# 1. Distance

### Parallax

• It is an object's apparent shift relative to more distance background as the observer's point of view (defined as baseline) changes.
• They are very small for even the closest stars.
• Thus, they are in arc seconds rather than degrees. ### Parsec

• Parsec = Parallax in arcseconds
• Earth is at 1 AU (150 million km) away from Sun creating a baseline (see picture).
• If an object supstends 1 arcsecond from this baseline

then the distance becomes 1 parsec (= 206265 AU = 3.1 x 1016 km). ### Nearest Neighbors

• Let's create a new scale:
• Sun - Earth = 1 m
• Then, end of Solar System = 50 m
• Proxima Centauri (alpha Centauri)
• p = 0.76 " (the largest) = 1.3 pc = 270 000 AU = 4.3 ly
• In new scale: 270 km
• Barnard's Star (next nearest)
• p = 0.55 " = 1.8 pc = 6.0 ly
• In new scale: 370 km  ### Stellar Motion

• True Motion has two components
• Transverse (location changes) - across the sky
• Radial (using Doppler shift) - along the line of sight

### Proper Motion

• Annual motion of a star across sky as seen from Earth; the value then corrected for parallax
• Take the transverse component
• Express the value in arcseconds per year = " / yr
• Barnard Star: moved 227 " during 22 years = 10.3 " / yr (largest) # 2. Luminosity

• Energy of a light source can be measured in two ways
• Apparent Brightness: at a certain distance from the source - contains distance
• Luminosity: right when the energy released into space (total energy) - distance is zero
• Luminosity = Absolute Brightness = Total Energy - Intrinsic to the source
• Apparent Brightness = Energy Flux striking a unit area at a distance

### Inverse Square Law So, to determine L, first measure object's apparent brightness, then d can be calculated ### Magnitude Scale - Apparent Magnitude

• It is related to our perception.
• It is a logarithmic scale:
• A change of 5 in magnitude
• corresponds to a change of a factor of 100 in apparent brightness.
• It is inverted:
• Larger magnitudes are dimmer.
• The original magnitude scale was defined so that
• the brightest stars in the night sky have magnitude 1,
• whereas the faintest stars visible to the naked eye have magnitude 6.
• Now, an increase of 1 in apparent magnitude corresponds to a decrease in apparent brightness by a factor of approximately 2.5.

### Therefore

• Observe apparent brightness (e.g in magnitude scale)
• Know / Measure the distance
• Calculate Luminosity = Total Energy # 3. Stellar Temperatures

• The color of a star is indicative of its temperature.
• Red stars are relatively cool,
• while blue ones are hotter.  • The blackbody curve is not symmetric.
• Observations at two wavelengths are enough to define the temperature

• Star (a) is very hot— 10,000 K—so its B intensity is greater than its V intensity.
• Star (b) has roughly equal B and V readings and so appears white. Its temperature is about 10,000 K.
• Star (c) is red; its V intensity greatly exceeds the B value, and its temperature is 3000 K ### Color Index

• CI = mB – mV
• CI = mB / mV
• Difference or ratio; both shows a comparison of two observations at two different wavelengths. ### Spectral Classification

• Colors ➤ Classification
• But we need spectral lines for more detailed classification
• Differences in star's spectra means differences in Temperature; not in composition
• But stars show "similar behaviors" at certain temperatures.
• Example: T > 25 000 K
• Strong absorbtion lines: He (singly ionized), O,N (multiply ionized)
• Because these can only occur in hot stars (large temperatures ➤ ionization)
• However, weak H absorbtion lines: because not much H left ➤ mostly ionized

Classification

• Spectral lines: mainly H line intensity (the early order)

➤ A, B, C ... (strongest to weakest)

• Then, they used surface temperature, Teff (order is shuffled)

➤ O, B, A, F, G, K, M (dropped the others)

• These are called spectral classes or spectral types
• Then each class subdivided into 10 parts designated by 0-9 (hotter to cooler)  # 4. Stellar Sizes

### Direct Geometry

• Measuring angular size of the observable disc i.e diameter of the object
• This can be done by interferometry
• This method gives hight resolution map of star's rotation

### Indirect Method

• Radiation ➤ Stefan – Boltzmann Law
• Flux : energy emitted /per/ unit area /per/ unit time

### F ∝ T4

• So, Luminosity ∝ surface area x T4

### L ∝ R2 x T4

• Large body radiate more energy than do small bodies having the same temperature • Mira:
• Tsurf 3000 K = T / 2
• L 1.6 x 1029 W = 400 L ➤ R = 80 R
• Mira is called a giant (10 - 100 R)
• Sirius B
• Tsurf 24 000 K = 4 T
• L 1025 W = 0.04 L ➤ R = 0.01 R
• Mira is called a dwarf (< R)
• Since 24 000 K object glows white ➤ white dwarf # 5. Hertzsprung - Russell Diagram (H-R)

A plot of luminosity against surface temperature (or spectral class), known as an H–R diagram, is a useful way to compare stars.

### H–R Diagram of Well-Known Stars

Plotted here are the data for some stars mentioned earlier in the text.

• The Sun:
• luminosity : 1 solar unit.
• temperature : 5800 K (read off the bottom scale); that of a G-type star.
• B-type star Rigel (at top left)
• temperature : 11,000 K
• luminosity more than 10,000 times that of the Sun.
• M-type star Proxima Centauri (bottom right)
• temperature : 3000 K
• luminosity less than 1/10,000 that of the Sun. ### H–R Diagram of Nearby Stars

Most stars have properties within the long, thin, shaded region of the H–R diagram known as the main sequence (MS)

• These are the 80 closest stars to us
• The points plotted here are for stars lying within about 5 pc of the Sun.
• Each dashed diagonal line corresponds to a constant stellar radius, so that stellar size can be indicated on the same diagram as stellar luminosity and temperature.
• The most stars are in the main sequence.
• White dwarf region: they are hot but not very luminous, as they are quite small. ### H–R Diagram of Brightest Stars

An H–R diagram for the 100 brightest stars in the sky is biased in favor of the most luminous stars

• They appear toward the upper left
• We can see them more easily than we can the faintest stars.
• These stars are all more luminous than the Sun.
• Two new categories appear here: the red giants and the blue giants.
• The brightest stars in the sky appear bright because of their enormous luminosities, not their proximity.

### In summary

• T (temperature) - increases - right to left
• L (Luminosity) - increases - bottom to top
• R (radius) - increases - bottom-left to top-right
• M (mass) - increases - along MS bottom-right to top-left ### H-R Diagram for 20 000 stars

• The main sequence is clear
• The red giant region is clear
• About 90% of stars lie on the main sequence
• 9% are red giants
• 1% are white dwarfs. # 6. Extending the Cosmic Distance Scale

### Spectroscopic Parallax

A measurement of the apparent brightness of a light source combined with some knowledge of its intrinsic properties can yield an estimate of the source's distance.

• Since most of the stars are in MS and they show L & T correlation, MS can tell us about average properties of most stars
• Get the spectrum
• Find the temperature (T)
• Use H-R and find corresponding luminosity (L)
• Use L to calculate the flux (F)
• Flux then can be related to distance (d) since it is 1 / d2 ### Luminosity Class

• The spectroscopic parallax calculation can be misleading if the star is not on the main sequence.
• The width of spectral lines can be used to define luminosity classes.
• The denser atmosphere of a main-sequence K-type star has broader lines (c) than a giant star of the same spectral class (b).

In this way, giants and supergiants can be distinguished from main-sequence stars:   # 7. Stellar Masses

• Mass determines a star's position on the MS (composition from spectroscopy fine tunes the position)
• Mass of an object can only be determined by observing its gravitational influence on some nearby body ie. Newton's Law ➤ Mass
• Many stars are in binary pairs:
• Measurement of their orbital motion allows determination of the masses of the stars.  Visual binaries can be measured directly. This is Kruger 60. Spectroscopic Binary

Properties of binary stars can be determined indirectly by measuring the periodic Doppler shift of one star relative to the other as they move in their orbits.

• The spectra diagrammed here show a so-called single-line system, in which only one spectrum (from the brighter component) is visible.
• An observer of the binary stars at right would see the spectrum redshifted for the visible component moving away (as at top right).
• Likewise, the spectrum is blueshifted for motion toward the observer (as at bottom right).

Eclipsing Binary

If the two stars in a binary-star system happen to eclipse one another, additional information on their radii and masses can be obtained by observing the periodic decrease in starlight as one star passes in front of the other. # 8. Other Properties Distribution of stellar masses. The more massive stars are much rarer than the least massive Mass Dependence

R-M : directly proportional.

L-M : L is much faster

Life time

• available fuel / fuel comsumption ∝ life-time = M / L
• due-to L-M relation (L ∝ M4 )
• life-time ∝ 1 / M3

# 9. Summary 