanmay
Exactly.
Anti-Capitalism
Hi! Anybody can tell me that what is the exactly the black hole?
sach Reply
l no
Jee
OK then tell me
sach
A black hole is a space where gravity pulls so much that even light cannot get out.
Pratiksha
This can happen when a star is dying
Pratiksha
ok👍
Pratiksha
Explain more
sach
ok wait
Pratiksha
when a star of mass greater than 20 times the mass of sun dies it forms a stellar mass black hole
Tanmay
it bends space time so much that you need to travel faster than speed of light to get out of it
Tanmay
the innermost stable orbit of a black hole is that orbit where only light speed particles can orbit it
Tanmay
accretion disk of black holes are the matter that it gained from the surrounding orbiting the black holes
Tanmay
black hole forms when the center of very massive star collapsed it upon it self
Pratiksha
relativistic jets come out of black holes when the angle between accretion disk is not 90 degrees(this might be wrong but I know that it has to do something with magnetic field)
Tanmay
@pratiksha you took a lot of time to type this I thought you are Going to type an essay
Tanmay
@Gurjinder you can ask it here
Tanmay
sorry, but the text was not going 😅
Pratiksha
OK thanks
sach
yr most welcome sis
Pratiksha
I am a student class 12
Tanmay
there are 3 types of black holes stellar mass intermediate mass and super massive black holes
Tanmay
yeah!
Pratiksha
super massive black holes are found at the center of most galaxies
Tanmay
My hometown is at Odisha but I study and live in up
Tanmay
I still don't know a lot of stuff especially the maths involved in astrophysics
Tanmay
becoming an astrophysics thanks for your words
Tanmay
oh cool what all subjects are you studying
Tanmay
oh it happens sometimes
Tanmay
sadly chemistry XD
Tanmay
are there any chapters only for astrophysics
Tanmay
thank you same to you
Tanmay
oh that's a lot of hard work I hope you get great achievements can you tell me what all you are currently studying and which channel
Tanmay
oh that's a good strategy thank you
Tanmay
super
Jee
positional astronomy?
Tanmay
can you tell what positional astronomy is
Tanmay
I am fairly good in trigonometry but really bad at graphs
Tanmay
Hello everyone.Looks a cool discussion
Mahesh
ok I will practice thanks
Tanmay
where are you studying @Gurjinder
Mahesh
ok I will learn more
Jee
At which college I meañ.
Mahesh
Good
Mahesh
I want to be a astronomer,astrophysicist,theortical physicist and a engineer
Mahesh
I am not that good at self study
Mahesh
@Gurjinder that's a lot of hardwork I have high respect for you
Tanmay
I am always curious
Mahesh
nothing about astrophysics?
Tanmay
Lol
Mahesh
is college better than school?
Pratiksha
l am in school
Jee
Mainly from feymann if you want a good grip of basics
Mahesh
me too
Pratiksha
I just completed so what.Learning has no age limit.
Mahesh
which standard
Jee
at least your physics teacher should know something about astrophysics
Tanmay
l no
Jee
I love the book on quantum mechanics
Mahesh
though I hate classical mechanics
Mahesh
yeah the education system in India just cares about Mark's I don't know what will happen to the new system
Tanmay
yeah! yr right
Pratiksha
I love the new system.
Mahesh
They are going to introduce
Mahesh
but I am happy to know that people around the world are destroying it
Tanmay
the old system
Tanmay
Hmmm
Mahesh
what is black hole is something and what can do a black hole is another thing
Kygo
A black hole is a singularity where matter is condensed and a gravity well that light cannot escape from.
Micheal
what is blue marble
Jee Reply
it is the another name of earth.
Sadman
hey friends how are you?
Sadia
why Mars is known as red planet?
Sadia
blue marble not blue planet,so it z not earth but the image of earth taken in 19's
Ethical
b/c of mars is known as red planet it have iron oxide or rust iron.
Dawit
you know when I was small, I had seen a comet ☄️ in the sky but I have a doubt that was that really a comet or meteors.. what is the difference between these two?
Jugal Reply
Do any one observe the moon every night?
Pratiksha Reply
I do look at it but with naked eyes
Tanmay
what do you see?
Shivam
sometime not every night as clouds are often there in the sky
Jugal
I look stars,Venus and moon by naked eye as I have no telescope yet
Jugal
how much does a telescope cost
Shivam
depending upon the resolution.....good resolution cost more
Jugal
I can some big dark spots from my naked eyes
Tanmay
A good telescope would cost 400$
Tushar
I have 16 inch dobsonian(truss tube).I'm observing Orion nebula and studying it.
Tushar
wow I am jealous.
Mahesh
why
Tushar
I have made this telescope by my own efforts
Tushar
16 inch is big enough to see Galaxies and star clusters
Tushar
wow you made it on your own
Mahesh
Yes
Tushar
But for that you need Some engineering skills and some proper guidance
Tushar
My Uncles helped me
Tushar
I'm from middle class family This project was funded by Uncles
Tushar
I am from a middle class family too
Mahesh
Uncommon For Me
Anti-Capitalism
What do you think will happen if we find humans rather than aliens on other planets?
Tanmay Reply
Hmmm
Mahesh
let's assume that those humans are just like us physically and mentally. Then what do you think will happen?
Tanmay
We and those humans form a conference like meeting and talk to improve the connection from both sides
Mahesh
But I think that the connection won't be trustworthy
Tanmay
In the same they think about us...
Mahesh
yeah you are correct they might send imposters to collect more data about us
Tanmay
The same way trump and Kim are acting in front of media as they sorted it out...
Mahesh
this would be a really tough situation
Tanmay
You think so...
Mahesh
If you go to Europe there are high chances of underwater creatures what about that..
Mahesh
underwater creatures what? I didn't understand what you are saying
Tanmay
You know europa
Mahesh
yes
Tanmay
The water on europa is more than the water in our earth that is able to sustain lige
Mahesh
So scientists predicted that there maybe underwater life in europa
Mahesh
oh yes I heard it
Tanmay
So in the time of crisis we can go to europa and bulid underwater colonies.
Mahesh
ok but a lot of people would have to be left behind
Tanmay
it is smaller than our moon
Tanmay
But we can bulid underwater floating cities . scientists said that not me..
Mahesh
Yeah and I have been wanting to ask you. Are you a girl or boy..
Mahesh
I am a boy
Tanmay
ok
Mahesh
yeah I don't know who they will leave behind and who will go to that
Tanmay
what do you think who all will be left
Tanmay
I didn't quite catch the question.Say it again..
Mahesh
what do you think, who will be left behind in earth in the time of disaster and who all would go to europa
Tanmay
I don't think all humans can be able to travel.So some humans and animals
Mahesh
yeah the astronauts would have select people so what do you think that on what basis they will select us
Tanmay
I think all the productive people would be allowed in the ship
Tanmay
Hmm I don't know.Maybe the people who are on the verge of dying and who are incapable for space travel are left behind I think.
Mahesh
yeah all the people capable of continuing our generation would be allowed. I hope that time shall not come
Tanmay
It will come.Time is inevitable
Mahesh
Do you know that Newton predicted that the world will end in 2060
Mahesh
some were saying that world would end in 2016
Tanmay
but I never heard Newton about 2060 one
Tanmay
But this Newton we are talking about not a astrologer or civilization calaender
Mahesh
I mean I never heard Newton saying that
Tanmay
He said it is one of his works I think..
Mahesh
but according to sanskrit calendar the earth will be doomed is a million years or so
Tanmay
as told by my teacher that the "kal yug" will end in a million years
Tanmay
kaliyug..
Mahesh
Not kal you
Mahesh
oh yes that
Tanmay
Hmm let's see.Did you watch meowise
Mahesh
Neowise
Mahesh
no I don't know what that is
Tanmay
comet neowise
Mahesh
no I never saw or heard about it
Tanmay
Really.It appeared in last month for a period of days
Mahesh
did it come at the end of August?
Tanmay
or somewhere in middle?
Tanmay
Yeah
Mahesh
Middle it is..
Mahesh
oh then I would have been able to see that but I didn't
Tanmay
I just searched it and it's orbital period is ginormous 6766 years
Tanmay
Yeah
Mahesh
That's why so many star grazers have clicked many photos of it all around the world
Mahesh
yeah I saw the photos
Tanmay
have you heard about doomsday clock?
Tanmay
kind of..
Mahesh
Btw what is the logic of a doomsday clock
Debabrata
some people have made a community which based on political stuff change the time of this clock
Tanmay
for me this is worthless to be honest
Tanmay
Are we living in a simulation?
Vedant Reply
it could be true.
Shivam
at this time there is no way for us to find out that we are in a simulation or not
Tanmay
We can say our universe is a simulation if we were able to do time travel and find out that no matter what you do future will remain same
Tanmay
It could be true because in every 100 years a pandemic comes and tests us,how are we doing in management against pandemic.There are some people that are extraordinary and they are discovering things that are essential for us to know,.There are alot of ufo sightings on earth every year.
Shivam
and there was a rumour that NASA had 2 two alien bodies in area51.Why NASA is not sending more missions on moon and not going there.They are just focusing on Mars but not on moon,when every one is focusing on lunar research.
Shivam
what is the basic idea I need to genarate astronomy?
SCIBASED Reply
I didn't understand the question
Mahesh
as per my opinion, the basic ideas that you need to generate the interest in Astronomy can only be said by looking at your interested topics in Astronomy..There are different topics like steller astronomy,planetary astronomy,etc
Jugal
Astronomy is only a branch for which you don't have to go somewhere to get something.. Just you have to turn your head towards the night sky in a pollution less area and then you can start your observations 👍👍
Jugal
how many types of astronomy branches are the?
Shivam
if I take astronomy as my career.Would I study all of them or I have to study them separately.
Shivam
what is cosmology?
Shivam
you have to choose a particular topic in astronomy (not all) and have to do research on it...👍👍
Jugal
ok
Shivam
cosmology is also a branch of physics in which one have to do a deep study about the past, present and future condition of the Universe. It includes the questions like how our Universe formed,what were the condition at that time,how life formed. A cosmologist can also predicted the future condition
Jugal
ok
Shivam
but which branch of astronomy is the best?
Shivam
tell me meaning of all branches? please
Shivam
there are many branches like planetary astronomy, astrometry,solar Astronomy, stellar Astronomy, galactic Astronomy,radio astronomy,etc
Jugal
you have to do research on a particular topic and also have to write a thesis or book on your topic in PhD to complete your research degrees...after that you would become a successful Astronomer 🔭🔭
Jugal
which do you think is the best?
Shivam
What does type I, II, III civilization mean? How do we calculate it?
Tanmay Reply
Type 1,2,3 civilization is the three categories of kardashvev scale which is a method of mesuring the tecnological advancement based on the amoumt of energy they can use
Sadman
Type 1 civilization can use the full energy which was came from there host star and their host planet
Sadman
type 2 civilization can use the hole energy of their host star
Sadman
type 3 civilization can use the energy of their entire host galaxy
Sadman
Ok but how can we calculate this is there any formula or some kind of energy output value or something?
Tanmay
A Type I civilization would have complete and total mastery over all the energy of its planet. A Type II civilization would be masters of all the energy produced by their home star, and a Type III civilization would own all the energy in their home galaxy.
Astronomy
The Kardashev scale is a method of measuring a civilization.
Astronomy
our civilization is 0.72
Mahesh
So the readings in Kardashev scale is based upon total energy consumption of a civilisation?
Tanmay
What is the lowest and highest value possible in this scale?
Tanmay
is time travel possible?
Shivam
Yes time travel is possible (in theory)...
Taj
Time is a physical property that can and IS affected by gravity... This has been PROVEN through experimentation via seismic clocks.
Taj
thanks for the answer
Shivam
no problem
Taj
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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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