>> Because this page is becoming too big, I've split it in two..all links will redirect you correctly ... you can
also navigate by page number.
Normally when you read non-technical book about Stars you mostly get pretty pictures and may be some explanation, but not much more.
I have been interested for long time about the details of how exactly we know the thing that we know about stars in more detailed way and finally I had some time
to pursue this interest. Let me take you with me in this exploration.
The first article will concentrate on how do we find about the different properties of the Stars, things like mass, distance, temperature etc... then the next
on how exactly normal stars generate energy, how they stay in equilibrium and don't explode for billions of years, then may be we will explore the
evolution of the stars.
This is not meant to be full thesis on Stars, just something more than simple star-overview. Again the idea is to fill a gap I think exists.
Because there is so many interconnections between how you can deduce one property of a star from another,
I want to point your attention toward the flowchart on the right, as you go trough the text below I'm almost sure you will get lost in this maze of information. I know I'm ;)
For this reason I created this shortcut diagram of some of the basic relations, so you can orient yourself of what you can derive from what and by what means.
For more detailed formulas
you can also consult the table at the end of the article.
Another helpful tool is this mini-toc, so you can easily jump around :
To give you some idea before we start with the meat of the matter, the things we can infer about stars come mostly from several major places, namely direct
observations for the closest stars using simple geometry, then comes the spectral analysis.
Also knowledge of the Universal law of gravitation and Kepler laws give us glimpse of the mass, periods of rotation, distances in gravitationally bound systems
And finally modeling the interior and exterior of stars from our knowledge of fundamental physics and principles of Quantum mechanics, ideal gases and similar
help us guess how exactly stars work and compare the results with our observations.
Now that you've got general idea what we will be looking for let start ...
I got in alot of trouble mixing data from different sources...so at the end I decided to only consult Wikipedia for source data. Also I'm limiting
myself to three decimal places and this round off in most of the cases causes some small discrepancies with the final values and Wikipedia. Anyway if you find any calculation error please let me
know .. all those numbers give me headache ;)
The first thing on our mind is how do we know how far away the stars are and because the distances we are talking about are staggering we will
first have to introduce some measurement units which are better suited at this task.
Distances in astronomy are normally measured in one of the following units :
- Astronomical Unit is equal to the distance between the Earth and the Sun :
`1 AU = 1.495978 * 10^11 m`
- Light year is equal to the distance light crosses in a year :
`1 ly = 9.46053 * 10^15 m = 6.324 * 10^4 AU`
- Parsec is the distance from Earth to an object which appears having a parallax angle of one arc second.
`1 pc = 3.085678 * 10^16 m = 3.261633 ly = 206265 AU`
- Arc-second `1'' = (1/60) * (1/60) * (1/360) = 1/1296000` of a circle
Let's now tackle the ways we can measure those distances.
The first and most obvious method for doing astronomical measurements is parallax. The good news is that it relies on simple geometry.
The bad news is that it is limited for ground telescopes to around ~150 ly and for space telescopes to ~1500 ly,
beyond those distances the angle of the parallax is too small to be distinguishable.
How does it exactly works. Check the diagram.
We simply look at an object from two different positions, then we find the angle between the two observations and finally use trigonometry
to calculate the distance.
Our eye-sight work in similar manner, that is how we get the perception of depth.
The radius (R) of the orbit of Earth around the Sun plays the role of the distance and is equal to 1AU as I mentioned earlier.
The angle (`theta`) is the half of the measured angle as measured between the two furthest points of Earth orbit.
Having those we can calculate the distance to the star, because we know that :
`tan theta = R/d`
we use the approximation for very small angles, which is :
`tan theta ~~ sin theta ~~ theta`
i.e. the ratio between the radius of Earth orbit and the distance to the Star is equal to this half-angle (called parallax).
Go ahead try to calculate sin() for small angles and see what you get.
Then the radius(R) by our definition is 1 AU, so the parallax-angle is :
`theta = 1/d`
or said in different way the distance is :
`d = 1/theta`
Remember `d` is in parsecs
Because this is the most accurate way of observation it is also often used for calibration for the other methods.
Let me mention something which I think is important in general, many of the results we get on this page will be some approximation. As you may expect we can not
measure precisely to the meter or millimeter for example the exact temperature or radius of the Sun or a star.
That aside there is many complementary methods which astrophysicists use to calibrate and recalibrate the measurements, so it is expected over time for us to be
more and more precise. And if there are errors they will be in the single digit percentages.
OK, now that we are clear, we can continue...
Let's test the parallax method and see what we get.
The closest star to Earth besides the Sun is Proxima Centauri
and it has 0.762 arc seconds
Which means if we apply the above formula we can calculate the distance to be :
`d = 1/0.762 = 1.31234 pc * 3.26 = 4.27 ly`
Then the recently discovered (in 2013) binary system of 2 brown dwarfs stars Luhman (WISE_1049-5319)
which took the spot of the third closest
stars (displacing Wolf 359 from third place) and have parallax 0.496 arc seconds
`d = (1/0.496) * 3.26 = 6.57 ly`
Easy peesy if somebody give us with the parallax we are good to go.
Variables : RR Lyrae, Cepheids
Type IA supernova
To tackle the calculation of the star mass the obvious way is to use Universal law of gravitation and Kepler laws.
In our first example with the Sun we will also use the centrifugal force.
It is basic physics calculation we know formula for one of the forces, then we find another force which is equal in magnitude and we just connect them
with equal sign and go from there...
Centrifugal force is :
`F = (mv^2)/r`
gravitation law is :
`F = (GMm)/r^2`
so we "equate" the two i.e. we find what happens when they are equal :
`(mv^2)/r = (GMm)/r^2`
then we divide both sides by the small mass, and express big-M, which is what we are looking for :
`M_o. = (r*v^2)/G`
(btw: the sign `M_o.` is used to signify the mass of the Sun. This way we can do calculation w/o manipulating mind-numbing exponential numbers)
Now that we have the formula, we would need a planet that rotate around the Sun for which we know the distance and velocity.
So we pick Earth which rotation speed around the Sun is : 29.78 km/s, let's plug that numbers into the equation :
`M_o. = ((29.78*10^3)^2*1.495*10^11)/G = 1.9865*10^30` kg
So the Sun is 1000++ more massive than Earth.
Brightness and Luminosity ... Magnitude ^
We all know that stars shine ... and I mean alot .. but we also know that as the distance increase the light attenuates according to the inverse square
law, that is because stars are ~spheres and dissipate energy in all directions, ergo this dependence.
So we have to distinguish between how much the star intrinsically shines and what we at our standpoint here on Earth perceive the "shininess" to be.
For this astronomers define two different measures.
Brightness (B) is the energy we intercept from Earth position i.e. how bright the star appear to us.
Luminosity (L) on the other hand is how much energy the star emits i.e. how bright the star really is.
The luminosity in most cases is calculated from the observed brightness, the knowledge we have that radiated energy is
dissipated over distance by the inverse square law and finally assuming the star to be a ideal black body
So assuming the star is ideal black body we can apply Boltzmann law to get the radiation per unit area and from there the Luminosity
`E = tau * T^4`
`E` : energy radiation of unit area (flux)
`tau` : Stefan-Boltzmann constant : `5.670373(21) * 10^-8 \ W/(m^2*K^4)`
`T` : temperature
applying it to a sphere we get :
`L = 4 pi R^2 tau T^4`
`L` : is luminosity of the star i.e. how much energy the star generates
So once we know the radius and the temperature we can calculate the Luminosity.
We can also calculate distance if we know the Brightness and Luminosity.
`d^2 = L / (4 pi B)`
here `B` is brightness.
Lets try our newly acquired knowledge on the Sun.
Energy production of the Sun ... Luminosity
Before we start here I have to explain what "Solar constant"
This is the measured energy of Sun radiation at Earth orbit (1AU) falling on area of `1 \ m^2` i.e. Brightness
per unit area.
The constant has been measured by satellites and is equal to : `1361 W/m^2`.
On the side note some quick info on the Solar panels. According to Shockley–Queisser limit the maximum theoretical efficiency of solar cell is
~33%, so in the best case scenario satellites in Earth orbit can generate at maximum ~450 W and this is in space, Earth based solar panels get orders of
magnitude less solar energy because of the Earth atmosphere. I think I read somewhere this to be in the range of ~100-200W.
Then next step is to find the ratio between the Sun-outer-surface and a surface of a sphere that extends to Earth orbit.
Once we have this ratio we just multiply it by the Solar constant and find the energy emitted by `1 m^2` at the Sun surface.
Then we multiply again this time by the total area of the Sun and we get the total energy production (Luminosity
So let's do the calculations :
Ratio - sun-to-earth-orbit
`S_e/S_s = (1.495*10^11)^2/(6.963*10^8)^2 = 46098.8` times bigger
`1361*46098.8 = 6.274 * 10^7 W/m^2` at the surface of the Sun per square meter
Formula for surface of a sphere is :
`S = 4 pi r^2`
Surface of the Sun is :
`S_s = 4*pi*(6.963*10^8)^2 = 6.092 * 10^18 m^2`
Finally Total energy emitted
`L = 6.274*10^7 * 6.092*10^18 = 3.822*10^26 W`
That is good but for stars light years away we normally don't have radius and distance information so readily,
thus we have to use the so called magnitudes which are not measured in Power units, but are index based on HR-diagram.
Apparent and Absolute magnitude ^
I find this part the most confusing, but is probably the most important one ... so we have no choice but to explore it.
I won't go trough the history of how it came to be. It is enough to say that we inherited from the Greeks the so called magnitude scale,
for measuring the brightness of the stars. The scale is unit-less and is reversed, what I mean by this is that
the brightest the star the lower the magnitude-number (on top of that with today's advancements in measurement precision the scale
had to extend below 0).
So a star with magnitude -1 is brighter than star with magnitude 5.
This causes alot of confusion, but it stuck, so we have to use it ;(, just think what is the opposite of a logical way of creating
a scale and you will be right.
That said, we have to figure out how to convert those magnitudes into Brightness/Luminosity and vice versa.
You may ask why would we burden our-self with this counter-intuitive scheme, the reason is mainly because when we start solving problems we
will most probably have access to information about the star magnitudes and from there we will derive Luminosity, not the other
way around. Plus having magnitudes give us access to the HR-diagram and from there we can extract other useful information about
the star, than just those two properties.
Let's define the terms first.
Apparent magnitude (m)
is measure describing the Brightness
of a star.
Absolute magnitude (M)
is the measure describing the Luminosity
of a star.
Next we have to quantify them. As you may suspect the scale was created when people were observing the sky with naked eye, so
naturally the granularity of the index were tailored to the dexterity of the human eye. The range set by the Greeks was from 1(brightness
But as we know what for the eye looks as linear increase in intensity of light is in fact exponential increase, said the other
way around our senses perceive light logarithmically.
So it was decided when more modern way of watching the skies become available that a star with magnitude 1 will be 100 times brighter than
a star with magnitude 6, therefore a difference of one magnitude corresponds to a factor of `5_sqrt(100) = 2.512` i.e. five steps where each consecutive
step is exponentially 2.512 time bigger than the previous one (`2.512^5 ~ 100`).
Similar to the way octaves works in music, just different scale factor
So we have magnitudes what do we do with them ... we can for example compare how much brighter one star is from another.
Magnitudes are also very often readily available compared to the luminosity and brightness as information.
And probably the best reason for usefulness of HR-Diagram as we will see soon.
So with this in mind lets derive some useful formulas.
We start with a way to find Brightness or Luminosity ratio knowing the magnitudes.
We will use :
`B_m`,`B_n` : brightness/Luminosity
`m`, `n` : magnitudes
We already defined that the ratio by which magnitudes increase with every step increment of 1 is 2.512 times bigger:
`2.512^x = 100^(x/5)`
so let's try to find the ratio of brightness'es (<< that sounds funny ;) :
`B_n/B_m = 100^(n/5) / 100^(m/5) = 100^((n-m)/5)`
But the magnitude scale is reversed, so we will "fix" the formula :
(1) `B_n/B_m == 100^((m-n)/5)`
Remember that `m > n` when `B_m < B_n`, because smaller magnitude means brighter star, urghhh..and so we would use `m - n`.
Let's now apply log to both sides.
We know that `log x^a = a log x`, so :
`log(B_n/B_m) = (m-n)/5 * log 100 = 0.4 * (m-n)`
(2) `m - n = 2.5 * log(B_n/B_m)`
This show us how difference in magnitudes relates to difference in brightness (or luminosity).
Lets try now to connect the magnitudes with distance in some way.
: If a star is "placed" 10 parsecs away from us its Apparent magnitude will be equal to its Absolute magnitude
Then using the connection with inverse square law we can say.
So by definition above if we use the model-Star properties to find the unit-magnitude :
`L = B * (d/10)^2`
which is equivalent to :
`L/B = (d/10)^2`
but if we reuse the ratio formula (2) then :
(3) `m - M = 2.5*log(L/B) = 2.5*log(d/10)^2 = 5*log(d/10)`
`m` : apparent magnitude (how we see it)
`M` : absolute magnitude (how it really is)
`B` : brightness (how we see it)
`L` : luminosity (how it really is)
`d` : distance in parsecs
from (3) we can infer these useful alternatives :
`m - M = 5*log d - 5`
`M = m + 5 - 5*log d`
`M = m + 5 + 5*log theta`
`d = 10^((m-M+5)/5)`
`theta` : parallax angle
`d` : is in parsecs
`m - M` is called distance modulus ! If it is < 0 stars are closer than 10 parsecs, if it is > 0 the star is further than 10 pc.
It just happens that the star GJ 75
has both apparent and absolute magnitude equal to 5.63, its distance modulus is 0 i.e.
it is exactly 10 pc away (you can double check this if you do the substitution in the distance equation).
Mass-Luminosity relation ^
Regarding the plasma as ideal-gas and the star as ideal-black-body in the beginning of the 19th century for the first time approximate
theoretical derivation were done for the relation between Mass
`L prop M^3.33`
But of course stars are very different, so more experimental approach gives the following relations :
`L_r = L/L_o.` : luminosity Star/Sun ratio
`M_r = M/M_o.` : mass Star/Sun ratio
|Average for main sequence stars||`L_r ~~ M_r^3.5`
|`M < 0.43 M_o.` ||`L_r ~~ 0.23 * M_r^2.3`
|`0.43 M_o. < M < 2 M_o.`||`L_r = M_r^4`
|`2 M_o. < M < 20 M_o.` ||`L_r ~~ 1.5 * M_r^3.5`
|`M > 20 M_o.` ||`L_r ~~ M_r`
The relation flattens for big stars. We would see this also later expressed in the HR-diagram...be patient ;)
We start with Wien law, here the basic idea is that if we detect the wavelength (in the spectrum of the Star radiation) at which the Star has maximum
Intensity i.e. most energy generated we can calculate the approximate temperature at the surface of the Star in a very straightforward way.
Wien law :
`lambda_max = k / T`
`lambda_max` : peak wavelength
`T` : temperature
`k` : constant of proportionality : `2.8977685 * 10^-3` mK
`T = k/ lambda_max`
Let's apply this in the case of the Sun :
Sun peak wavelength is : `501.3 nm = 501.3 * 10^-9 m` i.e. frequency : 598 THz
`T = (2.898 *10^-3) / (501.3*10^-9) = 5780 K`
We can also use another way of finding the temperature by using Stefan-Boltzman law, we mentioned already.
`E = tau * T^4`
which is :
`T = (E/tau)^(1/4) = ((6.247 * 10^7) / (5.67 * 10^-8))^(1/4) = 5767.54 K`
That is the temperature of the Sun photosphere.
We can use the same trick for Stars i.e finding the `lambda_max` and then calculate the temperature, the problem is that this method may sometimes be
inaccurate, because the light from stars when crossing interstellar space get absorbed, diffused ...etc.
But astrophysics have found another way around this using the black body radiation phenomena. The idea is that if you acquire the Intensity of light coming
from a star in two (or more) wavelengths-bands of the spectrum and compare them you can deduce the black-body curve.
( For more detailed discussion of it check QM Intro : Black body radiation
Astronomers take measurement of Brightness with blue and red filter and calculate a ratio, then compare the result with calibrated data and thus find the
We can even do very quick observation of the picture displayed on the right, if we subtract the Intensity of the filtered light, we can deduce that when
`B-V > 0`, the star is a hot star, and if `B-V < 0` we are talking about colder star.
If you want to find the Temperature from the BV color index, use the following formulas.
If B-V > -0.0413 then use :
`T = 10^((14.551 - (B-V))/3.684)`
If B-V < -0.0413 then use :
`T = 10^(4.945 - sqrt(1.087353 + 2.906977* (B-V)) `
If you want to convert the other way around Color/temperature to B-V, then :
If T < 9141 K
`B-V = -3.684 * log_10(T) + 14.551`
If T > 9141 K
`B-V = 0.344 * log_10(T)^2 - 3.402 * log_10(T) + 8.037`
(If you are working with magnitudes of course it will be the other way around, remember reversed logic of the magnitudes!!)
We will start from the luminosity formula again :
`L = 4 pi R^2 tau T^4`
it is easy to see that if we have luminosity (energy generated by the star) and temperature we can find the radius.
solving for `R` will give us :
`R_o. = sqrt(L/(4 pi tau T^4)`
Let substitute the values :
`5.670 373 * 10^-8 W m^-2 K^-4`
`R_o. = sqrt((3.822*10^26 W) / ( 4 * pi * 5.670*10^-8 W/(m^2*K^4) * (5778 K)^4)`
`R_o. = 6.937 * 10^8 m`
almost there :)
The Sun is OK, but how do we go about other Stars light years away.
It is a little bit harder, but still doable.
For our example we will pick the star Epsilon Eridani, I find the name to be very cute :) and on top of that is one of the nearby stars,
just 10.48 ly away.
To get started we will need some information :
Epsilon Eridani :
Absolute magnitude(`M_e`) : +6.19
Temperature (`T_e`) : 5084 K
Absolute magnitude(`M_o.`) : +4.83
Temperature (`T_o.`) : 5778 K
Radius (`R_o.`) : `6.963*10^8 m`
This is the radius we are looking for, I'm giving it here so that we can see how close it is to our result once we are done with our calculations :
Epsilon Eridani radius (`R_e`) : `0.735 * R_o.`
So we start again with the luminosity, but we are looking for the ratio between Eridani and the Sun - `L_e/L_o.`.
`L_e/L_o. = (4*pi*R_e^2*T_e^4) / (4*pi*R_o.^2*T_o.^4) = (R_e^2*T_e^4) / (R_o.^2*T_o.^4)`
we are interested of the radius ratio :
`(R_e/R_o.)^2 = (L_e/L_o.) / (T_e^4/T_o.^4) = (L_e*T_o.^4)/(L_o.*T_e^4)`
`R_e/R_o. = sqrt(L_e/L_o.) * (T_o./T_e)^2`
but substituting formula (1) for the luminosity ratio :
`R_e/R_o. = sqrt(100^((m-n)/5)) * (T_o./T_e)^2`
this is our final formula, let's fill the data :
` = sqrt(100^((4.83-6.19)/5)) * (5778/5084)^2 = 0.69 * R_o.`
we are 7% off the correct value.. nice..
[ PAGE 2
I'm nowhere near with finishing this article ... as I have time I will try to add some more explanation and other examples.
Some more graphs and images would be good too...
As you can see star characteristics we can glance are many and there is myriad ways of finding them, I hope with this article I was able to show how using basic algebra
you can discover them and possibly gain some understanding of the the mighty furnaces giving us light and life.
And now summarized in one place the basic relations that we used tr-ought...
|`L ~~ M^3.5`||Mass-luminosity ratio||Average for main sequence stars
|`E = tau T^4`||Stefan-Boltzmann law (flux)||Luminosity-temperature relation (per unit area)
|`L = 4 pi R^2 tau T^4`||Luminosity of a star||Boltzmann law applied to a sphere.
|`R = sqrt(L/(4 pi tau T^4)`||Radius from Temperature and Luminosity||the above formula rearranged
|`F = (G M m)/r^2`||Universal gravitational law||
|`G M T^2 = 4 pi R^3`||Kepler 3rd law||
|`lambda_max = k / T`||Wien law||Find temperature from spectrum.
|`m - M = 5 log d - 5`||Distance from apparent and absolute magnitudes||Distance in parsecs
|`(m_1 + m_2) * T^2 = (d_1 + d_2)^3 = R^3`||Mass of stars in binary system||Kepler 3rd law
|`M = (r*v^2)/G`||Centrifugal force : `F = (mv^2)/r` and G-law||
Next time we will explore how Stars work, the principles involved and some physics rule that stays behind it.