Wednesday, June 19th 2002. Reading: Cosmic Perspective Chapter 15
Use trigonometric parallax (triangulation) for nearby stars.
Parallax = angle subtended (i.e., covered) by 1 A.U., the distance from Earth to the Sun. (1/2 of the diameter of Earth's orbit).
As distance increases the parallax angle decreases. Ultimately limited by the smallest angles measurable (resolved). The atmosphere limits resolution ability, must go to space --> Hipparcos satellite.
The distance at which 1 A.U. subtends 1" (one arc second) is called 1 parsec (pc). (parallax arc second).
1 pc = 3.26 Light-years. Distance to nearest star, Alpha-Centauri:
Best angle measurements of Hipparcos were down to p = 0.002" ---> d = 500 pc (1,630 light-years).
< 0 --> Blueshift: moving toward us
> 0 --> Redshift: moving away from us
measured as an angular speed (typically 0.1"/yr for nearby stars), but to convert to physical speed (km/s) need to know the distance
We can also infer the temperature of the surface from the specific spectral lines that are seen. Recall that spectral lines arise from transitions of an electron from one level to another in an atom. The electron may only emit or absorb very specific wavelengths of light for specific transitions. So, for example, say we see an absorption line of hydrogen which indicates that an electron has absorbed a photon and jumped from level 4 to level 7 in the atom. For the electron to already be excited enough to be in level 4 the atoms in the star's surface must be moving fast and hitting each other very often. This kind of motion is in fact how we define temperature. So this would be a pretty hot star.
Stefan-Boltzmann Law: Recall that we had a relationship between the temperature of a thermal emitter and how much energy per unit surface area per unit time it emitted.
So if we want to know how much total energy per second a thermal emitter emits we need to multiply by its total surface area. Stars are spherical, so SA = 4R2. Thus the luminosity, L, of a star is related to its radius, R, and surface temperature, T.
Inverse Square Law of Light:
When you observe a star from a distance, d, you do not directly observe its luminosity because you only intercept a small fraction of the energy that left its surface. You observe the star's brightness, b. Brightness has unit of energy per unit surface area detected per unit time. Since light spreads away from a star in a spherical shell you can see that the amount of energy per unit surface area per unit time observed at a distance is just the Luminosity of the star divided by the surface area of a shell at the distance of the observer. This gives us the inverse square law of light.
This says that if you move twice as far from a star (from some starting position) the star will become 4 times as dim.
It also says that if you know the distance to a star, and you measure its brightness, you can infer its luminosity. Then you can measure its temperature from spectrum and hence get its size.
The spectra of stars are varied, but there are trends. In the early part of the 20th century a classification scheme was devised for stars based on their spectra. The scheme originally was based only on the relative strengths of Hydrogen lines in the stars' atmosphere. Type A stars have strongest hydrogen lines, type O the weakest.
The different classes were then later arranged in order of decreasing surface temperature (some where rejected due to redundancy). From hottest to coolest the order is:
The mnemonic most commonly used to remember this sequence is
Make up your own and email it to me!
|Property||How it is determined|
|Color||-Compare brightness in two different E&M spectrum bands.
-Examine spectra of star
|Temperature||Use color or spectra|
|Distance||-Directly measured via parallax
-Indirectly measured via method of standard candles
|Luminosity||-use brightness and distance
-Eclipsing Binary: Use Temperature and Size
|Radial Velocity||Doppler shift of spectral lines|
|Transverse Velocity||proper motion and distance|
|Rate of Spin||Doppler broadening of Spectral Lines|
|Size||-Luminosity and Temperature
-directly measure in an Eclipsing Binary
|Mass||-Use Kepler's 3rd Law in Binary System.
-Infer from Luminosity and Temperature.
-Infer from spectral lines.
|Chemical Composition||Spectral Lines|
|Strenth of Magnetic Field||Spectral Lines|
|Age||Main-Sequence Turn-off Point in a cluster
Now we'd like to understand what it all means. We need to examine the raw data in some meaningful way to see if any patterns emerge. Then we'll be able to make hypotheses and test them out with further observations.
In science, the concept of a plot or diagram is a very powerful one. It allows you to visually represent physical characteristics and look for qualitative and quantitative relationships between them.
In 1912 a pair of astronomers (by the names of Hurtzprung and Russell) independently devised a plot of some of the characteristics of a large number of stars. They plotted spectral class vs. luminosity of a large sample of stars and the resulting pattern was most interesting. The stars did not randomly scatter about the plot but rather fell into several well defined zones.
Main SequenceMost stars fall on a slightly curvy diagonal path (top left to bottom right). We call this the Main Sequence. The Sun is a Main Sequence, G star. We usually plot the Luminosity of stars in units of the Sun's luminosity (L = 3.9 x 1026 Joules/s). The plot spans a rather large range in luminosity. We see that there are stars with luminosities of only 10-4L and there are stars with huge luminosities of as much as 106L.
This leads us to consider what is different about these stars such that they have the same surface temperatures as stars on the main sequence but have different luminosities. The answer comes from the Stefan-Boltzmann Law:
Hence, we call the stars in the upper right Red Giants and the stars in the lower left White Dwarfs.
Further analysis also finds that stars tend to have larger radii on the hotter (blue) end of the main sequence and smaller radii on the cooler (red) end of the main sequence.
This relationship is only true for Main Sequence stars. There are also trends with the red giants and white dwarfs but they are not as simple. Generally, more luminous Red Giants are more massive.
The Sun has enough fuel to live on the Main Sequence for 10 Billion years. Stars with greater mass have more fuel to burn, but they also have much higer Luminosities; they have to in order to maintain hydrostatic equillibrium. A star with twice the mass of the Sun has a luminosity which is 16 times greater than the Sun. So despite having twice the mass to fuse it will use it up 16 times faster than the Sun. It's lifetime will therefore be 2/16 = 1/8th that of the Sun: Just over a Billion years. More massive stars live even shorter lives. A star with a mass 30 times the Sun's would live on the Main Sequence for only a few million years. This is a very short time in a Cosmic sense.
Stars with very low mass have such low luminosities that they may live for 10s - 100s of billions of years on the Main Sequence.
The H-R diagram of a cluster of Stars can be very instructive. Clusters contain stars which are all at the same distance from us. Thus differences in brightness translate directly into difference in Luminosity. This in fact is one way to measure distance, using the observed brightness of main-sequence stars in a cluster and comparing them to the known luminosities of similar main sequence stars in clusters with known distances.
The stars are also assumed to all be the same age. The theory of star formation would have that all the stars in the cluster formed from the same nebula that began to contract gravitationally.
There are two basic types of Clusters in the Galaxy: Open, which have several hundred widely spaced stars; and Globular, which have millions of stars all tightly packed into a glob.
If we make an H-R diagram of a cluster and examine only the main sequence we would see that not all stars would be represented there. The top will likely be missing some stars. Where did they go?
They died, and left the main sequence. Since the high mass stars live the shortest lives they will be the first to shuffle off the mortal coil of the main sequence.
The main sequence will then peel away from top to bottom with time. And the point that it has peeled down to allows us to determine the age of the cluster. The stars at the top of the main sequence are just about to die and turn off the main sequence. So we call that point the turn off point.
Since we know what the lifetimes of stars in that location of the main sequence are then we know how old the cluster is.
This is nicely illustrated by overplotting the Main Sequences of clusters of different ages.
Clusters with lots of hot blue stars are young. Clusters with yellow stars just turning off the main sequence are getting on in age. And those clusters with red stars turning off the main sequence are ancient.
Open clusters tend to be young, while globular clusters contain some of the oldest stars in the Universe.
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