Models for velocity decrease in HH34

The conservation of the energy flux in turbulent jets that propagate in the interstellar medium (ISM) allows us to deduce the law of motion when an inverse power law decrease of density is considered. The back-reaction that is caused by the radiative losses for the trajectory is evaluated. The velocity dependence of the jet with time/space is applied to the jet of HH34, for which the astronomical data of velocity versus time/space are available. The introduction of precession and constant velocity for the central star allows us to build a curved trajectory for the superjet connected with HH34. The bow shock that is visible in the superjet is explained in the framework of the theory of the image in the case of an optically thin layer.


Preliminaries
The velocity evolution of the HH34 jet has recently been analysed in [SII] 2, (672 nm), frames and Table 1 in [4] reports the Cartesian coordinates, the velocities, and the dynamical time for 18 knots in 9 years of observations. To start with time, t, equal to zero, we fitted the velocity versus distance with the following power law where v and x are the velocity and the length of the jet, 0 v is the velocity at 0 x x = and α with its relative error is a parameter to be found with a fitting procedure. The integration of this equation gives the time as a function of the position, x, as given by the fit ( ) where 0 x is the position at 0 t = . The fitted trajectory, distance versus time, is from which we can conclude that the velocity decreases with increasing distance, see Figure 1.
The time is derived from Equation (2) and Table 1

Two Simple Models
When a jet moves through the interstellar medium (ISM), a retarding drag force drag F , is applied. If v is the instantaneous velocity, then the simplest model assumes , n drag F v ∝ (6) where n is an integer. Here, the case of 1 n = and 2 n = is considered. In classical mechanics, 1 n = is referred to as Stoke's law of resistance and 2 n = is referred to as Newton's law of resistance.

Stoke's Behaviour
The equation of motion is given by The velocity as function of time is The time as function of distance is obtained by the inversion of this equation The velocity as a function of space is The numerical value of B is where 1 v is the velocity at point 1 x ; the data of Table 1 gives B = 0.0009549 ( Figure 2).

Newton's Behaviour
The equation of motion is The velocity as function of time is  where 0 v is the initial velocity. The distance at time t is The time as function of distance is obtained by the inversion of the above The velocity as function of the distance is ( ) The numerical value of A is where 1 v is the velocity at point 1 x ; the data of Table 1 gives A = 5.68381834.

Energy Flux Conservation
The conservation of the energy flux in a turbulent jet requires a perpendicular section to the motion along the Cartesian x-axis, A ( ) 2 A r r = π (19) where r is the radius of the jet. Section A at position 0 x is ( ) where α is the opening angle and 0 x is the initial position on the x-axis. At position x, we have International Journal of Astronomy and Astrophysics ( ) The conservation of energy flux states that x , see Formula A28 in [16]. More details can be found in [17] [18]. The density is assumed to decrease as a power law where 0 ρ is the density at 0 x x = and δ a positive parameter. The differential equation that models the energy flux is ( ) The velocity as a function of the position, x, Figure 3 reports the velocity as a function of the distance and the observed points.
We now have four models for the velocity as a function of time and Table 2 reports the merit function 2 χ , which is evaluated as and the velocity as function of time  Figure 4 reports the trajectory as a function of time and of the observed points.
The rate of mass flow at the point x, ( ) and the astrophysical version is

The Back Reaction
Let us suppose that the radiative losses are proportional to the flux of energy ( ) ( ) ( ) 3 1 , 2 x where  is a constant that is thought to be 1  . By inserting in the above equation the considered area, ( ) A x , and the power law density here adopted the radiative losses are By inserting in this equation the velocity to first order as given by Equation (25), the radiative losses, The sum of the radiative losses between 0 x and x is given by the following integral, L, The conservation of the flux of energy in the presence of the back-reaction due to the radiative losses is x The real solution of the cubic equation for the velocity to the second order,  The presence of the back-reaction allows us to evaluate the jet's length, which can be derived from the minimum in the corrected velocity to second order as a function of x, The solution for x of the above minimum allows us to derive the jet's length,

The Extended Region
To deal with the complex shape of the continuation of HH34 (e.g. see the new region HH173 discovered by [19]), we should include the precession of the source and motion of the host star, following a scheme outlined in [20]. The various coordinate systems are ( ) , , . The vector representing the motion of the jet is represented by the following 1 3 × matrix: where the jet motion L(t) is considered along x axis. The jet axis, x, is inclined at an angle prec Ψ relative to an axis ( ) 1 x , and therefore the 3 3 × matrix, which represents a rotation through z axis, is given by: The jet is undergoing precession around the ( ) The last translation represents the change of the framework from ( ( ) which is co-moving with the host star, to a system ( ( ) On assuming, for the sake of simplicity, that 0 The three components of the previous 1 3 × matrix A represent the jet's motion along the Cartesian coordinates as given by an observer who sees the star moving in a uniform motion. The point of view of the observer can be modeled by introducing the matrix E, which represents the three Eulerian angles , , Θ Φ Ψ , see [21]. A typical trajectory is reported in Figure 7 and a particularised point of view of the same trajectory is reported in Figure 8 in which a loop is visible.

Image Theory
This section summarises the continuum observations of HH34, reviews the transfer equation with particular attention to the case of an optically thin layer, analyses a simple analytical model for theoretical intensity, reports the numerical algorithm that allows us to build a complex image and introduces the theoretical concept of emission from the knots.

Observations
The system of the jet and counter jet of HH34 has been analysed at 1.5 μm and 4.5 μm, see Figure 3 in [3]. The intensity is almost constant,

The Transfer Equation
For the transfer equation in the presence of emission only see, for example, [22] or [23], is where I ν is the specific intensity, s is the line of sight, j ν is the emission coefficient, k ν is a mass absorption coefficient, ρ is the density of mass at position s, and the index ν denotes the frequency of emission. The solution to where ν τ is the optical depth at frequency ν : We now continue to analyse a case of an optically thin layer in which ν τ is very small (or k ν is very small) and where the density ρ is replaced by the concentration ( ) C s of the emitting particles: where K is a constant. The intensity is now The increase in brightness is proportional to the concentration of particles integrated along the line of sight.

Theoretical Intensity
The flux of observed radiation along the centre of the jet, c I is assumed to scale as ( ) ( ) 0 0 0 0 2 ; , , , ; , , , , where Q, the radiative losses, is given by Equation (33). The explicit form of this equation is This relation connects the observed intensity of radiation with the rate of energy transfer per unit area. A typical example of the jet of HH34 at 4.5 μm is reported in Figure 9.

Emission from a Cylinder
A thermal model for the image is characterised by a constant temperature and density in the internal region of the cylinder. Therefore, we assume that the number density C is constant in a cylinder of radius a and then falls to 0, see the simplified transfer Equation (51). The line of sight when the observer is situated at the infinity of the x-axis and the cylinder's axis is in the perpendicular position is the locus parallel to the x-axis, which crosses the position y in a Cartesian x-y plane and terminates at the external circle of radius a. A similar treatment for the sphere is given in [24]. The length of this locus in the optically thin layer approximation is ( ) A typical example of this cut is reported in Figure 10 and the intensity of all the cylinder is reported in Figure 11.

Numerical Image
The numerical algorithm that allows us to build a complex image in the optically thin layer approximation is now outlined.   The orientation of the object is characterised by the Euler angles ( ) , , Φ Θ Ψ and therefore by a total 3 3 × rotation matrix, E, see [21]. The matrix point is represented by the following 1 3 × matrix, B,  The intensity 2D map is obtained by summing the points of the rotated images.
A typical result of the simulation is reported in Figure 12, which should be compared with the observed image as given by Figure 13.

The Mathematical Knots
The trefoil knot is defined by the following parametric equations: The visual image depends on the Euler angles, see Figure 14.
The image in the optically thin layer approximation can be obtained by the numerical method developed in Section 6.5 and is reported in Figure 15.
This 2D map in the theoretical intensity of emission shows an enhancement where two mathematical knots apparently intersect.  Figure 15. Image of the trefoil with parameters as in Figure 14, the side of the box in pc is 1 and the radius of the tube in pc is 0.006.

Laws of motion:
We analysed two simple models for the law of motion in HH objects as given by the Stoke's and Newton's behaviour, see Section 3. A third law of motion is used for turbulent jets in the presence of a medium whose density decreases with a power law, as given by Equation (23). The model that is adopted for the turbulent jets conserves the flux of energy. For example, Equation (25) reports the velocity as function of the position. The 2 χ analysis for observed theoretical velocity as function of time/space, see Table 2, assigns the smaller value to the turbulent jet.
Back reaction: The insertion of the back reaction in the equation of motion allows us to introduce a finite rather than infinite jet's length, see Equation (39).
The extended region: The extended region of HH34 is modeled by combining the decreasing jet's velocity with the constant velocity and precession of the central object, see the final matrix (45).
The theory of the image: We have analysed the case of an optically thin layer approximation to provide an explanation for the so called "bow shock" that is visible in HH34. This effect can be reproduced when two emitting regions apparently intersect on the plane of the sky, see the numerical simulation as given by Figure 12. This curious effect of enhancement in the intensity of emission can easily be reproduced when the image theory is applied to the mathematical knots, see the example of the trefoil in Figure 15.