Radial Oscillations of levitating atmospheres Deepika Bollimpalli - - PowerPoint PPT Presentation

Nicolaus Copernicus Astronomical Center, Warsaw.

Radial Oscillations of levitating atmospheres

Deepika Bollimpalli POLNS Conference

27th March’18

Levitating atmospheres

M

NS

Radiation pressure Gravity

• Likely formed in neutron stars with Super-Eddington luminosity

In Newtonian Theory, gravity and radiation force fall of as 1/r2, whereas in General theory

• f Relativity, both have a different radial dependence.

Levitating atmospheres

M

NS

Radiative force Gravity

= r0 2M 1-λ2

• Likely formed in neutron stars with Super-Eddington luminosity

In Newtonian Theory, gravity and radiation force fall of as 1/r2, whereas in General theory

• f Relativity, both have different radial dependence.

r0 r0

λ is the Eddington parameter

• We assume a static, spherically symmetric spacetime

, where

Levitating atmospheres - Hydrostatic equilibrium

• Euler equation
• Continuity equation

Optically thin limit

• Hydrostatic equilibrium

Polytropic atmospheres

• We assume polytropic equation
• f state,
• Background solution,

NS r r0 Fluid variables : X = X(r)eit Continuity & Euler equations Differential equation/ Eigenvalue problem Eigenvalues/ Eigenfrequencies

Analytical approach

=

Trick is

Eigenvalue problem

• Real part of the eigenvalue problem is a Gegenbauer differential equation
• Imaginary part yields

which gives the damping coefficient, and the Gegenbauer relation gives,

Undamped Oscillations ( =0)

• Ratio of breathing mode frequency to the fundamental mode frequency
• Frequency of an eigenmode

Undamped frequencies

• f the ten first normal

modes of the thin atmospheres as a function of the atmosphere location

Undamped frequencies

Radial Oscillations

Damped Oscillations

• Underdamped
• Critically damped
• Overdamped

> 0 >

Damped frequencies

Damped frequencies of the ten first normal modes of the thin atmospheres as a function of the atmosphere location

Critically damped

Observational interests

• For a given λ or r0, the frequency of the mode increases with the number and most
• f the frequencies of these oscillations are in the desirable range of 300-600 Hz
• bserved for the frequencies of oscillations during Type I X-ray bursts.
• These oscillations could also have applications to the ULX systems with neutron

stars that often attain their super-Eddington luminosities due to accretion process.

• Levitating atmospheres can deflect the light rays coming from the central compact

source and this can significantly affect the appearance of the central object in the

• bservations (Rogers et. al. 2017).
• An analogue of these atmospheres had been found to occur in the coronae of the

accretion disks (Fukue 1996 ).