ASTR 001 HW 5

What happens to a low-mass star after it exhausts its core helium? Why can't it fuse carbon into heavier elements?

The star dies. Carbon fusion requires temperatures near 600 million K. Degeneracy pressure stops that from happening.

Measuring Stellar Mass. The spectral lines of two stars in a particular eclipsing binary system shift back and forth with a period of 6 months. The lines of both stars shift by equal accounts, and the amount of the Doppler shift indicates that each star has an orbital speed of 80,000 m/s. What are the masses of the two stars? Assume that each of the two stars traces a circular orbit around their center of mass.

p^2=[4(pi)^2]a^3/[G(M(1)+M(2))] M(1)=5.15x10^20kg M(2)=5.15x10^20kg

Calculating Stellar Radii. Sirius A has a luminosity of 26L(Sun) and a surface temperature of about 9400 K. What is its radius?

r=sqrt[L/(4(pi)(sigma)T^4)] sigma=5.7x10^(-8)watt/(m^2xK^4) 4.38x10^(-5)m

Solar Mass Loss. Estimate how much mass the Sun loses through fusion reactions during its 10-billion-year life. You can simplify the problem by assuming the Sun's energy output remains constant. Compare the amount of mass lost with Earth's mass.

1.30x10^27 kg of mass lost 225.8 times the mass of Earth

Algol's Orbital Separation. The Algol binary system consists of a 3.7M(Sun) star and a 0.8M(Sun) star with an orbital period of 2.87 days. Use Newton's version of Kepler's third law to calculate the orbital separation of the system. How does that separation compare with the typical size of a red giant star?

7.19x10^9 m This separation is close to or larger than the diameter of the typical red giant star.

In what sense is a black hole like a hole in the observable universe? Define the event horizon and the Schwarzschild radius, and describe the three basic properties of a black hole.

A black hole is like a hole in the observable universe as we can not see inside it. This is due to the event horizon, which is the area of a black hole where escape velocity exceeds c and nothing can escape. The Schwarzschild radius is the radius of the event horizon. The basic properties are mass, electric charge, and angular momentum.

What are neutrinos? What was the solar neutrino problem, and how was it solved?

A neutrino is a subatomic particle with a very tiny mass produced in the second step of the proton-proton chain. The solar neutrino problem is that there was a large discrepancy in the flux of solar neutrinos predicted and measured and it was solved by postulating that neutrons had mass and oscillated between three different types.

What is a nova? Describe the process that creates a nova and what a nova looks like.

A nova is when a white dwarf gains back enough hydrogen to start fusion again and become a star. The flash causes a bright light flash that lasts few weeks.

What is a planetary nebula? What happens to the core of a star after a planetary nebula occurs?

A planetary nebula is a shell of gas (ejected outer layers of a star) expanding away from carbon core and ionized by the ultraviolet radiation of the hot carbon core, which makes it glow - core cools ----> glow stops ----> gas ejected into space ----> core is a white dwarf

What do we mean by a star's spectral type, and how is spectral type related to surface temperature and color?

A spectral type (OBAFGKM) of a star is determined from the spectral lines present in a star's spectrum. Stars displaying spectral lines of highly ionized elements will be hot because it takes high temperatures to ionize atoms. The color of the spectral lines are determined by the type of radiation that the star releases. The hottest stars release blue light and are on the O end of the spectral types and the coolest stars release red light and are on the M end.

Describe the mass, size, and density of a typical neutron star. What would happen if a neutron star came to your hometown?

A star containing about 1½ solar masses of material compressed into a volume approximately 6 mi (10 km) in radius. (1 solar mass equals 4.4 × 10^33 lbm or 2.0 × 10^33 kg.) Neutron stars are one of the end points of stellar evolution and are the final states of stars that begin their lives with considerably more mass than the Sun. The density of neutron star material is 10^14 to 10^15 times the density of water and exceeds the density of matter in the nuclei of atoms. Neutron stars are pulsars (pulsating radio sources) if they rotate sufficiently rapidly and have strong enough magnetic fields. If a neutron star came close to the earth, we would end up splattered across the face of the star.

How is a star's apparent brightness related to its luminosity? Explain the inverse square law for light.

A star's apparent brightness is related to its luminosity inversely as luminosity is constant, as distance from the light source increases, the apparent brightness decreases by luminosity divided by the distance squared. Apparent Brightness=Luminosity/(4(pi)x(distance)^2) As light moves farther away from it's source it must cover more area and thus apparent brightness decreases by the inverse square.

What processes may cause a white dwarf supernova? Observationally, how do we distinguish white dwarf and massive star supernovae?

A white dwarf supernova is caused by carbon fusion starting, the carbon fuses all at once causing an explosion. A white dwarf supernova happens when an object with very little hydrogen in its outer layer collapses so, there is little or no hydrogen absorption in the spectrum. A massive star supernova typically has a fair amount of hydrogen left in its envelope, or at least in its immediate surroundings.

Explain why H-R diagrams look different for star clusters of different ages. How does the location of the main-sequence turnoff point tell us the age of the star cluster?

Clusters at different ages have different turnoff points. Age of the cluster is equal to the lifetimes of stars at the main sequence turnoff point.

What event initiates a supernova, and why is a neutron star or a black hole left behind? What observational evidence supports our understanding of supernovae?

Degeneracy pressure supports the inert iron core for a short time before it collapses. Gravity eventually succeeds in pushing the star's electrons past the quantum mechanical limit. The electrons combine with protons and become neutrons, releasing neutrinos. The degeneracy pressure disappears. The iron core of the dead star collapses into a ball of neutrons just a few kilometers wide. Neutron degeneracy pressure stops the collapse. The collapse causes a massive explosion of energy and the outer layers fly out into space. The result is the core remains as a neutron star, unless it is still too massive, in which case it becomes a black hole. We have historical records of what were likely supernovae explosions and the Crab Nebula is a nebula and neutron star where one such explosion is recorded to have happened.

What happens to the electron speed in a more massive white dwarf, and how does this behavior lead to the white dwarf limit for mass?

Electron speeds are faster in more massive white dwarf stars. The electrons can't travel faster than the speed of light and so white dwarves are unable to stably exist above 1.4 solar masses, the amount of mass required for the electrons to go at the speed of light. The white dwarf limit is that they can't rotate faster than once a second.

What is the helium fusion reaction, and why does it require much higher temperatures than hydrogen fusion? Why will helium fusion in the Sun begin with a helium flash?

Helium fusion reaction is when 3 Helium nuclei become one carbon nucleus. Helium has two protons which means greater positive charge. This is needed more speed for strong force to overcome electromagnetic repulsion resulting in greater temperature.

What is the interstellar medium? What is its chemical composition, and how do we measure it?

Interstellar medium is the matter between the stars and is composed of gas and dust. It is gas and dust, 75% hydrogen and 25% helium by mass. We measure it by studying spectra of interstellar gas clouds.

The Luminosity of Alpha Centauri A. Alpha Centauri A lies at a distance of 4.4 light-years and has an apparent brightness in our night sky of 2.7x10^(-8) watt/m^2. Recall that 1 light-year=9.5x10^12 km=9.5x10^15 m. a) Use the inverse square law for light to calculate the luminosity of Alpha Centauri A.

Luminosity=5.91x10^26 watts Apparent Brightness=Luminosity/[4(pi)d^2]

What is the difference between nuclear fission and nuclear fusion? Which one is used in nuclear power plants? Which one does the Sun use?

Nuclear fission is the process of splitting the large nuclei of atoms which is used in nuclear power plants. Nuclear fusion is the process of combining or "fusing" two nuclei into a large nucleus which is what the Sun uses to create energy.

Why does nuclear fusion require high temperatures and pressures?

Nuclear fusion requires high temperatures because at high temperatures, the collisions between atoms are more energetic because the nuclei are moving at higher speeds due to heat. Nuclear fusion requires high pressures because at high pressure, because with out it, the atoms would escape and explode preventing fusion.

How do red giants manufacture carbon-rich dust grains, and why are these important to life?

The convection in a low-mass star in its final stages of life dredges up carbon from the core and brings it to the surface. Because the carbon can then be lost via the stellar winds, these stars seed the interstellar medium with carbon, including the carbon that is used for life on Earth.

Summarize some of the observational evidence supporting our ideas about how heavy elements form in massive stars.

One piece of evidence that supports our theories about how heavy elements form in high-mass stars is the chemical composition of older stars. Our theory predicts that the older stars should have fewer heavy elements in their compositions. Observations indicate that this is so. Another piece of evidence supporting our theories is the relative abundances of the various elements. For example, since the helium-capture reactions are an important series of reactions in high-mass stars, we expect to see more elements with even numbers of protons than odd numbers of protons. This predicted pattern agrees with the observations quite well.

Explain how the presence of a neutron star can make a close binary star system appear to us as an X-ray binary. Why do some of these systems appear to us as X-ray bursters?

The accretion disk of the neutron star causes the system to emit lots of x rays. The x-ray bursters happen when some neutron stars get enough hydrogen to fuse enough helium to start helium fusion. The helium fusion stops and the hydrogen plies up again, starting the process over.

Why is mass so important to a star's life? How and why do we divide stars into groups by mass?

The birth mass determines the fusion progression within the core of the star. We divide these into groups because lower mass stars usually live longer than higher mass stars and they can produce different elements through fusion

What do we mean by solar activity? Describe key features including sunspots, prominences, flares, and coronal mass ejections.

Solar activity is the changing features on the surface of the Sun like storms and sun spots. Sun spots are cooler spots of the Sun that appear darker than the surrounding area of the Sun and are the result of certain magnetic fields and dissolve as their magnetic fields weaken. Prominences are giant loops of gas trapped in the Sun's chromosphere and corona that can last for weeks. Flares are dramatic solar storms that emit bursts of ultraviolet light and X-rays in addition to charged particles moving at the speed of light. Flares are likely the result of twisting and knotted magnetic fields that cannot bear the tension and release the tension by reorganizing as evidenced by close location to sunspots. Coronal mass ejections are huge bubbles of highly energetic charged particles from the Sun's corona and have large magnetic fields that can reach Earth.

Why can the lives of close binary stars differ from those of single stars? Describe the Algol paradox and its resolution.

The Algol paradox is that the star Algol doesn't seem to have evolved according to stellar evolution models. That's because Algol is a binary star, and shares material with its companion.

How do we know that pulsars are neutron stars? Are all neutron stars also pulsars? Explain.

The first indication was the very short period of some pulsars. Nothing else could rotate that fast without flying apart. The neutron star theory of pulsars has been further advanced by the study of a binary neutron star/pulsar, which has shown behavior predicted for such a system by general relativity theory. Probably not all neutron stars are pulsars. There are several mechanisms by which a neutron star may emit pulses, but the necessary conditions won't exist for all neutron stars. However, we can detect only those pulsars that emit their energy in our direction.

What two forces are balanced in gravitational equilibrium? What does it mean for the Sun to be in energy balance?

The outward push of internal gas pressure and the inward pull of gravity are balanced within the Sun in gravitational equilibrium. Energy balance is when the rate that fusion releases energy in the Sun's core and the rate that the Sun's surface radiates energy in space are equivalent.

How do we use stellar parallax to determine a star's distance, and how can we then determine its luminosity?

The stellar parallax is smaller for stars that are farther away, because when the stars are farther away, the angle between the stars is smaller so the distance of a star is inversely related to the inverse of the angle of stellar parallax. d(in parsecs)=1/p(in arcseconds) d(in light-years)=3.26x[1/p(in arcseconds)] Once we know the distance of a star, we can determine its luminosity using the inverse square law for light.

What is the sunspot cycle? Why is it sometimes described as an 11-year cycle and sometimes as a 22-year cycle? Are there longer-term changes in solar activity?

The sunspot cycle is the cycle in which the average number of sunspots on the Sun rises and falls to a solar maximum and solar minimum (which takes about 11 years) and frequency of prominences, flares, and coronal mass ejections follow this cycle. When the sunspot cycle is at its solar maximum, the entire Sun's magnetic field flips (turning magnetic north into magnetic south) which occurs ever 11 years, but it takes 22 years (two cycles) to return the magnetic field to its initial state. In the longer term, the sunspot cycle has varied in length although comparisons are difficult due to lack of technological consistency over time and across those long periods.

Density of a Red Giant. Near the end of the Sun's life, its radius will extend nearly to the distance of Earth's orbit. Estimate the volume of the Sun at that time using the formula for the volume of a sphere (4πr^3/3). Using that result, estimate the average matter density of the Sun at that time. How does that density compare with the density of water (1 g/cm3)? How does it compare with the density of 1 g/cm^3 Earth's atmosphere at sea level (about 10^(-3) g/cm^3 )?

Volume of Sun=4.5x10^33 m^3 --> Sun is 1.5x10^11 m from Earth Density of the Sun at this expansion is 0.44x10^(-6) g/cm^3 which is significantly less dense than water and Earth's atmosphere. -->Mass of the Sun is 1.99x10^30 kg

Describe the mass, size, and density of a typical white dwarf. How does the size of a white dwarf depend on its mass?

White dwarf, in astronomy, a type of star that is abnormally faint for its white-hot temperature (see mass-luminosity relation). Typically, a white dwarf star has the mass of the sun and the radius of the earth but does not emit enough light or other radiation to be easily detected. The existence of white dwarfs is intimately connected with stellar evolution. A white dwarf is the hot core of a star, left over after the star uses up its nuclear fuel and dies. It is made mostly of carbon and is coated by a thin layer of hydrogen and helium gases. The physical conditions inside the star are quite unusual; the central density is about 1 million times that of water.

A Black Hole I? You've just discovered a new X-ray binary, which we will call Hyp-X1 ("Hyp" for hypothetical). The system Hyp-X1 contains a bright, B2 main-sequence star orbiting an unseen companion. The separation of the stars is estimated to be 20 million kilometers, and the orbital period of the visible star is 4 days. a) Use Newton's version of Kepler's third law to calculate the sum of the masses of the two stars in the system. (Hint: See Mathematical Insight 15.4.) Give your answer in both kilograms and solar masses (M(Sun) = 2.0 * 1030 kg). b) Determine the mass of the unseen companion. Is it a neutron star or a black hole? Explain. (Hint: A B2 main-sequence star has a mass of about 10M(Sun).

a) 1.37x10^34 grams

The Lifetime of the Sun. The total mass of the Sun is about 2x10^30 kilograms, of which about 70% was hydrogren when the Sun formed. However, only about 13% of this hydrogen every becomes available for fusion in the core. The rest remains in layers of the Sun where the temperature is too low for fusion. a) Use the given data to calculate the total mass of hydrogen available for fusion over the lifetime of the Sun. b) The Sun fuses about 600 billion kilograms of hydrogen each second. Based on your result from part a, calculate how long the Sun's initial supply of hydrogen can last. Give your answer in both seconds and years. c) Given that our solar system is now 4.6 billion years old, when will we need to worry about the Sun running out of hydrogen for fusion?

a) 2x10^30 kg x 0.70=1.4x10^30 kg 1.4x10^30 kg x 0.13=1.82x10^29 kg b) [1.82x10^29 kg]/[6x10^11 kg/s]=3.03x10^17 seconds 3.03x10^17 seconds=9.62X10^9 years c) 9620000000-4600000000=5020000000 5.02 billions years

Neutron Star Density. A typical neutron star has a mass of about 1.5M(Sun) and a radius of 10 kilometers. a) Calculate the average density of a neutron star, in kilograms per cubic centimeter. b) Compare the mass of 1 cm3 of neutron star material to the mass of Mount Everest (≈5x10^10kg).

a) 7.16x10^26 kg/km^3 =7.16x10^17kg/m^3 =7.16x10^11kg/cm^3 b) The mass of neutron star material is denser thatn the mass of Mount Everest.

Stellar Data. The table below gives basic data for several bright stars; M(v) is absolute magnitude and m(v) is the apparent magnitude. Use these data to answer the following questions. Include a brief explanation with each answer [Hint: Remember that the magnitude scales runs backward, so brighter stars have smaller (or more negative) magnitudes.] a) Which star appears brightest in our sky? b) Which star appears faintest in our sky? c) Which star has the greatest luminosity? d) Which star has the least luminosity? e) Which star has the highest surface temperature? f) Which star has the lowest surface temperature? g) Which star is the most similar to the Sun? h) Which star is a red supergiant? i) Which star has the largest radius? j) Which stars have finished fusing hydrogen in their cores? k) Among the main-sequence stars listed, which one is the most massive? l) Among the main-sequence stars listed, which one has the longest lifetime?

a) The star that appears brightest in our sky is Sirius with the most negative m(v). b) The star that appears faintest in our sky is Regulus with the most positive m(v). c) The star with the most luminosity is Anteres with the most negative M(v). d) The star with the least luminosity is Alpha Centauri A with the most positive M(v). e) The star that has the highest surface temperature is Sprica B1. f) The star with the lowest surface temperature is Antares M1. g) The star most similar to the Sun is Alpha Centauri A because the Sun is a G2 V star. h) Supergiants are class I and red stars are of M or K spectral type. Anteres fits both qualifications so it is a red supergiant. i) As we move up in luminosity class (to smaller numbers), stars get bigger. Since Anteres is the only supergiant, it must have the largest radius. j) Any star that has left the main sequence has finished core hydrogen burning so stars of I-IV luminosity class have finished hydrogen burning: Antares, Canopus, and Aldebaran. k) On the main sequence, the more massive stars are also hotter. The hottest main sequence star is Spica so it must also be the most massive. l) The least massive stars must live the longest. The main sequence star with the least mass is Alpha Centauri A based on having the coolest spectral type.

Parallax and Distance. Use the parallax formula to calculate the given distance to each of the following stars. Give your answers in both parsecs and light-years. a) Alpha Centauri: parallax angle of 0.7420" b) Procyon: parallax angle of 0.2860"

d(in parsecs)=1/p(in arcseconds) d(in light-years)=3.26x[1/p(in arcseconds)] a) 1.35 parsecs 4.40 light-years b) 3.497 parsecs 11.399 light-years

Escape Velocity from a Red Giant. What is the escape velocity from a red giant with a mass of 1M(Sun)and a radius of 100R(Sun)? How does that velocity compare with the escape velocity from the Sun? Describe how your results help account for the fact that red giants have strong stellar winds.

escape velocity is defined as v = sqrt( 2 G M / r ). G=6.673x10^-11 M[sun]=1.9891x10^30 r[sun] = 1.392×10^9 /2 v[giant] = sqrt(2 x 6.673x10^-11 x 1.9891x10^36 / (140 x 1.392×10^9 / 2)) = 5.22x10^7 m/s v[sun] = sqrt(2 x 6.673x10^-11 x 1.9891x10^30 / (1.392×10^9 / 2)) = 6.176x10^5 m/s v[giant] / v[sun] = 84.5 solar wind particles must be much more energetic to escape the giant