Log T Log L xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx most massive least massive main sequence
(b) Least massive: not hot enough to begin nuclear reactions [1]; most massive: also extremely
luminous, blowing off outer layers to lose mass (Eddington Limit) [2].
(c)Small movement left to right as star ages, otherwise star remains at same location on M-S
throughout its H-burning lifetime. [2]
(d) using mass-luminosity relation M
α∝ L, and taking α = 3.0 (though any α in range 3.0−3.5
acceptable)
L
L
⊙=
M
M
⊙ α[1]
Plugging in numbers:
For M
= 0.1M
⊙H⇒ L/L
⊙= 0.1
3= 10
−3[1]
For M
= 50M
⊙H⇒ L/L
⊙= (50)
3≈ 10
5(e) Lifetime τ
= energy available ÷ luminosity ∝ M/M
α. Hence τ
∝ M
−2(see notes) [2] So
τ
τ
⊙=
M
⊙M
For M
= 0.1M
⊙H⇒ τ = 100τ
⊙= 10
12years [1]
For M
= 0.1M
⊙H⇒ τ = τ
⊙/2500
= 4 × 10
6years [1]
4.29
e How are stellar surface temperatures, Te, measured? What is the sequence of stellar spectral types from hot tocool? In what part of this sequence would you see spectral lines of (a) helium,
(b) molecules? [5]
A spherical star of ionised hydrogen has a mass M, radius R, and uniform density ρ. Given that gas pressure is P = 2nkT , where n is the proton number density, show by considering the pressure needed to balance the gravitational attraction of two halves of the star that its core temperature Tcdepends approximately on M and R
according to
[7]
Tc= G Mmp
6k R .
Give an expression for the luminosity L of a star in terms of R and Te, and given also the mass-luminosity
relation L∼ M7/2on the Main Sequence (MS), show that the surface temperature of a MS star depends on M and R as
[5]
Te∼ M
7/8
R1/2.
Solution: no solution available
4.30
e Summarise how a large gas cloud turns into a star, stating at what point it becomes s Main Sequence star. Whyis a protoplanetary disk produced? [3]
Estimate the temperature reached when a cloud of one solar mass has collapsed to the size of the Sun. [2]
Solution: no solution available
4.31
e Sketch two Hertzprung-Russell diagrams, one for a young star cluster, and one for an old star cluster, labelling the axes, the regions of main stellar types, and the locations of stars of 0.1, 1, and 10 solar masses. [4]State why the two diagrams differ. [1]
Solution: no solution available
4.32
e Using the Stefan-Boltzmann law L= 4π R2σT4relating luminosity L with temperature T and stellar radius R, sketch a model H-R diagram, showing clearly how lines of constant radius arise, and why they have gradient ofminus 4. [4]
Using a second drawing, sketch an observed H-R diagram, showing the position of the Sun, and marking
clearly the main sequence, white dwarf and red giant regions. [3]
The proton-proton chain is the main nuclear fusion reaction for main sequence stars. The reaction can be summarised as
411H→42He+ energy
in which the mass of a helium nucleus is 3.972 times the mass of a proton. Assuming that the Sun initially is composed entirely of hydrogen, and that only 10% of its mass is available for hydrogen fusion, estimate the main
sequence lifetime of the Sun. [6]
Solution: no solution available
4.33
e List the spectral classes in the Harvard Scheme for stellar classification, and state the basis on which the scheme is founded. In which class are:(a) molecular lines strong?
(b) hydrogen lines most prominent? [5]
Solution: no solution available
4.34
e The Stefan-Boltzmann law can be expressed in the form L ∝ ST4where S is the surface area of the emitter at temperature T , and L is its luminosity. Explain how this equation can be used to link the size and effective temperature of stars. Ensure that you explain the meaning of effective temperature. State further how the conceptof radiation flux arises, by considering the inverse square law. [4]
Given that the fluxes of radiation from two objects may be characterised in terms of a difference in appar- ent magnitude, according to the equation
m1− m2= −2.5 log(F1/F2),
demonstrate that for two stars at the same distance from an observer, with equally perfect viewing conditions,
m1− m2= −5 log(R1/R2)− 10 log(T1/T2),
where R1, R2are the radii of the stars, and T1, T2are the corresponding effective temperatures. [3]
An eclipsing binary star system comprising of a larger, brighter star with a smaller, dimmer companion can be identified from its characteristic light curve. Sketch this light curve, identifying the primary and secondary
minima, and explaining briefly how they arise. [4]
For such a system, show that the ratio of the stellar temperatures can be given by
T1 T2 = F − FP F− F S 14 ,
where F, FPand FSare respectively the flux received from the binary system when no eclipse is present, the
flux received during a primary eclipse, and the flux received during a secondary eclipse. [6]
Solution: no solution available
4.35
e What nuclear process sets in after H to He burning runs out in main-sequence (MS) stars? [1] What readjustments in the stellar structure cause this to happen? [1] What is the heaviest element reached in the fusion sequence of massive MS stars? [1] Estimate the energy released when the 2 solar mass core of a massive star collapses to a radius of 10km andname the resulting observational phenomenon. [2]
[G= 6.7 × 10−11N m2kg−2, solar mass = 2× 1030kg]
Solution: no solution available
4.36
e In the Harvard spectral classification scheme, state the spectral type that is characterised by the following line emissions:(a) strong molecular bands
(c) dominant metallic lines.
Describe very briefly how such spectral lines form, and why they are useful for stellar classification. [5]
Solution: no solution available
4.37
e Name and briefly describe the three outer layers of the Sun. [5]Solution: no solution available
4.38
e Sketch the structure of the Sun, indicating the approximate temperatures of the distinct radial regions. [2]What process is believed to heat the corona? [1]
Estimate the temperature at the centre of the Sun given that it has mass 2.0× 1030kg and radius 7.0× 108m. [2]