Locating a star cluster on a Hertzsprung-Russell diagram with other color index

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Recently I took pictures of the NGC 884 cluster in the B and I bands and selected some 30-what stars from them. I've reduced the data to a colour-magnitude diagram (absolute magnitude in the I band $M_I$ vs. (B-I) colour index). I am now asked to designate where the stars I've selected from this cluster are located on the Hertzsprung-Russell diagram and the approximate age of the stars.

My problem is the following: any HR diagram that I can find is either expressed in effective temperature instead of colour index, or is in the colour index (B-V). Also I can only seem to find empirical formulas for the effective temperature that use the (B-V) colour index, not the (B-I) one.

Does anyone know where to find an HR diagram that uses the (B-I) colour index?

With an absolute magnitude of zero and a colour (any colour) of zero, your stars are around spectral type A0 (by definition). The only thing that can confuse this is reddening/extinction.

If there is appreciable reddening/extinction, then your stars could be more luminous and hotter (O/B stars), but there isn't any way to tell from your data.

Hertzsprung-Russell diagram

There are several forms of H-R diagram. In most observational applications the absolute visual magnitude, M V, is the vertical axis and the color index, BV , the horizontal axis, color index being related to but more easily measured than spectral type. These are color-magnitude H-R diagrams. When studying a cluster, whose stars are all at the same distance, apparent rather than absolute magnitude is used. Other studies use bolometric magnitude against effective temperature – theoretical H-R diagrams – or luminosity against color index – color-luminosity H-R diagrams.

The H-R diagram is of great importance in studies of stellar evolution. Diagrams obtained on the basis of theoretical calculations can be tested against observationally determined diagrams. They can be drawn for the brightest stars (see illustration), for stars in a particular locality such as the solar neighborhood (mainly small cool main-sequence stars), for pulsating variables, for globular clusters, etc. The two broad stellar populations – populations I and II – can be demonstrated by the H-R diagrams of a young open cluster (no giants), a somewhat older open cluster (a few giants), and a much older globular cluster (many giants and supergiants).

The H-R diagram can also be used for distance determination by both main-sequence fitting for stellar clusters and by spectroscopic parallax for individual main-sequence stars, the star's spectral type fixing its position on the diagram and thus indicating its absolute magnitude and hence its distance modulus.

B. PROPERTIES OF THE STARS

APPARENT BRIGHTNESSES (MAGNITUDES)

• Astronomers measure brightnesses of stars using a refined version of the magnitude system introduced by the Greek astronomer Hipparchus about 130 BC. Hipparchus ranked the visible stars from first to sixth magnitude, with first being brightest.

As a result of this definition, a 1 magnitude difference in brightness corresponds to a factor of 2.51 in flux.

INTRINSIC BRIGHTNESSES (ABSOLUTE MAGNITUDES OR LUMINOSITIES)

• The magnitude scale just discussed measures the apparent brightness of stars---i.e. how they appear from the Earth. It does not refer to their intrinsic brightnesses.

Alpha Centauri (in the southern hemisphere) is the nearest star. It is at a distance of 1.3 parsecs.

A parsec is a convenient unit for the typical distances between stars near the Sun. It is defined in terms of the size of the Earth's orbit. One parsec is 3.1 x 10 13 km(!), 206,000 times the distance to the Sun, or about 3.25 light years (one light year is the distance light travels in a year).

Since most stars lie at distances far greater than 10 parsecs, their absolute magnitudes are much brighter than their apparent magnitudes.

TEMPERATURES

• We have learned to measure the surface temperatures of stars using their electromagnetic spectra, following experiments first done by the physicist Kirchhoff in the 19th century.

Bad Philosophy Footnote: Click here for a description of one of the worst, but not one of the last, faulty prognostications about science by a philosopher, in this case the claim that we could never know the temperatures of the stars.

For background on the electromagnetic spectrum, see these notes from ASTR 1210.

(Does this fact, combined with the principle of natural biological selection, suggest a reason why our eyes are most sensitive to yellow-green light?).

MASSES

Masses of stars are measured mainly by applying Newton's laws of motion and gravitation to stars which are in orbit around one another, i.e. binary or "double" stars.

Newton showed that the time taken to complete one orbit (the "period") by any object in a gravitational orbit around another is related to the combined mass of the two objects. By measuring the orbital sizes and periods of binary stars (and also their distances from us), we can therefore determine their masses. During the 19th and early 20th centuries, small telescopes were often used for this kind of study.

You can find lists of brighter binary stars in each constellation here.

Star clusters: Anything but simple

The heated debate on the importance of stellar rotation and age spreads in massive star clusters has just become hotter by throwing stellar variability into the mix.

A quiet revolution has been sweeping the field of star-cluster astrophysics. A decade ago, we were reasonably convinced that we understood the formation and evolution of the massive, well-populated star clusters that can be used as a statistical tool for studies of stellar evolution. Groups of stars characterized by a common age and chemical composition were considered ‘simple stellar populations’, given that all of their stars had presumably formed from the same progenitor molecular gas cloud at approximately the same time. Admittedly, the oldest galactic building blocks, the globular clusters, were known to exhibit evidence of multiple stellar generations 1 , but clusters younger than a few billion years appeared to obey our simple models. Fast forward a decade, and we now know that the majority of the 1–3 billion-year-old star clusters in the nearest galaxies, the Magellanic Clouds, are anything but simple. Indeed, writing in The Astrophysical Journal Letters, Ricardo Salinas and co-workers show that a significant population of pulsating stars can have a measurable effect on our interpretation of stellar evolution within such clusters 2 .

Locating a star cluster on a Hertzsprung-Russell diagram with other color index - Astronomy

Astronomy - Online Labs

Websites suitable for Astronomy labs K0 Constellations CLEA Liftoff J-Track EAAE/ESO/EU Event Stellar Parallax K1 Planetary motion (Jupiter) A View of the Moon Hipparcos: Education Resources Chemistry Wisconsin Spectroscopy (link doesn't work) Variable Star - AAVSO K2 CLEA Photometry and Spectroscopy labs Solar images ZEBU Virtual Lab UCSB RAAP Labs for the PC Noon Observation Project K3 CLEA - Mercury Results MapBlast DLR Solar Spectrum K4 Proper Motion Binary Stars & Evolution animations Fermilab [email protected] K5 Ages of Star clusters

Lab K0 CONSTELLATIONS.

Compare photos to maps of the constellations.
On star maps, you see only the brightest stars of a constellation. This is deceiving, because the stars might be rather faint in reality - e.g. Ophiuchus and Ursa Minor, or smaller than you think - Triangulum and Delphinus. Therefore you should become familiar with the constellations how they really look like in a sky with little light pollution.

The photos are not labeled, but instead lettered from A to S.

Lab K1 PLANETARY MOTION

Objective: This lab shows that (a) planet(s) "wander"(s) across the sky (the sky-background is comprised of the stars), whose motion is apparent over the course of some weeks/months. Stars, however, have always the same positions with respect to each other (excluding Earth's parallax and precession and the star's proper motion, which are of no concern to casual observing as we do it).

Planets: Jupiter and Saturn (for summer 2000 until spring 2001), Jupiter (for fall 96), Mars (for spring, summer and fall 97). This lab usually done for Jupiter in Taurus in 2000/01 only.

When comparing the light of planets with that of stars, what is an obvious difference? How is this difference produced?

Note: please do not print the photos! It would be a waste of ink and the resulting print will be terrible anyway. Do these online labs with the pictures on the screen.

Jupiter and Saturn in Taurus in 2000/01 July_28,_Aug_6, 11 Aug_20,_Sep_3, 24 Oct_20, 29,_Nov_15 Nov_24,_Dec_6,_Jan_3 Jan_14, 24,_Feb_12 Feb_18, 22,_March_16

Each link to Jupiter contains up to 5 images.
J U P I T E R
May - Aug 96 Aug - Nov 96 November 96

Each link to Mars contains up to 10 images.
M A R S
Aug - Sep 96 Nov 96 - Jan 97 February 97 March 97 March 97
April 97 May 97 June 97 July 97 August 97
September 97 October 97

The software uses these telescopes:

The latter two are at Kitt Peak National Observatories (20 miles west of Tucson, AZ).

CLEA (Contemporary Lab Exercises in Astronomy - Gettysburg College, PA, and National Science Foundation) provides several computer labs for students. Two of them let you analyze starlight.

With Photometry you determine the apparent brightness of stars, with Stellar Spectra you determine their spectral type.

CLEA provides the starfield of the Pleiades cluster, for which plenty of information is already available, most importantly an HR-diagram with data from former students.

All stars in the Pleiades are at approximately the same distance (give or take a few lightyears - insignificant at 400 lightyears) from our solar system. This assumption is justified for several reasons. First, they are all clustered, i.e. lots of fairly bright stars in a very small region. Second, positional measurements over years showed that they all move towards the same vantage point (we won't do this exercise - too little time). Third, the labs that you do will show that all stars lie on a line (the Main Sequence) in the HR-diagram - even by plotting apparent magnitude on the y-axis - another very good indication for the same distance. Fourth, all open clusters (and of course globular clusters) show similar patterns in their HR-diagrams when plotting apparent magnitude. Fifth, these nice patterns also strongly suggest that stars in a cluster formed at the same time. If that wasn't the case (i.e. not same distance nor same age) we'd find stars scattered all over the HR-diagram. But we don't . so the assumptions same distance and same age must be correct.

Photometry gives you the apparent brightness. With the HR-diagram (and another calibrated one, e.g. Hipparcos), you'll be able to determine the Pleiades' absolute brightness and with these two, you get the distance (fairly accurate). We are also able to determine the age of the Pleiades (ballpark figure).

We also determine the star's color index B-V, i.e. by how much the brightnesses in the Blue and Visual (yellow) differ, which gives us the spectral type as well (see last table in my Stellar Evolution Appendix).

Stellar Spectra gives you the spectral type as well (B-V and spectral analysis should confirm each other). This in turn - with your knowledge about absorption lines - reveals a star's surface temperature (see first table in my Stellar Evolution Appendix), and perhaps a Doppler-shift, rate of star rotation, strength of magnetic field, chemical composition of its atmosphere (these we won't do).

Surface temperature and absolute brightness give us the size (radius) of each star (Stefan-Boltzmann law). The luminosity-mass relationship (for main sequence stars only) give us a star's mass.

And a good astrophysics book on stellar structure and evolution (e.g. by R. Kippenhahn and A. Weigert) would give us the star's structure, core temperature, mode of nuclear fusion, life expectancy, evolution, etc. etc.

Color index B-V is the brightness comparison between Blue and Visual (yellow):

(mB - mV) = - 2.5 log ( count_B / count_V ) .

Apparent magnitude m is the apparent brightness (as seen from Earth) measured in magnitudes:

Magnitude-Brightness relation (m1 - m2) = - 2.5 log ( count_1 / count_2 ) , compared to another star (that's Alcyone in our case).

HR-diagram , i.e. Hertzsprung-Russell diagram, plots absolute magnitude versus temperature (or color index, or spectral type). In our case, we plot apparent magnitude versus color index.

Do this together (2 or 3 people). It'll make this much easier.

Access CLEA's Classification of Stellar Spectra.
- Run Take Spectra
- Click Dome to open it
- Click Tracking , so that the telescope follows the star's apparent movement
- these are not the Pleiades, so click on Field and choose the Pleiades

- Later on you will take data of a faint star. You may want to use a telescope with a larger aperture. So click on Telescopes, Request Time, then 4.0 m Mayall. Most likely you won't get it (but it's worth a shot). Then try Telescopes, Request Time, then 1.0 m KPNO. If you're lucky, click on Telescopes, Access .

- Play a little around
- Click and hold down on E, N, W, or S
- click on slew rate and it changes from 4 to 8, 16, 1, 2, back to 4, with that you move faster or slower across your starfield

- Do a test on Electra (type in its coordinates: 3h 41m 56s, 23d 57' 55"), the second brightest star.
- Center Electra in the red rectangle, which shows you the size of the monitor
- Click on Monitor
- Move Electra in between the red vertical bars
- Take Reading, click on Start/Resume Count
- After a short time (10 - 20 sec.) you notice that the spectrum doesn't change its shape anymore (if you get totally useless information, you probably haven't put the star in between the lines - so Return (do not save)
- Stop Count
- Save as Ele
(- write down Electra's HD number)
- Note that the spectrometer measures between 3900 Å and 4500 Å, i.e. in the far-violet part of the spectrum only
- Return (write down the coordinates) and Run , then Classify Spectra
- Load Unknown Spectrum Saved Spectra (*.CSP) Electra's spectrum
- Config Display Grayscale "Photo"
- Load Atlas of Standard Spectra Main Sequence (minimize this MS window to keep the screen from overcrowding)

- Now you can analyze Electra's spectrum
- You can Load Spectral Line Table (but don't have to - notice that your screen gets crowded)
- click and hold while moving around in the spectrum shows you which elements produce spectral lines --- compare to spectral line diagrams in your book (after playing with that, minimize this Spectral Line Identification window)
- Click Down (or Up) and compare your spectrum (in the middle) with the main sequence spectra
- change back to Config Display Intensity Trace , which works better because now you can click on Difference : the closer the wiggly red line is to a horizontal the better is the match.

- determine Electra's spectral type
- I determined B 2

Also, write down Electra's HD # as the star's name (HD # should appear during measurement and classification).

Access CLEA's Photometry.
- Start
- Click Dome to open it (you see the Pleiades)
- Click Tracking , so that the telescope follows the stars' apparent movement

- Continue the test on Electra
- Find and center Electra in the red rectangle (or use its coordinates), which shows you the size of the monitor
- Click on Monitor
- Move Electra into the red circle (the size of the photometer)
- Note that the Filter is set on V (visual, which measures primarily yellow)
- Click on Take Reading , which automatically takes 3 readings for 10 sec. each (therefore you wait for 30 seconds). Take reading means that the photometer counts the number of photons received from Electra.
- Write down Mean/Sec , which is the average # of photons per sec. (not per 10 sec.)
- Compare this result to V = 2,600,000 (my reading from 3-13-97)
- Change the Filter to B (blue)
- Take Reading , write down Mean/Sec , compare to B = 2,877,000 (A. Veh, 3-13-97)

Do not do the following on your own as I don't want you get frustrated. Leave this analysis to me.
- use the color index B-V relation to determine (mB - mV) and compare to my (mB - mV) = - 0.11 mag
- with your B-V and (see Lecture notes, Stellar Evolution, Appendix) determine Spectral Type and Surface Temperature
- compare to my results B 8 and T = 12,000 K
- use Electra's and Alcyone's V-counts and the Magnitude-Brightness relation to determine the magnitude difference (m1 - m2)
- with Electra's V1 = 2,600,000 and Alcyone's V2 = 5,636,000 you get (m1 - m2) = 0.8 mag- add this to Alcyone's mV = 3.0 mag and you get an apparent magnitude of mV = 3.8 mag for Electra
End of "do not do this on you own."

Now that you're familiar with this software, here are your actual measurements.

- toggle back to Stellar Spectra
- get Back to the telescope window (you need to leave the classification window)
- get the telescope view of all of the Pleiades (maybe you need to leave Monitor ).
- choose a fairly faint star (the bright stars are most likely taken up) !
- Take Reading , then click on Start/Resume Count
- after a short while, the pattern of the spectrum will become apparent - time to stop. If this is not happening, it means that the signal/noise ratio is very low, and you better get a larger telescope or move to a slightly brighter star than this one.
- follow the above steps (the ones you did for Electra), take spectrum, classify, measure its V and B in photometry
- don't forget to write down the coordinates as well

After that turn in your work. Your data will add one more star to this HR-diagram (April 12, 1998).

Here is a routine for the TI-83 to graph the UBV data, fit a Black-Body curve to it and determine the correct temperature.

3 STO> dim( L IN)
<365,440,550>STO>L NM
6.26 E -34 STO> H:1.38 E -23 STO> B:3 E -8 STO> C
For(J,1,3)
Disp "UBV":Input U:U STO>L IN(J)
End
L IN/ L IN(3) STO>L IN
Repeat 5>7
Disp "TEMPERATURE":Input T
550^3.5*(e^(H*C/550*10^(9)/(B*T))-1) STO> K
"K/X^3.5/(e^(H*C/X*10^(9)/(B*T))-1)" STO> Y 1
DispGraph:Pause :End

Documentation: execute the routine exit it right away (this was just for defining the lists IN and NM [intensity and nanometers]) insert these lists as x and y in a StatPlot change the window settings to x: 200, 700 and y: 0, 2 now start the routine it asks you for UBV three times (type in the measured UBV from CLEA photometry in that order) the three data are scaled, so that L IN(3)="V"=1 all the time and the V datapoint will also always appear on the Black-Body curve supply an estimated temperature the curve and data are plotted and you judge how good the curve hit enter to try a new temperature (you're in an endless loop - exit with ON ).

PS My exponent for the wavelength in the Planck curve is 3.5 because that fits most data (in order to achieve the correct temperature when checking the B-V in a table). Lab K3 CLEA - Mercury Results

Mercury - 1998  DATE t [min] D f [Hz] sep. [deg] R [mill. km] February 20 23.0 - 6,300 2.4 60 March 13 17.9 + 101,000 14.9 46 April 10 9.7 - 28,300 7.4 64 April 28 12.4 - 91,400 26.9 70 May 22 18.8 - 93,500 20.3 57 June 10 21.9 + 2,600 1.1 46

A radar beam is sent out from Earth which is reflected and then after some time t [min] received on Earth. Due to the moving Mercury, the frequency f = 430 MHz is Doppler-shifted by D f [Hz]. The line-of-sight velocity vo is determined via the appropriate Doppler-equation: vo = ( D f / 2 f ) c with the speed of light c = 300,000 km/s.
The orbital velocity is obtained using the geometry of Mercury's orbit with respect to Earth. For a very good account see Hoff . Only the equations shall suffice here.

Lab K4 Proper Motion of Barnard's star

All images are from the above website. Hipparcos was a satellite launched by ESA (European Space Agency). It made the most precise astrometric measurements (position and distance) of 100,000 stars and their proper motion (perpendicular to our line of sight in contrast, radial velocity along our line of sight is determined via the Doppler effect - but Hipparcos had no spectroscopic objectives).

Objective: Determine how fast Barnard's star moves.

Determine (measure, estimate and calculate) how many degrees Barnard's star moves per year. The six images are 20 years apart, starting with the year 2000, ending with 2100. During this time frame the other stars stay put. The coordinates of the star on the bottom are R.A.=175822 and Dec.=+035709 , the star at the upper right has R.A.=175605 and Dec.=+051017 . (Coordinates are in (+degrees) hours, (arc)minutes, (arc)seconds, each has two digits.)
When done compare this star field from the next century to the year 14000 (Barnard's star is long gone).
Of course, you can do the whole thing quicker by accessing ESA's Hipparcos website and extracting the coordinates of Barnard's star directly.

Lab K5 Ages of Star clusters

• examining their HR diagrams,
• determining their turn-off points (read off the B-V value on the x-axis),
• comparing that B-V value to my SEA-6 (determining the corresponding spectral type),
• crossreferencing that to SEA-5 by finally determining the MS life time for that particular spectral type which equals the age of the star cluster.

The Messier Object Index at SEDS. I found the above HR diagrams by accessing this web site first.

After reading your textbook and my "Measuring Stars" script, reason why the turn-off point gives the age of a star cluster.

Locating a star cluster on a Hertzsprung-Russell diagram with other color index - Astronomy

In order to better understand how stars are constructed, astronomers look for correlations between stellar properties. The easiest way to do this is make a plot of one intrinsic property vs. another intrinsic property. An intrinsic property is one that does not depend on the distance the star is from the Earth (e.g., temperature, mass, diameter, composition, and luminosity). By the beginning of the 20th century, astronomers understood how to measure these intrinsic properties. In 1912, two astronomers, Ejnar Hertzsprung (lived 1873--1967) and Henry Norris Russell (lived 1877--1957), independently found a surprising correlation between temperature (color) and luminosity (absolute magnitude) for 90% of the stars. These stars lie along a narrow diagonal band in the diagram called the main sequence. This plot of luminosity vs. temperature is called the Hertzsprung-Russell diagram or just H-R diagram for short.

Before this discovery astronomers thought that it was just as easy for nature to make a hot dim star as a hot luminous star or a cool luminous one or whatever other combination you want. But nature prefers to make particular kinds of stars. Understanding why enables you to determine the rules nature follows. A correlation between mass and luminosity is also seen for main sequence stars: Luminosity = Mass 3.5 in solar units.

The mass-luminosity relation for 192 stars in double-lined spectroscopic binary systems.

The hot, luminous O-type stars are more massive than the cool, dim M-type stars. The mass-luminosity relationship tells about the structure of stars and how they produce their energy. The cause of the mass-luminosity relation will be explored further in the next chapter.

The other ten percent of the stars in the H-R diagram do not follow the mass-luminosity relationship. The giant and supergiant stars are in the upper right of the diagram. These stars must be large in diameter because they are very luminous even though they are cool. They have a huge surface area over which to radiate their energy. The white dwarfs are at the opposite end in the lower left of the diagram. They must be very small in diameter (only about the diameter of the Earth) because even though they are hot, they are intrinsically dim. They have a small surface area and so the sum of the total radiated energy is small.

The H-R diagram is also called a color-magnitude diagram because the absolute magnitude is usually plotted vs. the color. The H-R diagram below is for all stars visible to the naked eye (down to apparent magnitude = +5) plus all stars within 25 parsecs. Luminous stars are easier to observe because they can be seen from great distances away but they are rarer in the galaxy. They tend to reside in the top half of the H-R diagram. Faint stars are harder to see but they are more common in the galaxy. They tend to reside in the bottom half of the H-R diagram.

Use the UNL Astronomy Education program's Hertzsprung-Russell Diagram module for another in-depth tutorial on the HR diagram via a graphical interface (link will appear in a new window).

Spectroscopic Parallax

1. Determine the star's spectral type from spectroscopy and measure the star's apparent brightness (flux).
2. Use a calibrated main sequence to get the star's luminosity. The Hyades cluster in the Taurus constellation is the standard calibrator.
3. Use the Inverse Square Law for Brightness to get the distance: unknown distance = calibrator distance × Sqrt[calibrator flux/unknown star's flux.]

How do you do that?

Distances to red giant and supergiant stars are found in a similar way but you need to investigate their spectra more closely to see if they are the very large stars you think they are. Their position in the calibrated H-R diagram is found and their apparent brightness gives you the distance. Also, this process can be used to find the distance of an entire cluster. The entire color-magnitude diagram for the cluster is compared with a calibration cluster's color-magnitude diagram. The calibration cluster is a known distance away. Some adjustments for the cluster's age and composition differences between the stars in the cluster and the calibration cluster must be made. Such fine-tuning adjustments are called main-sequence fitting''.

Open Star Clusters

The icon shows the Southern open cluster NGC 3293.
Open clusters are physically related groups of stars held together by mutual gravitational attraction. Therefore, they populate a limited region of space, typically much smaller than their distance from us, so that they are all roughly at the same distance. They are believed to originate from large cosmic gas and dust clouds (star-forming diffuse nebulae, or star-forming regions) in the Milky Way (or other parent galaxy), and to continue to orbit the galaxy within or near their parent galaxy's disk. In many clouds visible as bright diffuse nebulae, star formation still takes place at this moment, so that we can observe the formation of new young star clusters. The process of formation takes only a considerably short time compared to the lifetime of the cluster, so that all member stars are of similar age. Also, as all the stars in a cluster formed from the same diffuse nebula, they are all of similar initial chemical composition.

1. the stars in a cluster are all at about the samedistance
2. the stars have approximately the sameage
3. the stars have about the samechemical composition
4. the stars have differentmasses, ranging from about 80-100 solar masses for the most massive stars in very young clusters to less than about 0.08 solar masses.

Over 1100 open clusters are known in our Milky Way Galaxy, and this is probably only a small percentage of the total population which is probably some factor higher estimates of as many as about 100,000 Milky Way open clusters have been given.

Most open clusters have only a short life as stellar swarms. As they drift along their orbits, some of their members escape the cluster, due to velocity changes in mutual closer encounters, tidal forces in the galactic gravitational field, and encounters with field stars and interstellar clouds crossing their way. An average open cluster has spread most of its member stars along its path after several 100 million years only few of them have an age counted by billions of years. The escaped individual stars continue to orbit the Galaxy on their own as field stars: All field stars in our and the external galaxies are thought to have their origin in clusters quite probably.

The first open clusters have been known since prehistoric times: The Pleiades (M45), the Hyades and the Beehive or Praesepe (M44) are the most prominent examples, but Ptolemy had also mentioned M7 and the Coma Star Cluster (Mel 111) as early as 138 AD. First thought to be nebulae, it was Galileo who in 1609 discovered that they are composed of stars, when observing M44. As open clusters are often bright and easily observable with small telescopes, many of them have been discovered with the earliest telescopes: As seen in the list below, there are 33 in Messier's Catalog, and another 33 others were also known in summer 1782. Note that all these early known clusters belong to our Milky Way Galaxy. Note that this counting includes the star-forming nebulae, as they contain clusters of recently formed stars.

In 1767, Reverend John Michell (Michell 1767) derived that clusters were most probable physically related groups rather than chance collections of stars, by calculating that it would be extremely improbable (1/496,000) to find even one cluster like the Pleiades anywhere in the sky, not to speak of the number of then-known open clusters moreover he presumed that all or at least most then-known nebulous objects actually were composed of stars. The finding of common proper motion by Mädler for the Pleiades and other stellar groups, including the Ursa Major Moving Cluster by Richard A. Proctor (Proctor 1869), further established the physical relationship between cluster stars. Finally, spectroscopy was needed to show the common radial motion (velocity) of the cluster stars, and to show that the stars perfectly match in a Hertzsprung-Russell diagram (HRD), indicating that they all lie at roughly the same distance. The final confirmation of the roughly common distance came only from the direct measurement of parallaxes for a number of nearby clusters, both from Earth-bound observatories and from ESA's astrometric satellite Hipparcos.

BUCKNELL UNIVERSITY

This one comes (quite literally) straight from class. Even though massive stars have more hydrogen (after all, they're more massive), their interiors are so much hotter than the interiors of low mass stars that they use up their hydrogen supply much more quickly. An O star of say, 10 solar masses, will use up its hydrogen supply in only 10 million years that's a Main Sequence lifetime that's about a thousand times shorter than that of the Sun.

Another "Question to Ponder" straight out of class. Nowhere in the question does it say that either of these stars is a Main Sequence star. If they're not both Main Sequence stars, then you can't assume that they have the same luminosity. If, for example, one of these stars is a red giant and the other is a Main Sequence stars, their luminosities might differ by a factor of a thousand or more.

Since you don't know their luminosities, you can't figure out their distances from the flux measurements, and therefore you can't tell for sure which cluster is closer.

Oh my! A question right out of the Observing Lab. I thought we just had to show up at the Observatory. I didn't realize that we had to pay attention. You do. Re-read Observing Lab #2, and beware of questions based on all three Observing labs on the final.

The color index, as defined in your textbook, in class, and in Lab #6, is the difference in the apparent magnitude of a star through different colored filters. It tells you the ratio of the fluxes at two different wavelengths, and from this, you can figure out the surface temperature of a star. Take a look at Lab #6 is you're puzzled by this.

Yet another "Question to Ponder." I wonder if you should review all of these before the final. White dwarfs are basically how spheres of really dense matter, which are radiating light and therefore slowly cooling off (if they're radiating light, they're losing energy, and if they're losing energy, they're getting colder, since temperature is energy per particle). When blackbody radiators cool off they also become less luminous, so a white dwarf will move rightward (i.e., toward lower temperature), and downward (i.e., toward lower luminosity) in the Hertzsprung-Russell Diagram.

Really the only way to directly measure the mass of an object in astronomy is to watch how something else moves around it. Binary star systems provide us with that opportunity to measure the mass of some stars. Without binary star measurements, we would still be guessing at the masses of most stars, and we would have no idea about the mass-luminosity relationship for Main Sequence stars.

The whole reason that Main Sequence stars can remain stable for as long as they fuse hydrogen is that the pressure caused by the energy generation in the core resists the in ward force of gravity. When a star stops fusing, its core loses the ability to push back, and the star becomes unstable.

No fusion takes place in the centers of white dwarfs that's why they're compressed to such high density by gravity. However, in these objects, the relentless push of gravity is balanced by electron degeneracy pressure, which is essentially the fact that electrons resist being pushed too close together.

The problem with seeing all of the stars in the galaxy is that the galaxy is a dusty place. There's a lot of stuff between the stars, and this stuff attenuates starlight. The light from distant stars has to pass through a lot of this stuff, and so is attenuated so much that we can't detect it.

Infrared photons, however, can penetrate much more effectively through this obscuring material. Consequently, the infrared light from distant stars is not attenuated as much, and enough of it gets through so that we can detect even the stars on the other side of the galaxy.

The only way to detect mass is by the motion of stuff around it. This is especially true for "dark matter," which is by definition non-luminous.

Question #1: A star cluster is a group of stars located together in space. All evidence indicates that these stars formed together at about the same time . Therefore, all of the stars in any one cluster have about the same age.

We can figure out the age of the cluster by looking at the cluster's Hertzsprung-Russell Diagram. Most of the stars in any cluster will appear on the Main Sequence, but depending on the age of the cluster, some stars will have evolved past their hydrogen fusing stage and will appear off the Main Sequence. Since massive stars use up their hydrogen supplies faster than low mass stars, they will be the first to depart from the Main Sequence. As a cluster ages, stars of lower and lower mass will finish their hydrogen fusing lifetime and evolve off the Main Sequence.

Therefore, we can determine the age of a cluster by looking for the most massive star that's still on the Main Sequence. The age of the cluster must be less than the time it takes this star to exhaust its core hydrogen fuel supply.

Therefore, the cluster whose H-R Diagram is depicted below left is younger than the cluster whose H-R Diagram is below right. More time must have passed for the left-hand cluster because stars of lower mass have already evolved off the Main Sequence.

Let's first start with the prevailing view. Before Shapley's work, it was widely believed that the Sun was located somewhere near the center of our galaxy. Why? Because if you count stars in the Milky Way, you find roughly the same number of stars in every direction, and therefore you conclude that we must be near the center of the system of stars. This conclusion was incorrect, mainly because of the effects of interstellar dust. The space between the stars in our galactic disk contains a fair amount of dust, and just as you can't see the mountains across the valley on a hazy day, you can't see distant stars in our galaxy. In both cases, the intervening material just isn't transparent enough.

Shapley was able to (literally!) get around the problem of dust absorption in the galactic plane by looking slightly above and below the galactic plane. He looked at globular clusters, which are giant clusters of 100,000 stars or more. For reasons that he didn't understand (and we're only just beginning to understand even now), globular clusters are found more often above and below the galactic disk than in it. Therefore, they weren't as subject to dust absorption, and Shapley could see most of the globular clusters in the galaxy, including many of the very distant ones.

When Shapley looked at the distribution of globular clusters on the sky, he noticed that there weren't roughly equal numbers of clusters in every direction. Instead, he found a substantial concentration of clusters in the direction of the constellation Sagittarius. He reasoned that the distribution of globular clusters should be centered on the center of our galaxy, and since it wasn't centered on us, we're not the center. By calculating the distance and direction to each of the globular clusters, Shapley was able to figure out that the center of the galaxy lay 25,000-30,000 light years away in the direction of Sagittarius.

• flux = luminosity/(4 x pi x distance 2 )
• flux = (4.45 x 10 29 W)/(4 x pi x (9.29 x 10 17 m) 2 )
• = 4.1 x 10 -8 W/m 2

Problem #2, Part a): We can figure out how much energy is produced in a nuclear reaction by determining how much mass "disappears." Of course, it doesn't disappear, but it is converted into energy, and that's what we're looking for.

• initial mass = 3 x (mass of one He nucleus)
• = 3 x 6.6488 x 10 -27 kg
• = 1.995 x 10 -26 kg
• mass converted to energy = (initial mass) - (final mass)
• = 1.995 x 10 -26 kg - 1.99 x 10 -26 kg
• = 5 x 10 -29 kg
• = energy = mass x c 2
• = 5 x 10 -29 kg x (3 x 10 8 m/s) 2
• = 4.5 x 10 -12 km m 2 /s 2 , or 4.5 x 10 -12 Joules

Problem #2, Part b): From Part a), we now know how much energy is produced by one reaction if we knew how many reactions it would take to convert this mass of helium into carbon, then we'd just multiply the energy produced in one reaction by the total number of reactions to get the total energy produced.

• number of He nuclei in one solar mass = (solar mass)/(mass of one He nucleus)
• = 2.0 x 10 30 kg/ 6.6488 x 10 -27 kg
• = 3.0 x 10 56 He nuclei
• number of reactions = (number of He nuclei available)/(number of He nuclei required for one reaction)
• = 3.0 x 10 56 / 3
• = 1.0 x 10 56 reactions
• total energy produced = (number of reactions) x (energy produced per reaction)
• = (1.0 x 10 56 reactions) x (4.5 x 10 -12 Joules/reaction)
• = 4.5 x 10 44 Joules

Problem #3: In this problem, we need to understand what a rotation curve is, and we need to be able to read the correct information from the provided graph. The rotation curve is a measure of the speed of material in our galaxy as a function of distance from the galaxy's center. This material is moving because of the influence of gravity, and if the galaxy is stable (and we think it is reasonably stable), then the motion of the material must be sufficient to oppose the pull of gravity.

The plot tells us that at a distance of 6 kpc from the center of the galaxy, objects move with a speed of 225 km/s.

Locating a star cluster on a Hertzsprung-Russell diagram with other color index - Astronomy

The Hertzsprung-Russell (H-R) diagram is a standard graph in astrophysics for studying stellar
populations. It plots ( effective) surface temperature (or equivalently spectral class or some
measure of colour such as B-V ) along the bottom axis and luminosity (or magnitude ) along the vertical
axis for a large number of stars. Note that when temperature is plotted on the horizontal axis, the axis is
reversed, running from high temperatures on the left to low temperatures on the right (to match the
historic choice of plotting spectral class from O to M). Be careful when magnitude is plotted on the
vertical axis, because the smaller the magnitude, the higher the luminosity, so magnitude will usually also
go in the opposite direction to normal for a graph, from high at the bottom to low at the top! The version
above plots luminosity in units of solar luminosity (one solar luminosity being equal to the luminosity of
the sun).

IS: instability strip, the stars in this region are represented by open circles
MS: main sequence MS - RD: main sequence red dwarfs
PN: the position of the central stars of planetary nebulae
RG: red giants
sD: sub-dwarfs.
sG: sub-giants
SG: supergiants
Sol: the position of the Sun or Sol (yellow circle)
WD: white dwarfs .

Often the H-R diagram for a star cluster is plotted, and each star cluster has a unique diagram
depending on the age of the cluster. When stars begin their main life after leaving their embryonic
stages as protostars, they enter the main-sequence (MS). Stars do not move along the MS, but more
massive stars join at the high temperature - high luminosity end and dwarfs at the lower end. As a star
ages, it moves upwards slightly, so the MS becomes a scatter (as shown) rather than a neat curve

Subdwarfs and Metallicity

These are very old stars that have low metallicities (population I stars). Metallicity is the fraction of
elements heavier than helium in a star's atmosphere. After the Big Bang, heavy elements were extremely
rare and the oldest stars are the first-born, made almost entirely of hydrogen and helium, though they
synthesise some heavier elements during their lifetimes. Stars like Sol (the Sun) have higher metallicities
as they are formed from the ashes or star-dust of generations of stars that lived and died before them
(these are population II stars). Stars with lower metallicity are less luminous and so form main sequences
lower down, however, since more massive and hotter stars are shorter lived, these old populations of
stars contain only dwarfs now (unless we are looking back very far in time) and so their MS is apparent
only as a group of so-called subdwarfs below the normal population II MS.

Subgiants, Red Giants and Supergiants

Subgiant stars are giants that are smaller than usual for their spectral class. Many are considered to be
stars in transition from the main sequence (core hydrogen-burning) to the red giant phase (shell
hydrogen-burning).

Red giant stars are stars that have left the main sequence. They have diameters of 10-1000 times that
of the Sun and surface temperatures of 2000-4000 K. These old stars have exhausted the supply of
hydrogen in their cores and instead burn hydrogen in a thin shell around the inert core. This causes the
outer layers to expand massively, cooling as they do so. Though cooler, they have a high luminosity due
to their size.

Supergiants are the largest and most luminous type of star. Red supergiant, or asymptotic giant branch
(ASG) stars, are old and very massive stars in their final centuries or days of life. They typically undergo
periods of instability as they are burning fuel in both a hydrogen shell and a helium shell. The presence
of two burning shells creates instabilities called thermal pulses.

Magnitude is a measure of the brightness of a celestial object. The apparent magnitude is a measure
of how bright the object appears from Earth (adjusted to give the value if the Earth had no atmosphere).
The lower the magnitude, the brighter the object, e.g. the very bright star Sirius has an apparent
magnitude of -1.47 (Sirius is the brightest star, other than the Sun, at visible wavelengths). On the
Pogson scale , a magnitude difference 0f 5 magnitudes corresponds to a hundred-fold difference in
brightness. Tjhis is because the scale is logarithmic to allow for the fact that th eye perceives intensity on
a logarithmic scale - an apparent doubling in brightness, as seen by the human eye, corresponds to a
ten-fold increase in actual brightness. (This property allows the eye to perceive brightnesses over a very
wide range of values). Stars differing by one magnitude differ in brightness by 2.512 fold (the Pogson
ratio
). Apparent magnitude measured by eye, in this way, is the apparent visual magnitude.

These days magnitude can be measured over a wider range by a variety of instruments and over a
specified range or band of wavelengths, narrow or broad. Photoelectric magnitudes (measured by a
photometer with a wavelength filter) are typically measured over one of three wavelength bands: U, B or
V (the UBV system ). U is ultraviolet (centred on 365 nm), B is blue light (centred on 440 nm) and V is
visual (centred on 550 nm, yellow-green light to which the eye is most sensitive). Other systems use
different sets of band-pass filters.

Apparent magnitude does not give a measure of an object's actual luminosity - how much energy the
actual object emits (or reflects), that is its intrinsic brightness (as measured from a set distance).
Bolometric luminosity is the total output over all wavelengths, but luminosity may also be measured or
calculated over a narrower range of wavelengths (so a star might be most luminous in the red or
ultraviolet part of the spectrum, for example). Apparent bolometric magnitude is a measure of the total
radiation received from the object and differes from the visual magnitude by an amount called the
bolometric correction . Absolute magnitude gives a measure of an object's intrinsic luminosity at a
standard distance of 10 parsecs (and requires a measurement or estimate of the object's actual
distance from the observer).

The B-V color index is the blue apparent magnitude minus the visual apparent magnitude. (U-B is also
commonly used). This gives an indication of the star's colour. A0 stars (compensating for Doppler shift,
i.e. unreddened) are given a value of zero. Since smaller magnitudes correspond to brighter objects: a
very hot star will emit more energy at blue wavelengths and so B-V will be negative. For a cooler, redder
star, B-V will be positive. (N.B. This 'counter-intuitive' scale arises because smaller magnitudes are
defined to be brighter!).

Collins Dictionary of Astronomy (2nd ed.), 2000. HarperCollins (pub). [A newer ed. may be available.]

Introductory Astronomy and Astrophysics, Zeilik and Smith (2nd ed.), 1987. CBS College Publishing. [A

The Cambridge Atlas of Astronomy, Audouze and Israel (eds.) (3rd ed.), 1994. Cambridge University
Press. [I expect a newer ed. is available!]

The Open University course texts for S381, The Energetic Universe, 2002.

Eclipsing binary star systems

© 2005 Pearson Prentice Hall, Inc

If the orbits of the binary stars happen to lie edge-on to us, the stars will completely or partially eclipse each other as one passes between us and its partner. We can make out the dip in the light when this happens, to gain information about how fast the stars are moving and the sizes of their orbits, and use this to calculate the masses of the stars.

© 2005 Pearson Prentice Hall, Inc

Combining the information gained from all of these methods, we are able to get an understanding of how the radius is related to the mass, as well as how the luminosity is related to the mass. It is not surprising that an increase of stellar mass correlates with an increase in radius as well as luminosity.

© 2005 Pearson Prentice Hall, Inc

We will go into much more detail about the evolution and final stages of stars in future chapters. For now, we will just note that the lifetime of a star is inversely proportional to the cube of its mass. A massive star is much more luminous than a low mass star. It burns hotter and faster, and uses up its fuel much more rapidly than a small low-mass star.

© 2005 Pearson Prentice Hall, Inc

This table provides a good summary of stellar characteristics and how they directly measured or indirectly calculated using other measurements.

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