Measuring the Stars. The measurement of distances The family of distance-measurement techniques used by astronomers to chart the universe is called
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1 Measuring the Stars How to measure: Distance Stellar motion Luminosity Temperature Size Evolutionary stage (H-R diagram) Cosmic distances Mass The measurement of distances The family of distance-measurement techniques used by astronomers to chart the universe is called For nearby stars we use Stellar Parallax method: The apparent displacement of foreground object relative to a distant background as the observer s position changes is called parallax. The distance at which parallax is 1", is defined as 1 parsec. 1 parsec = 3.3 light yrs = 206,265.U. The Solar Neighborhood Motions of the Stars To us, stars appear fixed in the sky. ut in reality, they are always in constant motion. Stellar motion can be divided into two components: (described by proper motion & distance) To calculate true space motion we have to combine : Plot of 30 closest stars to the Sun within 4 pc of the Earth. Stellar motions are an important tool for studying the structure of our home galaxy, the Milky Way. 1
2 Proper Motion It is the annual motion of the star across the sky perpendicular to our line of site. To measure this movement, photos of the sky taken 20 to 50 years apart are compared. No proper motion. (It means distance is only a part of this effect.) Radial Velocity The radial velocity of a star is how fast it is moving directly towards or away from us. It is measured using the Doppler Shift of the star's spectrum. The Doppler Effect e.g Sound - Pitch of train whistle drops as it passes - this also happens with light. n observed wavelength or frequency will differ from the emitted (real) one if there is a relative motion between the emitter and the observer. More distant stars tend to have smaller Proper Motions. Difficult to measure beyond 1000 pc from the Earth. Typical proper motion is ~0.1 /year (only few stars have greater than 1 /yr). Largest: /yr (arnard's Star). In all cases, the Radial Velocity is Independent of Distance. True Space Motion The quantity we really interested in is the true motion of the star through space in 3-dimensions. Radial Velocity (v r ): We measure this using the Doppler Shift of stars spectrum. Transverse Velocity (v t ): We measure this from star s Proper Motion and Distance. Formula is: Transverse Velocity True Velocity Radial Velocity Case Study: Proper motions in the ig Dipper Due to the proper motions of the stars that make up this familiar constellation, its shape changes slowly over time. It takes many thousands of years, however, for the effects to be visible to the naked eye. We can now use the Pythagorean Theorem to derive the True Space Velocity (v): [Credit: R. Pogge, OSU, 2
3 Luminosity and pparent rightness Luminosity: It is the intrinsic property of the star, so, is independent of distance and location. It is the measure of total energy output the star. Measured in Power Units: Energy/second emitted by the object (e.g., Watts or ergs/sec or joules/sec etc.) It is sometimes also referred as absolute brightness. Inverse Square Law of rightness The pparent rightness () of a source is inversely proportional to the square of its distance (d): The pparent rightness is also directly proportional to the luminosity: pparent brightness: It is the measure of how bright an object appears to be as seen by a distant observer (i.e brightness that we see here on the Earth). It depends on the distance to the object and hence it is not the measure of total energy but of energy flux. Measured in Flux Units: Energy/second/area received from the object. [Credit: R. Pogge, OSU] In Words: 2-times Farther = 4-times Fainter 3-times Farther = 9-times Fainter The apparent brightness of a star depends upon: How far away it is (Distance). How bright it really is (Luminosity). It means, luminosity, i.e. actually energy output can be measured by knowing apparent brightness and distance. Measuring pparent rightness The process of measuring the apparent brightness of objects is called Photometry. Two ways to express apparent brightness are: 1. Stellar Magnitudes 2. bsolute Fluxes (energy per second per area) Magnitude System Traditional system: Dating to classical times (Hipparchus of Rhodes, c. 300C). Rank stars into 1st, 2nd, 3rd, etc. magnitude. 1st magnitude are brightest stars, 2nd magnitude are next brightest, and so on... The faintest stars visible to the naked eye are 6th magnitude. Modern System: Modern version quantifies magnitudes as: 5 steps of magnitude = factor of 100 in Flux. Examples: 10th mag star is 100x fainter than a 5th mag star. 20th mag star is 10,000x fainter than a 10th mag star. Computationally convenient, but somewhat obtusely defined (it is backwards: larger magnitudes = fainter stars). [Credit: R. Pogge, OSU] 3
4 Measurement of Luminosity or (bsolute rightness): star s luminosity (absolute magnitude) is its apparent magnitude when it is placed at a distance of 10 pc from the observer. Examples: Sun s apparent magnitude is absolute magnitude is Polaris s apparent magnitude is 2.5 absolute magnitude is Temperature: Temperature of the Stars Intensity: amount or strength of radiation. This is a basic property of radiation. The blackbody curve shifts toward higher frequencies and greater intensities as the temperature increases but the overall shape of the curve doesn t change. Electro-magnetic Spectrum Radiation laws Wien s law wavelength of 1 peak emission temperature Stenfan s law total energy emission temp 4 From: "stronomy! rief Edition," J.. Kaler, ddison- Wesley,
5 Color and the lackbody Curve Wien s law and blackbody curve will give us temperature. Why blue stars are hotter? Shorter wavelength (or higher frequency) Why red stars are cooler? Longer wavelength (or lower frequency) Stellar Spectra Detail analysis indicated that the seven star shown have very similar element abundances more or less as of Sun. So, the differences are only due to the different temperatures. ccording to the spectrum stars are classified into following categories (also called temperature sequence): It means: If (blue) filter picks up more radiation, star is hotter. If V (visual) filter picks up more radiation, star is cooler. (pparent magnitude in apparent magnitude in V) is called color index. (See table 17.1, it is important) Spectra of different stars arranged in order of decreasing surface temperature. Scientists used to classify stars based on their hydrogen line intensities, but now they classify stellar spectra based on temperature. O F G K M L T HIGHER TEMP. LOWER TEMP. 30,000 K 20,000 K 10,000 K 7,000 K 6,000 K 4,000 K 3,000 K K < 1300 K (Failed star) Spectral Classification Important: see table 17.2 of your text. stronomers further subdivide each spectral classification into 10 subdivisions. e.g Sun is G2 star, arnard star is M5. G1 G2 G3 G4 G5 G10 Hotter star cooler star (so on for other classes too) Stellar Sizes From Stefan s law we know that: energy emitted per unit time per unit area is directly proportional to the fourth power of surface temperature. Energy/sec/area Temperature 4 Or Energy/sec rea x Temperature 4 nd we know, area of a sphere is 4 R 2, so: This is called radius-luminosity-temperature relationship. 5
6 Example 1: Two stars are the same size, (R =R ), but star is 2x hotter than star (T =2T ): Therefore: Comparison of Sizes of some well know stars In words: "If two stars are the same size, the hotter star is brighter." Example 2: Two stars have the same effective temperature, (T =T ), but star is 2x bigger than star (R =2R ): Therefore: In words: "If two stars have the same effective temperature, the larger star is brighter." (ref: R.W. Pogge (OSU)) Star comparable in size to or smaller than the Sun Up to 100 times larger than the Sun More than 100 times larger than the Sun We can estimate the star s: the effective temperature, T, from its spectral type or black body curve. the luminosity, L, from its apparent brightness & distance. the size or radius, R, from radius-luminosity-temperature relationship. Is there any correlation between temperature and luminosity? Yes, and this correlation was first studied, independently in 1912 by a Danish astronomer Eljnar Hertzsprung and an merican astronomer Henry Norris Russell for nearby stars. Now it is know as HR diagram which is the plot between temperature from hottest to coolest along the horizontal axis, and luminosity from faintest to brightest along the vertical axis. The HR Diagram (lso called Color-Magnitude Diagram) Credit: R.W. Pogge (OSU) 6
7 Main Sequence 90% stars, including the Sun, lie along a diagonal the band in the H-R Diagram called the Main Sequence (MS) Stars. Most of them are (K and M stars with low luminosity) few are (Hot, luminous stars) very few are (Very hot and very luminous stars) lue Giants Red Dwarfs. Credit: D. E. Soper ( White dwarfs and Red Giants Credit: White Dwarfs (These are stars that are much fainter and smaller than Main Sequence stars of the same temperature.) Red Giants (These are stars that are much brighter and bigger than MS stars of the same temperature.) HR diagram Credit: Spectroscopic Parallax This is another method of distance determination, and it is valid up to several thousand parsecs from Earth. For the main sequence stars, luminosity can be read directly from the HR diagram if we know temperature. This whole process of using the stellar spectrum to infer distances by first determining temperature and then luminosity is called stellar parallax. Limitation: valid only for main-sequence stars. 7
8 Luminosity Class Width of the lines in the spectrum gives us the information about the density of stars. So, by studying line width we can distinguish the main sequence stars from other types of stars like giants or dwarfs etc. This categorization is known Luminosity Class. Ia Ib II III IV V right supergiants Supergiants right giants Giants Subgiants Dwarfs (MS stars) The full spectral classification of a star tells us the approximate location of the star on the H-R Diagram: The Sun: G2v etelgeuse: M2Ib Rigel: 8I Sirius: 1v (1 Main-Sequence star) ldebaran: K5III (K5 Giant star) Stellar Masses Masses of stars are measured by studying the gravitational effect of inary stars on each other inary Stars pparent inary Stars: Chance projection of two distinct stars along the line of sight. Often at very different distances. True inary Stars: pair of stars bound together by gravity. Orbit each other about their center of mass. etween 20% and 80% of all stars are binaries. Types of inaries Mass determination by Visual inaries Can see both stars and follow their orbits over time. Cannot separate the two stars, but see their orbit motions as Doppler shifts in their spectral lines. Can separate the stars, but the orbit is oriented in such a way that the total brightness drop when they periodically eclipse each other. Two stars orbiting about their center-of-mass. Procedure: 1) Measure the period, P, by following the orbit. 2) Measure semi-major axis, a, and the Mass Ratio, M 1 /M 2, from the projected orbit on the sky. 3) Use Kepler s 3 rd law and separate Masses. Limitation: We need to follow an orbit long enough to trace it out in detail: This can take decades. Need to work out the projection on the sky. Semi-major axis depends on distance. (ref: R.W. Pogge (OSU)) 8
9 Mass determination by Spectroscopic inaries Most binaries are too far away to be able to see both stars separately. ut, you can detect their orbital motions by the periodic Doppler shifts of the spectral lines. Determine the orbit period & size from the pattern of orbital velocities. Limitation: Cannot see the two stars separately: Semi-major axis must be guessed from the orbit motions. Can't tell how the orbit is tilted on the sky. (ref: R.W. Pogge (OSU)) Mass determination by Eclipsing inaries Two stars orbiting nearly edge-on to our line-of-sight. See a periodic drop in brightness as one star eclipses the other. From a combination of visual and eclipsing binaries, masses are known for about 150 stars. Range: ~0.1 to 50 Solar Masses Combine with spectra which measure orbital speeds. With the best data, one can find the masses of the stars without having to know the distance!!! Limitation: Eclipsing inary stars are very rare. Partial eclipses yield less accurate numbers. The atmospheres of the stars soften the edges. Close binaries can be tidally distorted. (ref: R.W. Pogge (OSU)) Stellar Masses and HR Diagram 9
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