Structure and Evolution of Stars Lecture 13: Homology Solutions (#1)

Transcription

1 Structure and Evolution o Stars Lecture 13: Homology Solutions (#1) Equations o Stellar Structure with mass as the independent variable Review o Observed relations between L, T and M Homology sel-similar solutions to stellar structure equations Provides dependencies o physical properties on mass Mass-Luminosity relationship Star o the Week #3: The current search or ailed supernovae Candidate #1 in NGC 696 Structure & Evolution o Stars 1

2 Equations o Stellar Structure Reached stage where we have set o dierential equations with unknowns (incorporating dependence o the pressure, opacity and energy generation rate in terms o density, temperature and composition rom our understanding o physics) and well-deined boundary conditions solution now possible in principle Note that there is no time dependence involved and thus the solutions to the equations can tell us about stellar structure but not, on their own, about stellar evolution The change in the composition o a star is critical or understanding how a star evolves. Change in mass with time due to nuclear reactions is very small (<1%) but mass-loss resulting rom exceeding Eddington Luminosity or cataclysmic events (supernovae are an extreme example) is also key Structure & Evolution o Stars

3 Equations o Stellar Structure Our amiliar equations o stellar structure or case o radiative energy transport and, or simplicity, opacity κconstant More tractable to solve using m as the independent variable rather than the radial distance r Have boundary conditions: dp dm dr dm dt dm Gm πr 1 πr ρ 3 ac κ 3 T L (πr ) r0 and L0 at m0 Ρ0 and T0 at mm dl dm ε ε ρt Structure & Evolution o Stars 3 0 n

4 Explaining the Main Sequence Observationally we have seen that there is a well-deined relation between luminosity and eective temperature or stars in the HRdiagram log L α logt + e where the slope, α, becomes steeper at higher L constant From measurement o stellar masses in binary systems have also established the existence o a mass-luminosity relation that is well approximated by L M Hypothesis is that the main-sequence is the locus o stars o dierent masses burning hydrogen in their cores. Can we now use the stellar structure equations to reproduce the observational data? β Structure & Evolution o Stars

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7 Homology Even assuming that stars on the main sequence have constant composition, are in hydrostatic and thermal equilibrium, possess constant opacity, with pressure due only to that o a perect gas still have a set o complex nonlinear dierential equations (Slide 3) that are diicult to solve or the variables r(m), P(m), ρ(m), T(m) and L(m) over range 0<m<M Instead, consider the relationship between variables as a unction o mass by using dimensional analysis o the equations Homology, or sel-similar, solutions to the equations describe how a particular solution to the equations scales as a unction o a dimensionless variable ractional mass in our case Structure & Evolution o Stars 7

8 Structure & Evolution o Stars 8 Homology ) ( ) ( ) ( ) ( ) ( ) ( ), ( ), ( ), ( ), ( L x L T x T x P x P R x r m L m T m m P m r M m x ρ ρ ρ Deine the dimensionless variable x, the ractional mass: Replace the ull solutions to the equations o stellar structure: with dimensionless unctions o x: 1 (x), (x) employing deinitions: where the coeicients N have the dimensions o the original unctions

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10 Homology Now substitute or x, r and P in the hydrostatic equilibrium equation to give: P M d dx GMx π 1 R x, 1 and are dimensionless the dimensions on both sides must be equal P GM R can separate the equation into parts, adopting a proportionality constant o unity d dx x π 1 ; P GM R Structure & Evolution o Stars 10

11 Homology Repeat the exact same procedure or all stellar structure equations along with PnkT or a perect gas to give a set o 5-pairs equations (PnkT gives the 3 rd equation as listed here) lh set consists o nonlinear dierential equations independent o M or the unctions 1, 5 rh set consists o algebraic equations that relate the dimensional coeicients P,R d dx d1 dx d dx d5 dx x π π ; n 3 (π Structure & Evolution o Stars ; 3 ; ; 5 P T 1 L ) ρ GM R M R µ m k ; 0 3 H L ε ρ T P ρ n ac T R κ M M

12 Homology Can combine the solutions o the algebraic and the dierential equations to provide behaviour o r,p as a unction o M Key point is that the shape o the proiles as a unction o the ractional mass, x, is the same in all stars, with the proiles diering only by a constant actor determined by the mass The similarity property is termed homology Solving only the algebraic equations can derive dependence o the physical properties on mass r P ρ T L 1 ( x) R 3 5 ( x) P ( x) T ( x) L ( x) ρ Structure & Evolution o Stars 1

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14 Homology Substitute or P and ρ (using irst two equations on previous but two slide) into the 3 rd equation gives: T µ m k H G M R Now substituting or T into the th equation gives: Relation is true at any value o the ractional mass but putting ractional mass equal to unity and we have the massluminosity relationship: L L ac µ m κ k M 3 H G M 3 Structure & Evolution o Stars 1

15 The Search or Failed Supernovae Candidate #1 in NGC 696 Gerke et al. 015 MNRAS Adams et al. 017 MNRAS Monitoring 7 galaxies with distance <10Mpc. Four years o data Ater ailed supernovae and quantitatively search or objects where luminosity has declined by more than 10 L sun Check viable via distance-modulus (Lecture ) d 10 (m-m+5)/5 pc - need a large telescope! Practical problems include alse positives due to limitations/eatures o the imaging/data and astrophysical supernovae imposters what might be examples? Structure & Evolution o Stars 15

16 Structure & Evolution o Stars 16 S Smartt (015, PASA, 3, 16 >30 progenitors)

17 Subaru optical image o NGC 696 Structure & Evolution o Stars 17

18 The Search or Failed Supernovae Candidate #1 in NGC 696 LBT in Arizona (two 8.m mirrors). CCD detector provides 3 by 3 arcmin images Well matched to covering most o galaxy, with depths R~6 mag with S/N~5 and seeing o 1 arcsec. Pixels 0.3 arcsec on a side Variable stars, due to many dierent phenomena, add conusion. Use a deep image in good seeing as the reerence. Main search-tool is to create a dierence -image by subtracting the reerence image rom each exposure. Number o epochs ~6 per year and galaxies have between 6-30 epochs Structure & Evolution o Stars 18

19 The Search or Failed Supernovae Candidate #1 in NGC 696 The only viable candidate rom the Gerke etal. search Have temperature rom V-R colour and luminosity rom magnitude and galaxy distance (5.96Mpc) Mass estimate 18-5 solar masses (rom models) Structure & Evolution o Stars 19

20 Follow-up Hubble Space Telescope observations in Adams et al. Existing HST precursor images (top), new 015 images (middle) and near-inrared images (bottom) Good or bad or the ailed supernovae hypothesis? Structure & Evolution o Stars 0

21 Photometric history, lightcurve, o the candidate. Y-axis lips rom linear to logarithmic at 10 Spitzer near-inrared magnitudes challenging given resolution! Observations by an amateur constrain peak brightness. Structure & Evolution o Stars 1

22 Photometry and model or progenitor (red) with dashed curve unreddened (without extinction) Now called N696-BH1! Need star (T80K and dust T1800K) L L sun Photometry now (black) with model or spectrum o star T1500K obscured by a dusty wind T1500K Qualitative explanation or shape/luminosity o obscured model? Structure & Evolution o Stars

23 Lecture 13: Summary Equations o stellar structure more tractable in orm with mass as the independent variable Homology, sel-similar, solutions allows the calculation o the dependence o the key physical parameters on the mass o the stars First well-deined dependencies to explain are the massluminosity and luminosity-temperature relationships or the main sequence Homology approach provides dependence o luminositymass relationship straightorwardly Observations o nearby galaxies oers best prospect o identiying ailed supernovae Structure & Evolution o Stars 3

24 Picture Credits Slide 6 Phillips, Wiley Slides 15 and 16 Astronomical Journal Slides 18- Luhmann et al. Structure & Evolution o Stars