Astrophysics: New Applications
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en-us2018 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemTue, 16 Jan 2018 13:02:03 GMTTue, 16 Jan 2018 13:02:03 GMTNew applications in the Astrophysics categoryhttps://www.maplesoft.com/images/Application_center_hp.jpgAstrophysics: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=185
Radiative Pressure
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A circular solar sail with 100% reflectivity and a radius (R) of 600 m. is at rest in the Earth's orbit. The sail is oriented so that its normal points directly to the Sun. The total mass of the sail and its payload "(m[sail]" ) is 900 kg. The sail moves toward Mars along a trajectory of increasing r. As it does so, its velocity (v) and acceleration (a) increase. Calculate the radiative pressure on the sail, a(r), v(r), v at the orbit of Mars, and how many days are required to reach Mars <BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Radiative Pressure" style="max-width: 25%;" align="left"/>A circular solar sail with 100% reflectivity and a radius (R) of 600 m. is at rest in the Earth's orbit. The sail is oriented so that its normal points directly to the Sun. The total mass of the sail and its payload "(m[sail]" ) is 900 kg. The sail moves toward Mars along a trajectory of increasing r. As it does so, its velocity (v) and acceleration (a) increase. Calculate the radiative pressure on the sail, a(r), v(r), v at the orbit of Mars, and how many days are required to reach Mars <BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154303&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberrySynchrotron Radiation: the Crab Nebula
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About 6500 years ago, a star in the constellation of Taurus exploded as a supernova. On July 4, 1054 CE, light from this explosion reached the Earth and was observed by Chinese astronomers, who made a record of their observation. The Crab Nebula has been identified as the detritus from the explosion; and in the approximately 960 years since the sighting of the supernova, the nebula has been expanding at a rate that roughly corresponds to its age.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Synchrotron Radiation: the Crab Nebula" style="max-width: 25%;" align="left"/>About 6500 years ago, a star in the constellation of Taurus exploded as a supernova. On July 4, 1054 CE, light from this explosion reached the Earth and was observed by Chinese astronomers, who made a record of their observation. The Crab Nebula has been identified as the detritus from the explosion; and in the approximately 960 years since the sighting of the supernova, the nebula has been expanding at a rate that roughly corresponds to its age.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154304&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryDuration of Synchrotron Radiation in a Typical Supernova Remnant
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Find an estimate for the length of time a typical supernova remnant can radiate, given its magnetic field strength.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Duration of Synchrotron Radiation in a Typical Supernova Remnant" style="max-width: 25%;" align="left"/>Find an estimate for the length of time a typical supernova remnant can radiate, given its magnetic field strength.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154305&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberrySirius A
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Sirius A (Alpha Canis Majoris) is close enough to Earth for accurate parallax measurement of its distance. This knowledge enables an accurate determination of its luminosity. Calculate its (A) mass, (B) radius, and (C) effective temperature, assuming (correctly) that it is a Main Sequence star with mass near that of the Sun.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Sirius A" style="max-width: 25%;" align="left"/>Sirius A (Alpha Canis Majoris) is close enough to Earth for accurate parallax measurement of its distance. This knowledge enables an accurate determination of its luminosity. Calculate its (A) mass, (B) radius, and (C) effective temperature, assuming (correctly) that it is a Main Sequence star with mass near that of the Sun.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154308&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryGraphing the Planck Function
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Make a plot of the Planck function for temperatures of 2000 K, 3000 K, 4000 K, 5000 K, 6000K, and 7000 K<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Graphing the Planck Function" style="max-width: 25%;" align="left"/>Make a plot of the Planck function for temperatures of 2000 K, 3000 K, 4000 K, 5000 K, 6000K, and 7000 K<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154309&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryCalculating the Zeeman Effect
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The Ap star BD+0°4535 has a powerful magnetic field of 2.1 T. Calculate the wavelengths of the H[alpha]
line caused by Zeeman splitting.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Calculating the Zeeman Effect" style="max-width: 25%;" align="left"/>The Ap star BD+0°4535 has a powerful magnetic field of 2.1 T. Calculate the wavelengths of the H[alpha]
line caused by Zeeman splitting.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154311&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryInverse Compton Scattering
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Consider an interstellar cloud of non-relativistic electrons at a temperature T. A large quantity of photons from the Cosmic Microwave Background with energies hν << kT flow continuously through the cloud.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Inverse Compton Scattering" style="max-width: 25%;" align="left"/>Consider an interstellar cloud of non-relativistic electrons at a temperature T. A large quantity of photons from the Cosmic Microwave Background with energies hν << kT flow continuously through the cloud.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154312&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryLine Broadening Due to Heisenberg's Uncertainty Principle
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Quantum effects based on the Uncertainty Principle cause a line damping profile known as a Lorentzian or Cauchy profile. This type of broadening is called natural broadening and is typically much smaller than Doppler broadening or pressure broadening, which are usually more significant in astrophysical objects. Find the amount of broadening due to the Uncertainty Principle.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Line Broadening Due to Heisenberg's Uncertainty Principle" style="max-width: 25%;" align="left"/>Quantum effects based on the Uncertainty Principle cause a line damping profile known as a Lorentzian or Cauchy profile. This type of broadening is called natural broadening and is typically much smaller than Doppler broadening or pressure broadening, which are usually more significant in astrophysical objects. Find the amount of broadening due to the Uncertainty Principle.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154315&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryLine Spectrum: Photon Energy
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A) Calculate the frequency of electromagnetic radiation corresponding to electron energy level 3. (B) Calculate the frequency of the spectral line resulting from an electron transition from n = 4 to n = 1. (C) Calculate the energy level corresponding to the longest Lyman wavelength. (D) Find the energy corresponding to the frequency of the Balmer line at 410 nm. (E) Find the energy level of an electron that has emitted a photon with wavelength of 486.1 nm from energy level 4. (F) Find the photon frequency required to move an electron from n = 1 to n = 3.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Line Spectrum: Photon Energy" style="max-width: 25%;" align="left"/>A) Calculate the frequency of electromagnetic radiation corresponding to electron energy level 3. (B) Calculate the frequency of the spectral line resulting from an electron transition from n = 4 to n = 1. (C) Calculate the energy level corresponding to the longest Lyman wavelength. (D) Find the energy corresponding to the frequency of the Balmer line at 410 nm. (E) Find the energy level of an electron that has emitted a photon with wavelength of 486.1 nm from energy level 4. (F) Find the photon frequency required to move an electron from n = 1 to n = 3.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154316&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryLimb Darkening: Eddington Approximation Versus Observational Data
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Calculate the amount of limb darkening of the Sun under the Eddington approximation and plot it together with the observational data given below.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Limb Darkening: Eddington Approximation Versus Observational Data" style="max-width: 25%;" align="left"/>Calculate the amount of limb darkening of the Sun under the Eddington approximation and plot it together with the observational data given below.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154317&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryOptical Depth
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Calculate (1) the optical depth of the whole Sun, (2) the distance of one optical depth of the outer layers of the Sun, and (3) the optical depth of an interstellar HI cloud in the Galaxy<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Optical Depth" style="max-width: 25%;" align="left"/>Calculate (1) the optical depth of the whole Sun, (2) the distance of one optical depth of the outer layers of the Sun, and (3) the optical depth of an interstellar HI cloud in the Galaxy<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154320&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberrySupernova Cas A
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Several parameters of the radio-bright supernova remnant Cas A, located approximately 3.4 kpc from Earth, are calculated in this worksheet. The supernova was observed in 1667, and its remnant has expanded to a diameter of 4 arcminutes. Relativistic electrons and protons are contained in the remnant, and the relativistic electrons are a source of synchrotron radiation.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Supernova Cas A" style="max-width: 25%;" align="left"/>Several parameters of the radio-bright supernova remnant Cas A, located approximately 3.4 kpc from Earth, are calculated in this worksheet. The supernova was observed in 1667, and its remnant has expanded to a diameter of 4 arcminutes. Relativistic electrons and protons are contained in the remnant, and the relativistic electrons are a source of synchrotron radiation.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154342&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryRadiative Flux
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Calculate the solar constant, the flux density of the Sun through a Johnson B filter, and the flux and temperature of the Sun.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Radiative Flux" style="max-width: 25%;" align="left"/>Calculate the solar constant, the flux density of the Sun through a Johnson B filter, and the flux and temperature of the Sun.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154299&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryPlanck's law, Stefan-Boltzmann law, Wien's law
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Based on Planck's law for radiance, calculate (A) the Stefan-Boltzmann law for flux, (B) Wien's displacement law and (C) the colour index for the following ranges of two colour filters: blue: 400-500 nm; red: 600-700 nm.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Planck's law, Stefan-Boltzmann law, Wien's law" style="max-width: 25%;" align="left"/>Based on Planck's law for radiance, calculate (A) the Stefan-Boltzmann law for flux, (B) Wien's displacement law and (C) the colour index for the following ranges of two colour filters: blue: 400-500 nm; red: 600-700 nm.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154301&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryLuminosity
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(A) Find the number which, when raised to the fifth power, equals 100. Then derive the formula relating magnitudes to fluxes. (B) Find the absolute magnitude of Sirius and the Sun. (C) Calculate the bolometric magnitude of Antares. Calculate its radius, given the corrected bolometric magnitude. Calculate the temperature of Rigil Kent. (D) Find the bolometric correction of Sirius.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Luminosity" style="max-width: 25%;" align="left"/>(A) Find the number which, when raised to the fifth power, equals 100. Then derive the formula relating magnitudes to fluxes. (B) Find the absolute magnitude of Sirius and the Sun. (C) Calculate the bolometric magnitude of Antares. Calculate its radius, given the corrected bolometric magnitude. Calculate the temperature of Rigil Kent. (D) Find the bolometric correction of Sirius.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154302&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryThe Sun
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For the Sun, calculate the (A) central pressure, (B) central temperature, (C) pressure scale height, (D) adiabatic sound speed for a monatomic gas, and (E) the adiabatic convection by Mixing-length Theory<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="The Sun" style="max-width: 25%;" align="left"/>For the Sun, calculate the (A) central pressure, (B) central temperature, (C) pressure scale height, (D) adiabatic sound speed for a monatomic gas, and (E) the adiabatic convection by Mixing-length Theory<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154310&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryBinary Star or Star with Exoplanet: Fitting a Curve to the Orbital Data
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The star 51 Pegasi shows a regular variation in redshift, indicating that it is orbiting around a common barycentre. This indicates the possibility that it is a single-line spectroscopic binary or that it has a planet of significant mass orbiting it. Fit a curve to the given data as a first step in determining which of these possibilities is the case.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Binary Star or Star with Exoplanet: Fitting a Curve to the Orbital Data" style="max-width: 25%;" align="left"/>The star 51 Pegasi shows a regular variation in redshift, indicating that it is orbiting around a common barycentre. This indicates the possibility that it is a single-line spectroscopic binary or that it has a planet of significant mass orbiting it. Fit a curve to the given data as a first step in determining which of these possibilities is the case.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154339&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryPulsating Star
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Given the relevant data of a pulsating star, calculate the star's rate of change in surface velocity, radius, and pressure.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Pulsating Star" style="max-width: 25%;" align="left"/>Given the relevant data of a pulsating star, calculate the star's rate of change in surface velocity, radius, and pressure.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154340&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryReimer's Mass-Loss Rate
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Reimers' (1977) mass-loss rate formula is an empirical formula that estimates the rate of mass loss for stars on the asymptotic giant branch (AGB).<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Reimer's Mass-Loss Rate" style="max-width: 25%;" align="left"/>Reimers' (1977) mass-loss rate formula is an empirical formula that estimates the rate of mass loss for stars on the asymptotic giant branch (AGB).<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154325&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert RoseberryMain Sequence Lifetime
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(A) Calculate the Main Sequence lifetime of the Sun. (B) Devise a formula in solar units relating the Main Sequence lifetime to stellar mass, and use it to calculate the Main sequence lifetimes of (i) a 17-solar-mass star, and (ii) a 0.34-solar-mass star.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.<img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Main Sequence Lifetime" style="max-width: 25%;" align="left"/>(A) Calculate the Main Sequence lifetime of the Sun. (B) Devise a formula in solar units relating the Main Sequence lifetime to stellar mass, and use it to calculate the Main sequence lifetimes of (i) a 17-solar-mass star, and (ii) a 0.34-solar-mass star.<BR><BR>For more information on how to do this problem, and for tables of relevant data, see <A HREF="http://www.canismajor.ca">www.canismajor.ca</A>.https://www.maplesoft.com/applications/view.aspx?SID=154326&ref=FeedMon, 25 Sep 2017 04:00:00 ZDr. Robert RoseberryDr. Robert Roseberry