Dynamical Friction on supersonic gravitating
and non-gravitating spheres in a gaseous medium
In 2015 and early 2016, I worked on MHD simulations of supersonic gravitating and non-gravitation spheres in a gaseous medium for my MSc thesis at the University of Tuebingen. Our setup was tailored to represent a binary star system during the common envelope phase to analyse the forces acting on the secondary star.
Extracts from my thesis have been included on this page to give an overview of the project. Some results have been included in Thun (2016).
Binary Star Systems
More than half of the stars we observe today are said to be stars in multiple star systems, the
most common variety are binary star systems, or binaries. During their evolution binary star
system might experience a common envelope phase
Classification of Binary Star Systems
The first recorded observation of a true binary star system dates back to an observation of
Johann Baptist Cysat in 1619. Binary star systems are stellar systems containing two gravitationally bound stars moving around their common centre of mass. One differentiates between detached binaries, semi-detached binaries and close contact binary star systems. On this page, I will use the following convention regarding the designation of the stars in such a system: The primary star is the the initially more massive star while the companion star is referred to as secondary star.
Detached Binary Star Systems
Detached binary star systems are systems containing two stars that are gravitationally bound but far enough apart so they do not influence each other’s evolution.
Semi-detached Binary Star Systems
Semi-detached binary star systems are systems containing two stars that are gravitationally bound with one of the stars filling its Roche lobe. Mass can escape via the Lagrange point L1 and spiral down towards the companion star. If the secondary star is small compared to the distance a separating them, the inflowing mass cannot be accreted. Instead the gas will form an accretion disk around the secondary star. Viscosity in the disk will then cause the gas in the accretion disk to loose energy, heating it through conversion of potential into thermal energy and causing it to spiral inwards. Mass will continue to be added to the accretion disk via the so-called hot spot as long as the primary star is still massive enough to fill its Roche lobe.
Close Contact Binary Star Systems
Close contact binary star systems are systems containing two stars that are gravitationally bound with both stars filling their Roche lobe. Effectively they now move through one common envelope.
The Formation of Binary Star Systems
In theory it is possible that binary star systems are formed when one star captures a, up
until then, unbound star moving past it. However this scenario seems rather unlikely and cannot explain the large amount of, partly very young, binary star systems observe. Instead
it is assumed that binary star systems form together in molecular clouds when high angular
momentum leads to a fragmentation of the cloud.
Illustration of a detached binary star system.
Illustration of a semi-detached binary star system.
Illustration of a close contact binary star system.
Illustration of a spherical object in a gaseous medium. An overdensity has formed behind the object leading to gravitational interactions between the wake and the object itself.
We assume that the object is instantaneously introduced into the medium. The object is spherical, has a mass of M and a propagation velocity of v . The medium through which it travels is presumed to be in hydrostatic equilibrium and can be a fluid, gas or, in a more abstract sense, a cluster of stars or galaxies. At the beginning we will find a random
distribution of particles inside the medium. Once the object begins to travel through the medium and perturb it, a wake will form behind it, resulting in the formation of an overdensity as shown in the figure. In the region yet in front of and so far undisturbed by the object, the distribution of particles is still random. However, behind the object the density has clearly increased. Gravitational interactions between the overdense wake and the object will lead to a dynamical (or gravitational) drag force influencing the object.
Grid Based MHD
All simulations were carried out with the MHD code PLUTO. PLUTO solves the equations of hydrodynamics on a grid using the finite volume method (FVM). In the scope of the FVM the computational domain is divided into intervals, from now on referred to as grid cells. A quantity q such as for example the density, is then integrated over the entire cell and updated after each time step by approximating the flux through the endpoints of the spatial interval. The concept reduced to one dimension is shown in the figure to the left.
Visualization of the Finite Volume Method.
Visualization of the numerical setup.
The figure to the right shows a visualisation of the numerical setup.
We place a spherical object at the centre of our coordinate system. We will use spherical coordinates in 2D and therefore any position within the computational domain will be specified by the coordinates r and θ, where r ∈ [0, R Domain ] and θ ∈ [0, π].
In the rest frame of the spherical object, the movement through the gaseous medium is simulated by introducing an inflow of gas with the density ρ and velocity v.
Forces Acting on the Gaseous Medium and a (Non-) Gravitating Sphere
The figure to the right shows the z-components of the individual forces as well as the total force acting on the sphere and medium in code units.
The individual force contributions are due to momentum transport through the outer boundary (black triangles), pressure on the inner boundary and therefore the surface of the sphere (blue triangles), and pressure on the outer boundary (green triangles). We consider the object itself to be impermeable, therefore no momentum transport through the inner boundary is taking place. The absolute value of all individual force contributions increases with
an increasing Mach number. The purely hydrodynamic drag force is negative, therefore pointing in the opposite direction to the motion of the sphere through the gaseous medium. As the system is considered to be in equilibrium, the total force should disappear. In our simulations the total
force does not add up to zero but seems negligible small.
In the figure to the left, the black triangles represent the force due to momentum transport through the outer boundary, the blue circles and the green triangles are forces due to pressure on the inner and outer boundary respectively and the cyan triangles represent the z-component of the cumulative numerical drag force due to the gravitational interaction between the sphere and the wake of gas behind it. The absolute values of all force contributions decrease with an increasing Mach number.
In the Mach number regime we are considering, the absolute value of the hydrodynamic force is small compared to the absolute value of the cumulative gravitational drag force. We can therefore assume that the hydrodynamic drag force on the sphere can be neglected in light of the much larger gravitational drag force. However, towards even larger Mach numbers, the absolute value of the hydrodynamic drag will increase while the gravitational drag will decrease leading to a crossover point. Regardless of the Mach number, both forces are negative and are therefore pointing in opposite direction to the movement of the sphere through the gaseous medium.
Individual contributions to the total force acting on a non-
gravitating sphere moving through a gaseous medium.
Individual contributions to the total force acting on a
gravitating sphere moving through a gaseous medium.
Parameter Study of the Adiabatic Index
Standoff distance normalised to the accretion radius of the
object versus the adiabatic index γ.
By preforming a parameter study and analysing the dependence of the standoff distance (the distance between shock front and object) on the adiabatic index, we find the relation shown in the figure to the right.
Using this result, Thun (2016) were able to develop an improved equation for the dynamical friction and confirm that the minimum radius of relevance is the standoff distance.
Supersonic motion of a non-gravitating sphere
in a gaseous medium
Supersonic motion of a gravitating sphere
in a gaseous medium.