A Stellar Astronomy Toolbox

In order to understand stars there are important aspects about stars that we would like to know.

These aspects we will call stellar parameters.

Here are some questions: Here are the stellar parameter names:

1) How big are stars? Radius or Mass

2) What are stars made of? Chemical Composition

3) How hot are stars? Surface temperature

4) How old are they? Age

5) How much energy do they produce? Luminosity

6) How far away are they? Distance

7) How do they move? Velocity

Stellar parameters are fundamental aspects of stars. They are determined by analyzing data by

either making calculations or through comparison to theoretical models. None of the stellar

parameters are directly measureable. They have to be determined using data which we will refer

to as observables.

Here are some stellar observables:

1) The color of a star

2) The brightness of the star as seen from Earth

3) Characteristics of the stellar spectral lines (This includes the identification, strength,

width, and location of spectral lines)

Defining stellar parameters and why they are important.

Size of the star: There are two ways we will define the size of a star in this class.

a) Radius. This refers to the size of the star in kilometers. This is similar to how much

space the star takes up (which we call volume). For a spherical star (the kind we will

be concerned with) the volume is related to the radius as follow:

𝑉 =4

3𝜋𝑟3

b) Mass is another way to characterize the size of a star. Mass is the amount of matter

in an object. In our everyday world when an object has a large volume it is normally

very massive also. For example an elephant is much bigger than a mouse in mass and

in volume. This is not always the case for stars. For stars such as main sequence

stars there is a direct relation between the mass and the volume. But there are also

evolved stars called giants, supergiants and white dwarfs for which there is not a

clear relationship between radius and mass.

c) How can objects with a large radius have a small mass, or an object with a small

radius have a large mass? The amount of matter contained in an object’s volume is

called its density. Density is defined as mass divided by volume.

𝐷𝑒𝑛𝑠𝑖𝑡𝑦 =𝑀𝑎𝑠𝑠

𝑉𝑜𝑙𝑢𝑚𝑒

Knowing how dense a star is tells us a great deal about the age and energy processes

occurring in the star.

Determining Mass: To determine the mass of a star we need to use Newton’s laws of motion

and law of gravity or else Kepler’s third law. The basic concept is this:

How long it takes for an object to complete one orbit around another object depends only on

the mass of the two objects and the distance between them.

𝑀 + 𝑚 𝑃2 =4𝜋2

𝐺𝑑3

Where d is the size of the orbit and P is the orbital period. The orbital period is just the time it

takes an object to orbit once. For instance, the Earth orbits the Sun in one year. So the orbital

period of the Earth is 1 year or 3.14 x 107 seconds.

Finding the distance between the objects is somewhat harder, especially for stars that are very far

away. We will find that in some cases we can measure the orbital velocity of an object. If we

know the velocity and we know the period we can find d because

𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝑡𝑖𝑚𝑒

If time is the orbital period, then distance is just the circumference (C) of the orbit.

𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒

𝑝𝑒𝑟𝑖𝑜𝑑=

2𝜋𝑑

𝑃

To compute the mass of stars we use binary stars. These are stars that orbit around each other.

The period can be determined as the time it takes to complete one orbit and the velocity can be

found using the Doppler shift (something we will discuss).

Stellar Brightness and Law of Gravity (Inverse square laws)

How bright an object appears to use depends on two variables. One variable is how much energy

the object is emitting. The other variable is the distance to the object. Here are two thought

experiments.

1. Suppose you have a 100 watt light bulb and a 40 watt light bulb. They are both the same

distance away from you. Which will appear brighter? (Note: a watt is a unit of power. It

is the amount of energy being emitted each second. Where energy is measured in Joules)

Answer: the 100 watt light bulb

2. Suppose you have two 100 watt light bulbs and one is a few feet away and the other is 50

feet away. Which one will appear brighter to you?

Answer: the one that is a few feet away.

So using these two observables, we can come up with an equation to describe brightness.

How bright an object appears is directly proportional to its luminosity. (In astronomy we

refer to power as luminosity). In other words, the more luminous an object the brighter it will

appear.

How bright an object appears is inversely proportional to its distance. In other words the

greater the distance the less bright the object will appear.

From this we can begin to guess an equation. Let B stand for brightness, L stand for luminosity

and D stand for distance. Then our equation must look something like this:

𝐵 𝑔𝑜𝑒𝑠 𝑙𝑖𝑘𝑒 𝐿

𝐷

There is one problem with our equation. That is how light spreads out when it leaves a source.

Imagine a flash bulb that gives one bright flash and then is gone. The light from the flash moves

out in all directions and is spread over the surface of a sphere. As the distance from the original

flash moves outward the sphere grows in size. The light is spread out over a larger and larger

surface area. See image below:

So we need to relate the distance from the flash to the surface area of this growing sphere. The

surface area of a sphere is

𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 4𝜋𝐷2

So the true brightness equation is:

𝐵 = 𝐿/4𝜋𝐷2

This is a very important equation in this class. Notice that brightness is something we can

readily measure here on Earth using a telescope and a digital camera. So,

1. If we know how far away a star is (D) and we measure its brightness (B) then we can

calculate how luminous (L) it is.

2. Or if we know how luminous a star is (L) and we measure its brightness (B) then we can

calculate how far away (D) the star is.

Newton’s Law of Gravity.

When Newton formulated his law of gravity he assumed that the gravitational force must

spread out the same way the light spreads out, over the surface of a sphere. He also realized that

the force of gravity must go like the product of the two masses. The law is:

𝐹 = −𝐺𝑀𝑚/𝐷2

Where D is the distance between two massive objects, M is mass of one object and m is mass of

the other. G is the universal gravitational constant and is the same everywhere in the universe.

Its value is G = 6.67 x 10-11

N m2/kg

2

Both the Brightness law and the Law of Gravity are called inverse square laws, because the

brightness (or the force) is inversely proportional to the square of the distance.

What is temperature?

We all use the word temperature often in our lives but you may not know how it is defined.

Temperature is defined in terms of kinetic energy. So before I continue lets talk about what

kinetic energy is.

Kinetic energy is the energy of motion. Any object which is moving, or in other words, has

velocity, also has kinetic energy. The amount of kinetic energy that an object has depends on

how fast it is moving. Kinetic energy is directly proportional to the velocity, which means that

the faster you move the more kinetic energy you have. To be more precise the amount of kinetic

energy actually goes like the square of the velocity. Kinetic energy also depends on how much

mass an object has. If you were running alongside a small child you would both have the same

velocity, but you are likely to be more massive than the child. That means that you actually have

more kinetic energy than the child does. So kinetic energy is directly proportional to the mass of

an object. Here is the equation:

𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 =1

2𝑚𝑣2

Where m is the mass and v is the velocity. You can see in this equation, that if v = 0 then you

have no kinetic energy.

Now let’s change the subject just a bit. You are right now surrounded by air molecules. They

are mostly Nitrogen and Oxygen molecules with a little carbon dioxide and methane thrown in.

If we think back to Galileo’s law, all objects fall at the same rate regardless of their mass, then

there should be no reason that air molecules shouldn’t fall to the ground at the same rate that a

book would fall. But we know that this is not the case. The atmosphere of the Earth is not lying

on the ground. This must mean that the molecules in the atmosphere are moving. In fact, the air

molecules around moving very rapidly and constantly colliding with each other. IF they are

moving, then they also have kinetic energy.

As you might expect, not all of the molecules are moving at the same speed. In fact some are

moving very fast and some are moving very slowly. When a fast moving one collides with a

more slowly moving one, the slower molecule can pick up speed and the fast molecule can slow

down. There are billions of molecules around you that are moving and colliding all the time.

This means it is virtually impossible to keep track of the velocity, or kinetic energy, of all the

individual particles. Fortunately, we can keep track of the average kinetic energy of a very large

group of particles.

The plot above shows the distribution of velocity for a bunch of gas molecules. On the vertical

axis is the number of particles, n, and on the horizontal axis is their velocities, v. Let’s first just

concentrate on the blue line. What blue line shows is that there are virtually no particles with

zero velocity. Then the number increases as the velocity increases. The number of particles

reaches a peak at around v = 650 m/s. Then it decreases once again. When you get to very high

velocities you can see that n becomes small once again.

So, there is a very large range of velocities. But it is possible to assign an average velocity to the

blue curve. The average velocity is very close to the peak in the curve. So v_ave ~ 650 m/s.

knowing this and knowing the mass of the molecules we could determine an average kinetic

energy.

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 =1

2𝑚(𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦)2

Now consider the other two curves. What do you think the average kinetic energy of the red

curve and the green curve are compared to the blue curve? Since the average velocity is near the

peak, it must be the case that the average kinetic energy of the molecules represented by the red

curve is less than the molecules represented by the green curve. And the green curve is less than

the blue curve.

Now on to temperature. Temperature is directly proportional to the average kinetic energy of a

group of particles. The equation is:

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 =3

2𝑘𝑇

Where T is the temperature and k is the boltzman constant (k = 1.38 x 10-23

J/K).

So what is important here? The higher the average kinetic energy of a group of particles, the

higher the temperature. They are directly proportional.

Why does a stove burner feel hot?

When a stove burner has a high temperature it means that the molecules in the burner have a

large average kinetic energy. Or in other words, the molecules are moving around very rapidly.

Your hand (on the other hand) is at a much lower temperature. This means that your hand

molecules are not moving nearly as fast. When your hand comes in contact with the stove, the

molecules in the burner smash violently into the slowly moving molecules in your hand. The

result is that your hand molecules begin to move very rapidly and you sense this as something

hot. If you leave your hand on the burner for even a couple of seconds, the molecules in your

hand will move so violently that they will be torn apart into smaller molecules or even individual

atoms. The result is a nasty wound on your hand.

How is a balloon like the Sun?

When you inflate a balloon you are packing a huge number of air molecules into the balloon.

These molecules have kinetic energy and are flying all around the inside of the balloon. Some of

them run into the interior walls of the balloon and exert a force on the balloon walls. The rubber

in the balloon does not stretch by itself. In order to make it stretch, a force has to be applied to

the balloon. When the balloon is inflated, the force that is acting against the rubber of the

balloon is the collisions of the air molecules against the inside of the balloon. As a result the

balloon expands until you stop adding more air molecules. IF you cool the temperature of the air

molecules in the balloon, then they stop exerting as much force on the walls and the balloon

shrinks. If you heat up the air molecules inside, then they move faster and exert a greater force

on the inner walls of the balloon forcing it to expand.

A balloon that is inflated, tied off and just sitting on a table, is in a state of force equilibrium.

The force of the rubber in the balloon, which is trying to shrink the balloon, is exactly balanced

by the force of the molecules inside that are banging off the walls.

The Sun can be thought of as a big ball of gas, sitting out in the vacuum of space. It is analogous

to a balloon in so much as there are hot atoms inside of it, moving with very high velocities.

Normally we would expect these gases to rapidly diffuse off into the vacuum of space, yet this

does not happen. The Sun remains the same year after year. There is clearly no rubber exerting

a force to hold the fast moving atoms inside the Sun, but just like a balloon, there is a force that

keeps the gas contained. This is the inward pull of gravity.

So the Sun is also in a force equilibrium, just like a balloon. The force of gravity is trying to

contract the Sun and the hot (or fast moving) atoms are trying to escape. This allows the

particles to exert a force that resist the contraction of the Sun. Just like the case of the balloon.

A problem to consider.

The Sun is giving off huge amount of energy in the form of radiant energy. This radiant energy

comes from the fast moving particles inside the Sun. Where is this radiant energy coming from?

What I mean by radiant energy, is the energy that is streaming out of the Sun in the form of

electromagnetic (EM) radiation. Let’s see what EM radiation is.

A charged particle has associated with it an electric field. This electric field is analogous to a

magnetic field and can be represented as lines of force. The image below is created by placing a

magnet under a piece of paper and sprinkling iron filings on top of the paper.

The magnetic lines of force are easily seen as outlined by the iron filings and represent lines of

force that extend from the north pole to the south pole of the magnet.

Electric field lines around charged particles are harder to see but very similar. The figure below

show representative electric field lines around a positive charge, negative chare and a neutral

charge.

When the particles are not moving the field lines extend directly out from the charge all the way

to infinity. Now, if a charged particle, such as an electron, is hit with a force, it will accelerate

momentarily and rapidly move. When this happens the field lines that are very far from the

electron do not know that the electron has moved. They remain just as they were before the

electron was hit. But up close to the electron, the field lines point directly where the electron

currently is. The result is a kink in the electric field lines as seen below.

This kink is due to the fact that it takes time for the change in position of the electron to be felt

by the distant electric field. The kink represents a change in the electric field around the electron

and it propagates outward at the speed of light.

This kink would not be very noticeable if it were not for the relationship between the electric and

magnetic fields. Here is the rule.

A changing electric field produces a magnetic field and a changing magnetic field produces an

electric field.

So the kink, which is a change in the electric field of the electron, produces a magnetic field.

This new magnetic field produces an electric field and so on. The result is an electromagnetic

wave of continuously changing electric and magnetic fields. This wave is what we call

electromagnetic radiation. Once the effect is set off, the electromagnetic wave propagates itself

through space by generating electric and magnetic fields. Electromagnetic waves that we can see

are called light.

To produce these electromagnetic waves requires energy. The energy that is carried in the

electromagnetic radiation I will refer to as radiant energy.

But where does this radiant energy come from? It comes from the kinetic energy of the electron,

or in other words, the energy of motion of the electron. This has very important consequences.

Energy Conservation

Energy can be neither created nor destroyed. It can only change forms. In this class we will

only deal with three types of energy.

Potential energy – stored energy

Kinetic energy – energy of motion

Radiant energy – energy in the electromagnetic wave

The total energy in a system is always constant, from energy conservation. But energy can

change forms.

ETotal = EPotential + EKinetic + ERadiant

Here is an example:

When a book is held up in the air it has potential energy. It is not moving and it is not radiating

so,

ETotal = EPotential

When it is dropped, it gains kinetic energy. The kinetic energy comes from the potential energy.

Just before hitting the floor, all the potential energy is turned into kinetic energy and the results

is,

ETotal = EKinetic

After the book hits the floor it is no longer moving and it has no farther to fall. This means that

the potential energy and kinetic energy of the book is zero. But the total energy has to remain

constant. Where does the energy go? The book vibrates the molecules in the floor which are

accelerating charges. When you accelerate charges they produce EM radiation, as we showed

just before. This propagates off into space as radiant energy.

ETotal = ERadiant

In all the cases above, ETotal does not change. Only the form of the energy changes.

What does all this have to do with the Sun?

Charged particles in the Sun are being accelerated. When this happens they produce EM

radiation which takes away energy and sends it out into space. The radiant energy comes from

the kinetic energy of the electrons in the Sun. Since energy is conserved this means that when

the electrons radiate, they lose kinetic energy. Less kinetic energy means the temperature goes

down (this is how temperature is defined). The expected result is that the electrons cool and the

Sun should shrink, since the force of gravity is not changing. But the Sun isn’t shrinking.

Something has to be replenishing the kinetic energy inside the Sun that is lost to radiant

energy.

Determining the Surface temperature of the Sun and other stars. (One of our crucial

stellar parameters)

So we know that particles in the Sun are moving around with high kinetic energies and radiating.

But how can we measure the temperature of something like the Sun or a star. It is impossible to

send a thermometer to the Sun and other stars in order to take their temperature. So we need

another way.

Luckily, the wavelength of the light that is coming from the Sun contains information about the

kinetic energy of the particles (or temperature). When the average kinetic energy of the electrons

in a star is extremely high, then the light that they produce has a lot of light with high energies.

This corresponds to short wavelength light because the energy in a wave is inversely

proportional to the wavelength of the light:

𝐸 = 𝑕𝑐/𝜆

If we spread the light from three different stars, one hot, one warm and one cooler, we see that

the hottest star has most of its light coming at shorter wavelengths (bluer light). A star that is

cooler has most of its light coming out at longer wavelengths (redder light). This is because in

the cooler star the particles do not have as high of kinetic energy as in the hotter star. So the

cooler star does not have as much available kinetic energy to turn into radiation. The light comes

out redder.

The peak intensity in each of the above radiation curves has a specific wavelength associated

with it. The hotter the star, the more λmax moves to bluer colors. This allows us to tell the

temperature by measuring the wavelength λ where the peak intensity occurs. The mathematical

relation that allows us to compute the surface temperature is called Wein’s Law.

𝑇 = 3,000,00/𝜆𝑚𝑎𝑥

It should not be surprising that the radiation curve has a peak wavelength where most of the light

is emitted, since the electrons in the star also have a peak velocity at which most of the electrons

move.

This is the procedure we use to measure the surface temperature in a star.

Chemical Composition (Another key physical parameter)

Particle-wave Duality

Quantum physics has shown that subatomic “particles” such as the electron, proton and neutron

can sometimes act like particles. They are localized and are able to have any kinetic energy that

is provided to them. Just like a baseball being thrown around. The strange thing is that

sometimes these “particles” act like waves. This means that they are not localized, and cannot

take any energy that is provided to them. Here is the rule for when an electron acts like a particle

and when it acts like a wave.

If an electron is free to move around, it acts like a particle.

If an electron is bound to an atom it acts like a wave, in particular, a standing wave.

Below is a diagram showing various standing waves on a string.

An electron acts like a 3-dimensional standing wave when it is bound to an atom. This means

that it oscillates like a wave. Standing wave can only have very discrete energies. The bound

electron can only accept a specific energy that allows it to oscillate in one of the higher energy

states. When the electron loses energy it can only lose the amount that allows it to oscillate as a

lower energy standing wave. Here are some computer generated energy levels for the hydrogen

atom.

The lowest energy level is the least complex in the upper right hand corner. When the electron

receives just the right amount of energy it can oscillate in one of the more complex wave forms

but nothing else. When the electron gives up its energy it emits a very specific wavelength of

light, which corresponds to the energy difference between the excited state and the lower energy

state.

The result is something like this:

The middle panel shows a hot gas with bound electrons. This gas can only emit at very specific

wavelengths which correspond to the energy difference between the energy levels. The result is

a handful of bright lines and darkness at all the other wavelengths. Contrast this to the

Continuum spectrum. This spectrum is created by free electrons which act like particles. They

can emit and all possible wavelengths and because of this there is light from the red end all the

way to the violet end (and in the infrared and ultraviolet regions as well).

The bottom spectrum is an absorption spectrum. This is a composite spectrum. The continuous

spectrum comes from the light bulb, which has free electrons. When the light passes threw a gas

that has bound electrons, the electrons can only absorb the wavelength of light that corresponds

to the energy of the transition. This is why the only light that is missing is at the wavelengths

where the emission lines occurred in the emission spectrum. Absorption is the reverse of

emission. For absorption, the electron is at a lower energy state and is excited to a higher state

when the light is absorbed.

Every element has a unique spectrum. In other words each of the elements on the periodic table

has electrons which have unique standing waves. This means that the spectrum of an element

has a unique “fingerprint” that allows us to identify the element that is present. Notice the

difference in the Hydrogen, Mercury and Neon emission spectrum.

nt

Because the emission occurs at different wavelengths for each element it is possible to determine

the composition of a radiating source, such as a star. Notice in the absorption spectrum above

has hydrogen absorption present.

So, by taking a spectrum of a star and looking at the wavelength of light which is absorbed, it is

possible to figure out the chemical composition of the star. Below is the absorption spectrum for

the Sun.

Determining the total Lifetime of the Sun

We know that the Sun is radiating a huge amount of energy out into the universe every

second. This is the luminosity of the Sun. The luminosity of the Sun is 3.8 x 1026

J/s. This

radiant energy comes from the kinetic energy of the particles inside of the Sun. Energy

conservation tells us that the total energy in a system is conserved (constant). So the kinetic

energy must be lost by the particles and turned into radiant energy. Since the radiant energy

leaves the Sun, never to return, it would be expected that the kinetic energy of the particles in the

Sun should decrease with time. In other words, the temperature would cool. However the result

of this cooling would be gravity overcoming the outward gas pressure and the Sun should shrink.

The Sun is not shrinking so there must be an energy source inside the Sun that continues

to re-invigorate the particles. This source is thermonuclear fusion in the core of the Sun. Below

is the p-p chain, which is the nuclear reaction that occurs in the core of the Sun to release this

energy.

Although this may look complicated, the entire reaction can be summed up as follows:

4H He + energy

Four hydrogen atoms are fused together to make helium and energy. The energy comes from

Einstein’s famous equation:

E = mc2

Where m is the mass that is converted to energy and c is the speed of light. When you add up the

mass of four Hydrogen nuclei, the total mass is greater than the mass of the Helium nuclei. This

is because a small fraction of the mass is turned into radiant energy. This energy is then

absorbed by electrons in the outer layers of the Sun and causes them to gain kinetic energy.

The entire computation of the life time of the Sun is outlined in the PowerPoint

presentation, Calculating the Lifetime of the Sun (Sixth Presentation) and I will not go through

it in detail here. But I will summarize the key points.

1) The energy leaving the surface of the Sun every second (Luminosity) must equal the

amount of energy being produced at the center of the Sun every second. If this were not

true the Sun would have to expand or shrink.

2) We calculate the luminosity of the Sun by using the apparent brightness and the known

distance to the Sun. L = B(4πd2)

3) We can use the luminosity to find out the number of reactions occurring each second in

the core of the Sun. (rate of fuel consumption)

4) We can determine the possible total number of reactions in the Sun by knowing the Sun’s

core mass. (amount of fuel)

5) We can then find the lifetime of the Sun by taking the ratio of available fuel to the rate of

consumption. (fuel/rate of consumption)

The answer is that the Sun’s lifetime is 10 billion years.

Stable Equilibrium

The sun is in a state of stable equilibrium between the inward pull of gravity, trying to shrink the

Sun and the outward gas pressure trying to expand the Sun. The reason that this is a stable

equilibrium is because the inward force of gravity controls the nuclear reaction rates in the core.

Reaction rates depend on two factors:

1) The temperature of the particles. This is because the hydrogen nuclei must move very

fast in order to get close enough together to stick. The nuclei have to overcome the

mutual repulsion. This is only possible if they are moving incredibly fast.

2) The density of particles. Nuclei can only fuse if they have a head-on collision. The

chance of a head-on collision is much higher if there are many nuclei packed into a give

volume. Just like in a classroom you are more likely to bump into someone when the

class room is full, as compared to when there are only 10 people in the classroom.

If the reaction rates were to slow in the core of the Sun, gravity would begin to shrink the core.

If the core shrinks both the density and temperature would increase. This increase the reaction

rate and more energy would be produced. This cases greater pressure in the core and the core re-

expands.

As long as there is available hydrogen in the core of the Sun, the Sun will remain in a stable

equilibrium. Continuing to emit the same amount of energy and remaining the same size.