Download A Stellar Astronomy Toolbox 9

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:
1)
2)
3)
4)
5)
6)
7)
How big are stars?
What are stars made of?
How hot are stars?
How old are they?
How much energy do they produce?
How far away are they?
How do they move?
Here are the stellar parameter names:
Radius or Mass
Chemical Composition
Surface temperature
Age
Luminosity
Distance
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:
4
𝑉 = 𝜋𝑟 3
3
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 =
4𝜋 2
𝐺
𝑑3
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/kg2
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
𝑚𝑣 2
2
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
𝑚(𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦)2
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:
3
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 = 2 𝑘𝑇
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.