# ASTROPHYSICS SUMMARY.

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ASTROPHYSICS SUMMARY

Constellation is a group of stars that form a pattern as seen from the Earth, but not bound by gravitation Stellar cluster is a group of stars held together by gravitation in same region of space, created roughly at the same time. Galaxy is a huge group of stars, dust, and gas held together by gravity, often containing billions of stars, measuring many light years across. Star is a massive body of gas held together by gravity, with fusion going on at its center, giving off electromagnetic radiation. There is an equilibrium between radiation/gas pressure and gravitational pressure

Stars’ and planets’ radiation spectrum is approximately the same as black-body radiation/ Plank’s law. Intensity as a function of wavelength depends upon its temperature Wien’s law: Wavelength at which the intensity of the radiation is a maximum λmax, is:

Luminosity (of a star) is the total power (total energy per second) radiated by an object (star). If we regard stars as black body, then luminosity is L = A σT4 = 4­πR2σT4 (Watts) Stefan-Boltzmann’s law A is surface area of the star, R is the radius of the star, T surface temperature (K), σ is Stefan-Boltzmann constant. (Apparent) brightness (b) is the power from the star received per square meter of the Earth’s surface b = L 4π 𝑑 (W/m2) L is luminosity of the star; d its distance from the Earth

What can we say about their relative sizes and temperatures?
Suppose I observe with my telescope two red stars A and B that are part of a binary star system. Star A is 9 times brighter than star B. What can we say about their relative sizes and temperatures? Since both are red (the same color), the spectra peak at the same wavelength. By Wien's law then they both have the same temperature. L = 4π R2 σ T4 (W) It must be that star A is bigger in size (since it is the same temperature but 9 times more luminous). How much? Star A is 9 times brighter and as they are the same distance away from Earth star A is 9 times more luminous: So, Star A is three times bigger than star B.

Thus we have for star C, and for star D By Wien's law
Suppose I observe with my telescope two stars, C and D, that form a binary star pair. ▪ Star C has a spectral peak at 350 nm - deep violet ▪ Star D has a spectral peak at 700 nm - deep red What are the temperatures of the stars? By Wien's law Thus we have for star C, and for star D

So that stars C is 4 times smaller than star D.
If both stars are equally bright (which means in this case they have equal luminosities since the stars are part of a pair the same distance away), what are the relative sizes of stars C and D? So that stars C is 4 times smaller than star D.

Magnitude Scale • Magnitudes are a way of assigning a number to a star so we know how bright it is
Apparent magnitude (m) of a celestial body is a measure of its brightness as seen from Earth. The brighter the object appears the lower its apparent magnitude. Greeks ordered the stars in the sky from brightest to faintest… Later, astronomers accepted and quantified this system. • Every one step in magnitude corresponds to a factor of 2.51 change in brightness. Ex: m1 = 6 and m2 = 9, then b1 = (2.51)3 b2

Absolute magnitude (M) of a star is the apparent magnitude that a star would have if it were at distance of 10 pc from Earth. It is the true measurement of a star’s brightness seen from a set distance. m – M = 5 log d 10 m – apparent magnitude M – absolute magnitude of the star d – its distance from the Earth measured in parsecs. • If two stars have the same absolute magnitude but different apparent magnitude they would have the same brightness if they were both at distance of 10 pc from Earth, so we conclude they have the same luminosity, but are at different distances from Earth !!!!!!!!!!!!!! • Every one step in absolute magnitude corresponds to a factor of 2.51 change in luminosity. Ex: M1 = – 2 and M2 = 5, then L1 / L2 = (2.51)7

Binary star is a stellar system consisting of two stars orbiting around their common center of mass.
The ONLY way to find mass of the stars is when they are the part of binary stars. Knowing the period of the binary and the separation of the stars the total mass of the binary system can be calculated (not here).

Visual binary: a system of stars that can be seen as two separate stars with a telescope and sometimes with the unaided eye They are sufficiently close to Earth and the stars are well enough separated. Sirius A, brightest star in the night sky and its companion first white dwarf star to be discovered Sirius B.

Spectroscopic binary: A binary-star system which from Earth appears as a single star, but whose light spectrum (spectral lines) shows periodic splitting and shifting of spectral lines due to Doppler effect as two stars orbit one another.

Eclipsing binary: (Rare) binary-star system in which the two stars are too close to be seen separately but is aligned in such a way that from Earth we periodically observe changes in brightness as each star successively passes in front of the other, that is, eclipses the other

The Hertzsprung–Russell (H – R) diagram(family portrait) is a scatter graph of stars showing the relationship between the stars' absolute magnitudes / luminosities versus their spectral types(color) /classifications or surface temperature. It shows stars of different ages and in different stages, all at the same time. main sequence stars: fusing hydrogen into helium, the difference between them is in mass left upper corner more massive than right lower corner. white dwarf compared to a main sequence star: • has smaller radius • more dense • higher surface temperature • energy not produced by nuclear fusion LQ = sun luminosity = × 1026 W

Techniques for determining stellar distances:
stellar parallax, spectroscopic parallax and Cepheid variables.

Stellar parallax • two apparent positions of a close star with respect to position of distant stars as seen by an observer from two widely separated points are compared and recorded; • the maximum angular variation from the mean, p, is recorded; • the distance (in parsecs) can be calculated using geometry tan p = Sun−earth distance Sun−star distance = 1 AU d for small angles: tan θ ≈ sin θ ≈ θ (in radians) d = 1 AU p if p = 1 sec of arc, d = 3.08x1016 m defined as 1 pc d (parsecs) = 1 p(arcseconds) limit because of small parallaxes: d ≤ 100 pc

Spectroscopic parallax: no parallax at all
Spectroscopic parallax: no parallax at all!!!! (a lot of uncertainty in calculations) • light from star analyzed (relative amplitudes of the absorption spectrum lines) to give indication of stellar class/temperature • HR diagram used to estimate the luminosity • distance away calculated from apparent brightness limit: d ≤ 10 Mpc Spectroscopic parallax is only accurate enough to measure stellar distances of up to about 10 Mpc. This is because a star has to be sufficiently bright to be able to measure the spectrum, which can be obscured by matter between the star and the observer. Even once the spectrum is measured and the star is classified according to its spectral type there can still be uncertainty in determining its luminosity, and this uncertainty increases as the stellar distance increases. This is because one spectral type can correspond to different types of stars and these will have different luminosities.

1 - Spica • Apparent magnitude, m = 0.98 • Spectral type is B1 • From H-R diagram this indicates an absolute magnitude, M, in the range: -3.2 to -5.0 m – M = 5 log d 10 d = 10 (m-M+5)/5 M= -3.2, d = 10 ( (-3.2) +5)/5 = pc M= -5.0, d = 10 ( (-5.0) +5)/5 = pc The Hipparcos measurements give d = pc

2 - Tau Ceti • Apparent magnitude, m = 3.49 • Spectral type is G2 • From H-R diagram this indicates an absolute magnitude, M, in the range: +5.0 to +6.5 m – M = 5 log d 10 d = 10 (m-M+5)/5 M= +5.0, d = 10 ( )/5 = 5.00 pc M= +6.5, d = 10 ( )/5 = 2.50 pc The Hipparcos measurements give d = 3.64 pc

So, to find out how far away Cepheid is:
Cepheid variables are stars with regular variation in absolute magnitude (luminosity) (rapid brightening, gradual dimming) which is caused by periodic expansion and contraction of outer surface (brighter as it expands). This is to do with the balance between the nuclear and gravitational forces within the star. In most stars these forces are balanced over long periods but in Cepheid variables they seem to take turns, a bit like a mass bouncing up and down on a spring. Left: graph shows how the apparent magnitude (the brightness) changes, getting brighter and dimmer again with a fixed, measurable period for a particular Cepheid variable. There is a clear relationship between the period of a Cepheid variable and its absolute magnitude. The greater the period then the greater the maximum luminosity of the star. Cepheids typically vary in brightness over a period of about 7 days. Left is general luminosity – period graph. So, to find out how far away Cepheid is: • Measure brightness to get period • Use graph absolute magnitude M vs. period to find absolute magnitude M • Measure maximum brightness • Calculate d from b = L/4πd2 • Distances to galaxies are then known if the Cepheid can be ascertained to be within a specific galaxy.

Thank you Francis