IB Physics

Option F - Astrophsyics

Stellar Types

star types / nucleosynthesis / stellar evolution / hertzsprung-russell diagram

Star Types

Binary Stars

A binary star is a stellar system consisting of two stars orbiting around their center of mass (which is clearly not part of either star!). For each star, the other is its companion star. Recent research suggests that a large percentage of stars are part of systems with at least two stars. Binary star systems are very important in astrophysics, because observing their mutual orbits allows their mass to be determined. The masses of many single stars can then be determined by extrapolations made from the observation of binaries.

Binary stars are not the same as optical double stars, which appear to be close together as seen from Earth, but may not be bound by gravity. Binary stars can either be distinguished optically (visual binaries) or by indirect techniques, such as spectroscopy. If binaries happen to orbit in a plane containing our line of sight, they will eclipse each other; these are called eclipsing binaries.

The components of binary star systems can exchange mass, bringing their evolution to stages that single stars cannot attain. Examples of binaries are Algol (an eclipsing binary), Sirius, and Cygnus X-1 (of which one member is probably a black hole).

A true binary is a pair of stars bound together by gravity. When they can be resolved (distinguished) with a powerful enough telescope (if necessary with the aid of interferometric methods) they are known as visual binaries. In other cases, the only indication of binarity is the Doppler shift of the emitted light. Systems in which this is the case, known as spectroscopic binaries, consist of relatively close pairs of stars where the spectral lines in the light from each one shifts first toward the blue, then toward the red, as it moves first toward us, and then away from us, during its motion about their common center of mass, with the period of their common orbit. If the orbital plane is very nearly along our line of sight, the two stars partially or fully occult each other regularly, and the system is called an eclipsing binary, of which Algol is the best-known example.


Visual Binary Stars

A visual binary star is a binary star for which the angular separation between the two components is great enough to permit them to be observed as a double star in a telescope. The resolving power of the telescope is an important factor in the detection of visual binaries, and as telescopes become larger and more powerful an increasing number of visual binaries will be detected. The brightness of the two stars is also an important factor, as brighter stars are harder to separate due to their glare than dimmer ones are.

A visual binary star system

The brighter star of a visual binary is the primary star, and the dimmer is considered the secondary. The position angle of the secondary with respect to the primary is measured, together with the angular distance between the two stars. The time of observation is also recorded. After a sufficient number of observations are recorded over a period of time, they are plotted in polar coordinates with the primary star at the origin, and the most probable ellipse is drawn through these points such that the Keplerian law of areas is satisfied. This ellipse is known as the apparent ellipse, and is the projection of the actual elliptical orbit of the secondary with respect to the primary on the plane of the sky. From this projected ellipse the complete elements of the orbit may be computed, with the semi-major axis being expressed in angular units unless the stellar parallax, and hence the distance, of the system is known.

Spectroscopic Binary Stars

A spectroscopic binary star is a binary star in which the separation between the stars is usually very small, and the orbital velocity very high. Unless the plane of the orbit happens to be perpendicular to the line of sight, the orbital velocities will have components in the line of sight and the observed radial velocity of the system will vary periodically. Since radial velocity can be measured with a spectrometer by observing the Doppler shift of the stars' spectral lines, the binaries detected in this manner are known as spectroscopic binaries. Most of these cannot be resolved as a visual binary, even with telescopes of the highest existing resolving power.

In some spectroscopic binaries the spectra of both stars are visible and the lines are alternately double and single. Such stars are known as double-line binaries. In others, the spectrum of only one of the stars is seen and the lines in the spectrum shift periodically towards the blue, then towards red and back again. Such stars are known as single-line spectroscopic binaries.

The orbit of a spectroscopic binary is determined by making a long series of observations of the radial velocity of one or more component of the binary. The observations are plotted against time, and from the resulting curve a period is determined. If the orbit is circular, then the curve will be a sine curve. If the orbit is elliptical, the shape of the curve will depend on the eccentricity of the ellipse and the orientation of the major axis with reference to the line of sight.

It is impossible to determine individually the semi-major axis a and the inclination of the orbit plane i. However, the product of the semi-major axis and the sine of the inclination (i.e. a sin i) may be determined directly in linear units (e.g. kilometres). If either a or i can be determined by other means, as in the case of eclipsing binaries, a complete solution for the orbit can be found.

Eclipsing Binary Stars

An eclipsing binary star is a binary star in which the orbit plane of the two stars lies so nearly in the line of sight of the observer that the components undergo mutual eclipses. In the case where the binary is also a spectroscopic binary and the parallax of the system is known, the binary is quite valuable for stellar analysis.

Eclipsing binary

Eclipsing binaries are variable stars, not because the light of the individual components vary but because of the eclipses. The light curve of an eclipsing binary is characterized by periods of practically constant light, with periodic drops in intensity. If one of the stars is larger than the other, one will be obscured by a total eclipse while the other will be obscured by an annular eclipse.

The period of the orbit of an eclipsing binary may be determined from a study of the light curve, and the relative sizes of the individual stars can be determined in terms of the radius of the orbit by observing how quickly the brightness changes as the disc of the near star slides over the disc of the distant star. If it is also a spectroscopic binary the orbital elements can also be determined, and the mass of the stars can be determined relatively easily, which means that the relative densities of the stars can be determined in this case.

Cepheids

Cepheids, also called Cepheid Variables, are stars which brigthen and dim periodically. This behavior allows them to be used as cosmic yardsticks out to distances of a few tens of millions of light-years.

In 1912, Henrietta Swan Leavitt noted that 25 stars, called Cepheid stars, in the Magellanic cloud would brighten and dim periodically. Leavitt was able to measure the period of each star by measuring the timing of its ups and downs in brightness. What she determined was that the brighter the Cepheid, the longer its period. In fact, Cepheids are very special variable stars because their period (the time they take to brighten, dim and brighten again) is

  • regular (that is, does not change with time), and
  • a uniform function of their brightness.

That is, there is relation between the period and brightness such that once the period is known, the brightness can be inferred.

Cepheids are reasonably abundant and very bright. Astronomers can identify them not only in our Galaxy, but in other nearby galaxies as well. If one requires the distance to a given galaxy one first locates the Cepheid variables in this galaxy. From these observations one determines the period of each of these stars. Leavitt's data states that a given period has a unique brightness associated to it. So from the period and Leavitt's plot we get the brightness at the distance of one light-year (see the image above). We can also measure the brightness on Earth. The brightness at the distance of one light-year will be larger than the observed brightness due to the fact that brightness drops like the square of the distance. From these numbers one can extract the distance to the stars. This method works up to 13 million light-years when Earth-bound telescopes are used; for larger distances these stars become too dim to be observed. Recently, space-based telescopes such as the Hubble Telescope, have used these stars to much farther distances. Looking at a galaxy in the Virgo cluster called M100, astronomers used the Cepheid variables observed there to determine its distance - 56 million light-years.

Red Giant

According to the Hertzsprung-Russell diagram, a red giant is a large non-main sequence star of stellar classification K or M; so-named because of the reddish appearance of the cooler giant stars. Examples include Aldebaran, in the constellation Taurus and Arcturus.

Structure of a red giant

They are stars of 0.4 - 10 times the mass of the Sun which have exhausted their supply of hydrogen in their cores and switched to fusing hydrogen in a shell outside the core. Since the inert helium core has no source of energy of its own, it contracts and heats up, and its gravity compresses the hydrogen in the layer immediately above it, thus causing it to fuse faster. This in turn causes the star to become more luminous (from 1,000 – 10,000 times brighter) and expand; the degree of expansion outstrips the increase in luminosity, thus causing the effective temperature to decrease. In stars massive enough to ignite helium fusion, an analogous process occurs when central helium is exhausted and the star switches to fusing helium in a shell, although with the additional complication that in many cases hydrogen fusion will continue in a shell at lesser depth — this puts stars onto the asymptotic giant branch. The decrease in surface temperature shifts the star's visible light output to the red — hence red giant. Stars of spectral types O through K are believed to become red giants (or supergiants in the case of O and B stars).

Planetary Nebulae

A planetary nebula is an astronomical object consisting of a glowing shell of gas and plasma formed by certain types of stars at the end of their lives. They are in fact unrelated to planets; the name originates from a supposed similarity in appearance to giant planets. They are a short-lived phenomenon, lasting a few tens of thousands of years, compared to a typical stellar lifetime of several billion years. About 1,500 are known to exist in the Milky Way Galaxy.

Planetary nebulae are important objects in astronomy because they play a crucial role in the chemical evolution of the galaxy, returning material to the interstellar medium which has been enriched in heavy elements and other products of nucleosynthesis (such as carbon, nitrogen, oxygen and calcium). In other galaxies, planetary nebulae may be the only objects observable enough to yield useful information about chemical abundances.

Images of different types of planetary nebulae

In recent years, Hubble Space Telescope images have revealed many planetary nebulae to have extremely complex and varied morphologies. About a fifth are roughly spherical, but the majority are not spherically symmetric. The mechanisms which produce such a wide variety of shapes and features are not yet well understood, but binary central stars, stellar winds and magnetic fields may all play a role.

Neutron Stars

A neutron star is one of the few possible endpoints of stellar evolution. A neutron star is formed from the collapsed remnant of a massive star after a Type II, Type Ib, or Type Ic supernova.

A typical neutron star has a mass between 1.35 to about 2.1 solar masses, with a corresponding radius between 20 and 10 km (they shrink as their mass increases) — 30,000 to 70,000 times smaller than the Sun. Thus, neutron stars have densities of 8 x 1013 to 2 x 1015 g/cm3; about the density of an atomic nucleus.

Artists impression of a neutron star

Since a neutron star retains most of the angular momentum of its parent star but has only a tiny fraction of its parent's radius, the moment of inertia decreases sharply causing a rotational acceleration to a very high rotation speed, with one revolution taking anywhere from one seven-hundredth of a second to thirty seconds. The neutron star's compactness also gives it high surface gravity, 2×1011 to 3×1012 times stronger than that of Earth. One of the measures for the gravity is the escape velocity, the velocity needed for an object to escape from the gravitational field to infinite distance. For a neutron star, such velocities are typically 150,000 km/s, about 1/2 of the velocity of light. Conversely, an object falling onto the surface of a neutron star would strike the star also at 150,000 km/s. To put this in perspective, if an average human were to encounter a neutron star, they would impact with roughly the energy yield of a 200 megaton explosion (a power equivalent to four times the Tsar Bomba, the biggest nuclear weapon ever detonated).

Current understanding of the structure of neutron stars is defined by existing mathematical models, which of course are subject to revision. On the basis of current models, the matter at the surface of a neutron star is composed of ordinary atomic nuclei as well as electrons. The "atmosphere" of the star is roughly one meter thick, below which one encounters a solid "crust". Proceeding inward, one encounters nuclei with ever increasing numbers of neutrons; such nuclei would quickly decay on Earth, but are kept stable by tremendous pressures. Proceeding deeper, one comes to a point called neutron drip where free neutrons leak out of nuclei. In this region there are nuclei, free electrons, and free neutrons. The nuclei become smaller and smaller until the core is reached, by definition the point where they disappear altogether. The exact nature of the superdense matter in the core is still not well understood. While this theoretical substance is referred to as neutronium in science fiction and popular literature, the term "neutronium" is rarely used in scientific publications, due to ambiguity over its meaning. The term neutron-degenerate matter is sometimes used, though that term incorporates assumptions about the nature of neutron star core material. Neutron star core material could be a superfluid mixture of neutrons with a few protons and electrons, or it could incorporate high-energy particles like pions and kaons in addition to neutrons, or it could be composed of strange matter incorporating quarks heavier than up and down quarks, or it could be quark matter not bound into hadrons. (A compact star composed entirely of strange matter would be called a strange star.) However so far observations have neither indicated nor ruled out such exotic states of matter.

Pulsars

Pulsars are rotating neutron stars that are observable as sources of electromagnetic radiation in radio wavebands. The radiation intensity varies with a regular period, believed to correspond to the rotation period of the star. Pulsars also create what is called the lighthouse effect, this is when the light from a pulsar is only seen at a specific position and not all of the time. Werner Becker of the Max-Planck-Institut für extraterrestrische Physik recently said, "The theory of how pulsars emit their radiation is still in its infancy, even after nearly forty years of work. There are many models but no accepted theory."

Chandra Observatory X-ray image of the Crab Nebula pulsar

Quasars

A quasar (contraction of QUASi-stellAR radio source) is an astronomical source of electromagnetic energy, including light, that dwarfs the energy output of the brightest stars. A quasar may readily release energy in levels equal to the output of hundreds of average galaxies combined. In optical telescopes, a quasar looks like a single point of light (i.e. it is a point source), and has a very high redshift. The general consensus is that this high redshift is cosmological, the result of Hubble's law, which implies that quasars must be very distant and hence very luminous.

Artists impression of a quasar

Some quasars display rapid changes in luminosity, which implies that they are small (an object cannot change faster than the time it takes light to travel from one end to the other; but see quasar J1819+3845 for another explanation). The highest redshift currently known for a quasar is 6.4.

The scientific consensus is that quasars are powered by accretion of material onto supermassive black holes in the nuclei of distant galaxies, making these luminous versions of the general class of objects known as active galaxies. No other currently known mechanism appears able to explain the vast energy output and rapid variability.

More than 100,000 quasars are known. All observed spectra have shown considerable redshifts, ranging from 0.06 to the recent maximum of 6.4. Therefore, all known quasars lie at great distances from us, the closest being 240 Mpc (780 million ly) away and the farthest being 4 Gpc (13 billion ly) away. Most quasars are known to lie above 1.0 Gpc in distance; since light takes such a long time to cover these great distances, we are seeing quasars as they existed long ago — the universe as it was in the distant past.

Although faint when seen optically, their high redshift implies that these objects lie at a great distance from us, making quasars the brightest objects in the known universe. The quasar which appears brightest in our sky is the ultraluminous 3C 273 in the constellation of Virgo. It has an average apparent magnitude of 12.8 (bright enough to be seen through a small telescope), but it has an absolute magnitude of −26.7. So from a distance of 10 parsecs (about 33 light-years), this object would shine in the sky about as bright as our sun. This quasar's luminosity is, therefore, about 2 trillion (2 × 1012) times that of our sun, or about 100 times that of the total light of average giant galaxies like our Milky Way.

The hyperluminous quasar APM 08279+5255 was, when discovered in 1998, given an absolute magnitude of −32.2, although high resolution imaging with the Hubble Space Telescope and the 10 m Keck Telescope reveal that this system is gravitationally lensed. A study of the gravitational lensing in this system suggests that it has been magnified by a factor of ~10. It is still substantially more luminous than nearby quasars such as 3C 273. HS 1946+7658 was thought to have an absolute magnitude of −30.3, but this too was magnified by the gravitational lensing effect.

Quasars are found to vary in luminosity on a variety of time scales. Some vary in brightness every few months, weeks, days, or hours. This evidence has allowed scientists to theorize that quasars generate and emit their energy from a very small region, since each part of the quasar would have to be in contact with other parts on such a time scale to coordinate the luminosity variations. As such, a quasar varying on the time scale of a few weeks cannot be larger than a few light-weeks across.

Quasars exhibit many of the same properties as active galaxies: Radiation is nonthermal and some are observed to have jets and lobes like those of radio galaxies. Quasars can be observed in many parts of the electromagnetic spectrum including radio, infrared, optical, ultraviolet, X-ray and even gamma rays. Most quasars are brightest in their rest-frame near-ultraviolet (near the 1218 angstrom Lyman-alpha emission line of hydrogen), but due to the tremendous redshifts of these sources, that peak luminosity has been observed as far to the red as 9000 angstroms, in the near infrared.

White Dwarfs

A white dwarf is an astronomical object which is produced when a low or medium mass star dies. These stars are not heavy enough to generate the core temperatures required to fuse carbon in nucleosynthesis reactions. After such a star has become a red giant during its helium-burning phase, it will shed its outer layers to form a planetary nebula, leaving behind an inert core consisting mostly of carbon and oxygen.

This core has no further source of energy, and so will gradually radiate away its energy and cool down. The core, no longer supported against gravitational collapse by fusion reactions, becomes extremely dense, with a typical mass of that of the sun contained in a volume about equal to that of the Earth. The white dwarf is supported only by electron degeneracy pressure. The maximum mass of a white dwarf, beyond which degeneracy pressure can no longer support it, is about 1.4 solar masses. A white dwarf which approaches this limit (known as the Chandrasekhar limit), typically by mass transfer from a companion star, may explode as a Type Ia supernova via a process known as carbon detonation.

Eventually, over hundreds of billions of years, white dwarfs will cool to temperatures at which they are no longer visible. However, over the universe's lifetime to the present (about 13.7 billion years) even the oldest white dwarfs still radiate at temperatures of a few thousand kelvins.

As a class, white dwarfs are fairly common; they comprise roughly 6% of all stars.

Artists impression of a white dwarf star

Almost all small and medium-size stars will end up as white dwarfs, after all the hydrogen they contain is fused into helium. Near the end of its nuclear burning stage, such a star goes through a red giant phase and then expels most of its outer material (creating a planetary nebula) until only the hot (T > 100,000 K) core remains, which then settles down to become a young white dwarf which shines from residual heat.

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Nucleosynthesis

Nucleosynthesis is the process of combining light elements into heavier elements, also known as fusion. Nucleosynthesis requires high speed collisions and high temperatures. Temperatures in the core are much higher than on the surface of a star. The main process for energy production in a star is nuclear fusion of hydrogen to helium. This is the process that occurs during most of a star's lifetime. After the hydrogen in the star's core is exhausted, the star can burn helium to form progressively heavier elements, carbon and oxygen and so on, until iron and nickel are formed. Up to this point the process releases energy. The formation of elements heavier than iron and nickel requires the input of energy. Supernova explosions result when the cores of massive stars have exhausted their fuel supplies and burned everything into iron and nickel. The nuclei with mass heavier than nickel are thought to be formed during these explosions.

In a stable star, the pressure of the gas pushing out from the center is equal to gravity pulling atoms towards the center, core. When these forces are equal, the star is at equilibrium. Once a star reaches equilibrium for the first time, it will start fusing hydrogen into helium.

Wherever there is a large cloud, nebula, of hydrogen in the universe, gravity pulls it together making it contract and get denser and hotter (GPE is converted into KE). If the temperature and pressure in the center of the protostar become high enough, fusion reactions ignite and the star enters the main sequence in the H-R-diagram. Where on the main sequence it appears (what spectral class it will have) depends primarily on its mass - the higher, the hotter.

The Sun generates energy in its core by converting hydrogen to helium.

In this process, a little mass is lost in that the helium nucleus is slightly less massive than 4 times the mass of 1 hydrogen nucleus. The above figure leaves off two electron neutrinos that are produced during the fusion process.

Several paths may be followed for the conversion of hydrogen to helium. The Sun mainly uses the so-called proton-proton (pp) chain. More massive stars, for example, use what is known as the carbon-nitrogen-oxygen (CNO) cycle.

Clouds of hydrogen and helium form into main sequence stars, where nuclear fusion takes place, fusing hydrogen to form helium. Further fusion only takes place in heavier stars, otherwise the pull of gravity forces the star to contract and cool to a red dwarf. If further fusion takes place the star becomes a red giant.

Red giants are formed when the hydrogen in the core of the star has fused into heavier helium and helium fusions occur to create berilium. Gravity causes the star to contract and heat up. The hydrogen around the core burns more fiercely and causes the outer part of the star to expand and cool down. Small red giants (1.4 solar masses - Chandraseka limit) can not withstand the pull of gravity, so it shrinks, becomes extremely hot, until it finally cools into a white dwarf. Larger red giants fuses until iron is formed, however, further fusion can not take place without energy input. Therefore, the star contracts and heat up because of the large kinetic energy in the particles, and creates supernova.

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Stellar Evolution

Our universe, for the most part, is cold and dark. However, the same entities that save the night sky from being completely flat and uninteresting also warm the Earth and make our existence possible. They are the stars. It is a curious thing, to consider the stars. How do objects that are hot; hot enough to emit visible light or even higher frequencies of electromagnetic radiation, form, seemingly spontaneously out of that which cold and dark? In other words, how would one go about making one’s own star?

Life cycle of the sun

The nebulae in space from which stars are created are actually the remains of a previous star that has reached the end of its lifecycle and died. Generally speaking, they consist of hydrogen and helium, and small amount of the other heavier elements. The nebula is cold and dark. To spark the transformation into a star, any number of events may happen. In the case of our sun, it is speculated that a nearby supernovae explosion was responsible.

The nebula, under the influence of gravity, begins to condense, and eventually, a protostar is formed. Such protostars can be observed in nebulas such as the horsehead nebula and the crab nebula. It is in this stage that the process of nucleosynthesis begins. Nucleosynthesis, in contrast to the nuclear processes that we are used to on Earth, is fusion, not fission. That is, instead of splitting a heavy nucleus, light nuclei are smashed together and fuse to produce a heavier nucleus, and gamma rays. It is called the proton-proton cycle. The star will continue to react its core of hydrogen into helium for all of its main-sequence lifetime.

Once the star runs out of helium, the core collapses, and, under the additional gravitational pressure, the helium in the core will start to undergo fusion. This causes the outer layers of the star to expand, however, the outer layers also cool, and the star becomes a red giant. The core continues to react, and elements such as carbon, neon, oxygen, silicon and iron are produced. It is here that the elements that compose our world are created. Without the stars then universe would be composed of hydrogen and little else.

When the star finally runs out of fuel completely; usually when the core becomes iron, the red giant star collapses. The next stage of the star is determined by the mass of that star and the Chandrasekar limit.

  • If a star is below 4 solar masses (Type G), it is less that the Chandrasekar limit, and when it collapse it form a white dwarf of 1.4 solar masses or less, along with a planetary nebula. The white dwarf star continues to cool and eventually becomes invisible.

  • If a star is above 4 solar masses (Type A, B, O), it is above the Chandrasekar limit, and instead of becoming a regular red giant, it becomes a super red giant. In this case, when the star dies, it takes a rather more spectacular path than the star below the Chandrasekar limit, becoming a supernova. Depending on the mass of the star, it will either go on become a black hole or a neutron star.

Occasionally, pulsars are formed by this process. Pulsars are basically cosmic sources of weak radio waves. They pulsate and a very rapid and precise frequency, hence the name pulsar. Pulsars have been theoretically linked with rotating neutron stars, which would be expected to emit a intense bean of radio waves in one specific direction. Due to the rotation, this is perceived as a pulse on Earth.

Quasars, or quasi-stellar objects, are a different matter altogether. They appear to be point sources of light and radio waves that are very far away. The redshifts involved are enormous, and place at the vey edge of the visible universe, indicating that the power output of these objects must be very high, especially considering their size. Quasars are not well understood, however, it is speculated that their cause is the presence of a super-massive black hole and that the energy emitted is a result of stars being absorbed by the black hole.

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Hertzsprung-Russell Diagram

The Hertzsprung-Russell diagram (usually referred to by the abbreviation H-R diagram or HRD, also known as a Colour-Magnitude (CM) diagram) shows the relationship between absolute magnitude, luminosity, classification, and surface temperature of stars. The diagram was created circa 1910 by Ejnar Hertzsprung and Henry Norris Russell, and represented a huge leap forward in understanding stellar evolution, or the 'lives of stars'.

Three different ways of seeing the H-R diagram

Hertzsprung-Russell diagram by Richard Powell. with permission. 22 000 stars are plotted from the Hipparcos catalog and 1000 from the Gliese catalog of nearby stars. An examination of the diagram shows that stars tend to fall only into certain regions on the diagram. The most predominant is the diagonal, going from the upper-left (hot and bright) to the lower-right (cooler and less bright), called the main sequence. In the lower-left is where white dwarfs are found, and above the main sequence are the red giants and supergiants. The Sun is found on the main sequence at luminosity 1 (magnitude approx. 5), around 5400K (Stellar Class G2).

There are two equivalent forms of the H-R diagram. One is the observer's form, which plots the colour index of the star on one axis and the absolute magnitude on the other axis. These two quantities can be derived from observations. The theoretician's form plots the temperature of the star on one axis and the luminosity of the star on the other. These two quantities can be calculated from computer models.

The H-R diagram is used to define different types of stars and to match theoretical predictions of stellar evolution using computer models with observations of actual stars.

H-R diagram can also be used to plot the evolution of a star from its birth as a protostar until its death as a white dwarf

Most of the stars occupy the region along the line called main sequence. During that stage stars are burning hydrogen. Next concentration of stars is on horizontal branch (helium fusion in the core and hydrogen burning in a shell surrounding the core). Another prominent feature is Hertzsprung gap located in the region between A5 and G0 spectral type and between +1 and -3 absolute magnitudes (i.e. between the top of main sequence and the giants in the horizontal branch). RR Lyrae stars can be found in the left of this gap. In the upper section of instability strip Cepheid variables are residing.

The H-R diagram is also used by scientists to help the figure out roughly how far away the stars are from Earth. This can be done if we know the apparent magnitude we can plot the star onto the graph using its spectral class and the type of star it is. We can then use the graph to deduce the absolute magnitude of the star. See more of this in the next section on Stellar Distances.

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