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 Issue A&A Volume 532, August 2011 A52 8 Stellar atmospheres https://doi.org/10.1051/0004-6361/201116866 21 July 2011

## 1. Introduction

It is now well established that the Hanle effect serves as a diagnostic tool to determine the deterministic or turbulent weak magnetic fields on the Sun (see e.g., Stenflo 1982; Trujillo Bueno et al. 2004). The Hanle effect is a magnetic modification of the Rayleigh scattering spectral line polarization. The scattering polarization signatures in strong resonance lines are particularly sensitive to the frequency redistribution mechanism used in their computation. In the case of Rayleigh scattering, the differences in Q/I computed using angle-averaged and angle-dependent partial frequency redistribution (PRD) functions is between 10% and 30% (Sampoorna et al. 2011, see also Faurobert 1988). In the presence of weak magnetic fields, Nagendra et al. (2002) showed that the Stokes U is very sensitive to the angle-averaging of the redistribution function. However, these authors used a computationally expensive numerical method to solve the polarized line transfer equation including angle-dependent PRD function and the Hanle effect. Our aim here is to present an approximate method based on the concept of the single scattering approximation (see Frisch et al. 2009; Frisch 2010; Anusha et al. 2010; Sampoorna et al. 2011) to solve the above-mentioned problem efficiently.

The problem of angle-dependent PRD is characterized by intricate coupling between angle and frequency variables. This makes the evaluation of scattering integrals, and the solution of polarized transfer equation a challenging problem. A decomposition technique to reduce the non-axisymmetric Stokes transfer equation to cylindrically symmetric one was developed by Frisch (2007), for the case of an angle-averaged PRD with the Hanle effect. This technique was extended by Frisch (2009) to handle angle-dependent PRD with the Hanle effect. Basically the Stokes vector (I,Q,U) is first decomposed into a set of six irreducible components . In the particular case of angle-dependent PRD, these components are also non-axisymmetric. An azimuthal Fourier expansion of the angle-dependent PRD

functions then allows us to further reduce into an infinite set of cylindrically symmetric Fourier coefficients . This method helps us to construct an infinite set of integral equations for the Fourier coefficients of the irreducible source vector components. As shown in Frisch (2010), the problem simplifies when the magnetic field is set to zero or assumed to be turbulent. In this case, the Fourier wavenumber k takes only the values k = 0,1,2, and four components are sufficient to represent the polarized radiation field. As shown in Sampoorna et al. (2011), the polarized radiation field can be calculated by solving a set of integral equations for the corresponding cylindrically symmetric irreducible source vector components. Several numerical methods are described in this reference : two accelerated lambda iteration (ALI) methods, associated with either a core-wing or a frequency-angle by frequency-angle technique, and one scattering expansion method, based on a Neumann series expansion in terms of the mean number of scattering events.

For the Hanle effect with an angle-dependent PRD, the numerical solution is possible only if one truncates the infinite set of integral equations for the Fourier coefficients of the irreducible source vector components. For the type II PRD function (frequency coherent scattering in the atomic rest frame) of Hummer (1962), Domke & Hubeny (1988) showed that, at least five terms in the azimuthal Fourier expansion are needed to get a good agreement with the exact value of the function. This implies that the azimuthal order k takes values 0,  ± 1,  ± 2,  ± 3, and  ± 4, for each combination of K and Q of the irreducible source vector. For the Hanle problem, K = 0 and 2, with  − K ≤ Q ≤  + K. Hence, we obtain a total of 54 coupled integral equations. Although they can be numerically solved, it is computationally demanding. To simplify this numerically difficult problem, we present an approximate method of solution based on the concept of a single scattering approximation.

In Sect. 2, we briefly recall the decomposition technique presented in Frisch (2009) for the Hanle effect with angle-dependent PRD. In Sect. 3, we derive approximate expressions for the Fourier coefficients of irreducible source vector components applying the single scattering approximation. A generic form for the Hanle scattering redistribution matrix is assumed in Sects. 2 and 3 (see Eq. (4) below). In Sect. 4, we generalize the results of Sects. 2 and 3 to the Hanle scattering redistribution matrix derived by Bommier (1997b, the so-called approximation-II). Section 5 is devoted to the comparison of results computed using the single scattering approximation with those computed using the perturbation method of Nagendra et al. (2002). Conclusions are presented in Sect. 6. Some technical details are given in Appendices A and B.

## 2. A decomposition method for Hanle effect with angle-dependent PRD

For the sake of clarity, we recall some important equations from Frisch (2009). The polarized transfer equation for the Stokes vector can be written in component form as (1)where Ω  (θ,χ) is the outgoing ray direction defined with respect to the atmospheric normal and μ = cosθ. The line optical depth τ is defined by    dτ =  −kl   dz, where kl is the frequency–averaged line absorption coefficient, and ϕ(x) is the normalized Voigt function. The frequency x is measured in units of the Doppler width, with x = 0 at the line center. The ratio of continuum to line absorption coefficient is denoted by r. The total source vector is given by (2)where Sc,i are the components of the unpolarized continuum source vector. We assume that Sc,0 = Bν0, where Bν0 is the Planck function at the line center, and Sc,1 = Sc,2 = 0. The line source vector can be written as (3)where Ω′  (θ′,χ′) is the direction of the incoming ray defined with respect to the atmospheric normal, and    dΩ′ = sinθ′   dθ′   dχ′. For simplicity, we assume that the primary source is unpolarized, namely that only G0(τ) is non-zero, and it is proportional to Bν0. In Sect. 3, we assume a generic form for the elements of the redistribution matrix in the presence of a weak magnetic field B given by (4)where r(x,x′,Θ) is any redistribution function with proper normalization conditions, such as the type I, II, or III PRD functions of Hummer (1962), and Θ is the angle between the incoming (Ω′) and outgoing (Ω) rays. In terms of the irreducible spherical tensors introduced by Landi Degl’Innocenti (1984), the Hanle phase matrix elements can be written as (see Chaps. 5 and 10 of Landi Degl’Innocenti & Landolfi 2004; or Frisch 2007; 2009) (5)For the coefficients we assume the simple form (6)where are coefficients that describe the effects of the magnetic field, ϵ the thermalization parameter, WK the depolarizability factors depending on the J-quantum numbers of upper and lower levels, (θB,χB) the polar angles defining the magnetic field orientation with respect to the atmospheric normal, and Γ = 2πνLgu/Aul in standard notation is the Hanle parameter.

The Stokes vector and the source vector can each be decomposed into six irreducible components and that are non-axisymmetric because of the angle dependence of the PRD function. To describe this angle dependence of the PRD function, we introduce an azimuthal Fourier expansion written as (see Eq. (15) of Frisch 2009) (7)where (8)It is easy to verify that . To describe the azimuthal dependence of the irreducible components , we introduce the Fourier decomposition (9)Assuming a similar expansion for the primary source term , it is easy to show that has an azimuthal expansion similar to Eq. (9).

The Fourier coefficients satisfy a non-local thermodynamic equilibrium (hereafter non-LTE) transfer equation (10)where the source term is given by Eq. (2) with and instead of Sl,i and Sc,i. Since the continuum is assumed to be unpolarized, . The Fourier coefficients of the line source vector are given by (11)where the primary source term , and k′ = k + Q′ − Q′′. For a given value of k, the values of k′ are clearly restricted to k − 4 ≤ k′ ≤ k + 4. The non-zero azimuthal Fourier coefficients are of the form (12)where K and K′ are both even or both odd. The are combinations of trigonometric functions that may be written as . It is easy to verify that satisfy the following conjugation property (13)We note that the range of k values extends a priori from  − ∞ to  + ∞, thereby giving rise to an infinite set of integral equations for . A solution is possible only if we truncate this infinite set. As already noted in Sect. 1, if we restrict k to k = 0,    ± 1,    ± 2,    ± 3,    ± 4, we obtain a set of 54 coupled integral equations. Before going into such a difficult numerical problem, we show in the following section how the problem can be simplified by using a single scattering approximation (see Frisch et al. 2009; Frisch 2010).

## 3. Approximate expressions for the Fourier coefficients

First we make the reasonable assumption that Stokes I is unaffected by the polarization. We can thus describe Stokes I with only, neglecting the contributions of the K ≠ 0 terms. For an angle-dependent PRD, is non-axisymmetric and can be Fourier expanded as in Eq. (9). However, as Stokes I is almost independent of the magnetic field (in the weak field Hanle regime that we consider), we make a further assumption that is independent of χ. Hence, only the k = 0 term contributes to and we can write (14)With these approximations, the irreducible component takes a simple form, namely (15)where , and the thermalization parameter ϵ = ΓI/(ΓR + ΓI), with ΓR and ΓI being the radiative and inelastic collisional de-excitation rates.

To obtain approximate expressions for the Fourier components of , we apply the single scattering approximation to Eq. (11), namely we keep in the right-hand side of this equation only the contribution from . This amounts to assuming that k′ = K′ = Q′′ = 0. Since k′ = k + Q′ − Q′′, we obtain k =  −Q′. Hence, only the components with k = 0, ± 1, ± 2 are non-zero. These 25 Fourier components are given by the following approximate expression (16)The explicit forms of the azimuthal Fourier coefficients are given in the Appendix A.

The irreducible components and their Fourier components for k ≠ 0 are complex numbers. We introduce here their real components. Following Frisch (2007, Eqs. (25) and (26)), we define the real components of to be (17)(18)For the Fourier components , we use the conjugation property given in Eq. (13) to define their real components as (19)(20)For k ≠ 0, using the conjugation property of (see Appendix A) and of (see Chapter 5 of Landi Degl’Innocenti & Landolfi 2004), we can deduce from Eq. (16) that (21)Combining Eqs. (17) and (18) with the azimuthal Fourier expansion of , we obtain (22)(23)(24)where (25)for k > 0 and Q > 0. There are 25 non-zero real components of . Explicit expressions are given in Appendix B.

## 4. The Hanle scattering redistribution matrix

In Sect. 2, the decomposition method is presented with an oversimplified redistribution matrix. Here we show how to extend it to a standard Hanle redistribution matrix that takes into account elastic collisions and that the Hanle effect only operates in the line core.

Using a QED approach, Bommier (1997a,b) derived the Hanle scattering redistribution matrix including the effects of elastic collisions for a two-level atom with unpolarized lower level. For practical applications, she also presents the so-called approximation-II and approximation-III, where the two-dimensional (2D) frequency space (x,x′) is decomposed into several domains, in each domain the frequency redistribution being decoupled from the polarization. The approximation-II uses the angle-dependent PRD functions and approximation-III the corresponding angle-averaged functions. The approximations-II and III of the Hanle redistribution matrix were considered in Nagendra et al. (2002, see also Nagendra et al. 2003), where a perturbation method is used to solve the polarized transfer equations. For approximation-III, a polarized ALI method based on the core-wing approach was later developed by Fluri et al. (2003).

For approximation-III, the domains depend on (x, x′), but for approximation-II they depend in addition on the scattering angle Θ between the incoming and outgoing rays (see Figs. 1 and 2 in Nagendra et al. 2002). Thus, one practical question that arises when applying the decomposition technique developed for angle-dependent PRD presented in Sect. 2 to the case of approximation-II, is how to reduce or remove the (χ − χ′) dependence appearing in the expression of the redistribution matrix given in Eqs. (90) − (98) of Bommier (1997b). Here, to circumvent this problem, we use the angle-averaged domains of approximation-III, together with the angle-dependent redistribution functions of approximation-II. With this simplifying assumption, which as shown in Sect. 5 yields a reasonable solution, it is straightforward to generalize all the equations given in Sects. 2 and 3 to handle the approximation-II of Bommier (1997b). The only difference is that now depends on both x and x′. Therefore in Eqs. (11) and (16), should be inside the frequency integral, and should be replaced by (26)where the index m ( = 1,2,3,4,5) stands for different (x, x′) frequency domains. The domains relevant to the type-III redistribution are represented by m = 1,2,3 and those relevant to the type-II redistribution by m = 4,5. Expressions for the can be found in Bommier (1997b, see also Nagendra et al. 2002; Frisch 2007; Anusha et al. 2011).

## 5. Results and discussions

Here we validate the single scattering approximation presented in Sect. 3, by comparing the solutions computed with this approximation, and those obtained by solving the full radiative transfer equation with an angle-dependent PRD and the Hanle effect. For the latter, we use a perturbation method developed by Nagendra et al. (2002). To obtain the single scattering solution, we first solve the scalar radiative transfer equation with the source term given by Eq. (15), using an ALI method based on the core-wing approach or frequency-angle by frequency-angle approach (see Sampoorna et al. 2011). The scalar intensity obtained in this way is then used in Eq. (16) or Eqs. (B.1) and (B.2), to compute the single scattered source vector. A formal solution then gives the corresponding single scattered radiation field.

 Fig. 1Comparison of single scattered (dashed lines) and multiple scattered (solid lines) solutions computed with the angle-dependent PRD. A 1D cut-off assumption is used with xc = 3. The model parameters are (a,   ϵ,   r,   ΓE/ΓR) = (10-3,   10-3,   0,   0) and the magnetic field parameters are (Γ,   θB,   χB) = (1,   30°,   0°). The line of sight is represented by μ = 0.11 and χ = 0°. Panel a) corresponds to T = 1, panel b) to T = 100, and panel c) to T = 104. Open with DEXTER

We consider isothermal, self-emitting plane-parallel atmospheres with no incident radiation at the boundaries. These slab models are characterized by (T,   a,   ϵ,   r,   ΓER), where T is the optical thickness of the slab and ΓE is the elastic collisional rate. The depolarizing collisional rate D(2) is assumed to be 0.5 × ΓE. For all the figures presented in this paper, a = 10-3, ϵ = 10-3, r = 0 (pure line case), and the line of sight is defined by μ = 0.11 and χ = 0°. The magnetic field parameters are taken as Γ = 1, θB = 30°, and χB = 0°.

In Fig. 1, we compare the single scattered solution (dashed lines) with the full radiative transfer solution (hereafter, multiple scattered solution; see solid lines) for T = 1 (panel (a)), 100 (panel (b)), and 104 (panel (c)). For brevity, we only show the Q/I and U/I profiles. The comparison is carried out for the type-II angle-dependent redistribution function of Hummer (1962), i.e., with ΓER = 0. We have used a one-dimensional (1D) cut-off assumption, setting the Hanle effect to zero for x > xc with xc = 3. Owing to this abrupt cut-off, spikes or dents are observed around x = 3 (see Fig. 1). Smoother curves can be obtained by using the more realistic 2D domains of Bommier (1997b) (see Fig. 3).

Figure 1 shows that the single scattering approximation can capture the main features of the Q/I and U/I profiles. It is the first time that this approximation has been employed for Stokes U. We can see that it is able to follow the very non-monotonic behavior of U/I. From a quantitative point of view, we observe at line center that the single scattering approximation underestimates the polarization for T = 1 and overestimates it for T = 100 and T = 104. The transition takes place around T = 100. As explained in Frisch et al. (2009) for the case of complete frequency redistribution (CRD), this change in behavior is related to the competition between a limb-darkened outgoing radiation and a limb-brightened incoming one (see also Trujillo Bueno 2001). In the near wings of Q/I, the single scattering approximation strongly underestimates the polarization. We recall that these wings are formed by Rayleigh scattering. For the angle-dependent case, Sampoorna et al. (2011) showed that the exact value of Q/I all along the profile can be obtained by an iterative method incorporating successive scattering orders (see Fig. 5 in this reference). It is likely that this method will also be able to improve the estimation of U/I.

In Fig. 2, we present the single scattered and multiple scattered solutions computed for T = 104, using the more realistic 2D domains of Bommier (1997b). Panel (a) corresponds to the collisionless case and panel (b) to an equal mix of type-II and type-III scattering. We recall that single scattered solutions are calculated with the angle-averaged domains corresponding to approximation-III and the multiple scattered solutions with angle-dependent domains corresponding to approximation-II. Figure 2 shows that this simplifying assumption gives solutions that compare reasonably with the corresponding multiple scattered solutions. The comparison between Figs. 2a and 2b shows that the single scattered solution can become a very good approximation to the multiple scattered solution in the presence of elastic collisions, particularly at the near wing maximum in Q/I profile. For angle-averaged PRD, we also have examples of single scattering solutions providing very accurate approximations (see Fig. 4 of Anusha et al. 2010).

 Fig. 2Comparison of single scattered (dashed lines) and multiple scattered (solid lines) solutions computed with approximation-II of Bommier (1997b). The model parameters are (T,   a,   ϵ,   r) = (104,   10-3,   10-3,   0) and the magnetic field parameters are (Γ,   θB,   χB) = (1,   30°,   0°). Panel a) corresponds to collisionless case (ΓE/ΓR = 0) and panel b) to an equal mix of type-II and III scattering (ΓE/ΓR = 1). Open with DEXTER

The comparison of Fig. 2a with Fig. 1c suggests that U/I is very sensitive to the type of cut-off approximation used. We now discuss this point in more detail. Figure 3 shows the Q/I and U/I profiles computed for the collisionless case, and with three types of frequency domains. The solid lines are computed using a 1D cut-off assumption (xc = 3), and dashed lines with actual 2D domains of Bommier (1997b). The dotted lines are computed using a square domain (). Such square domains have been used by Fluri et al. (2003, see their Fig. 2b). Figure 3a shows the single scattered solution, while Fig. 3b shows the multiple scattered solution. Clearly Q/I is insensitive to the type of frequency domains used, while U/I is quite sensitive, particularly in the transition region (3 ≤ x < 5).

## 6. Conclusions

The solution of a polarized line transfer equation with an angle-dependent PRD and the Hanle effect is computationally very demanding. To reduce the complexity of the numerical problem, Frisch (2009) developed a decomposition technique for angle-dependent Hanle problem. In this technique, the non-axisymmetric polarized radiation field is decomposed into an infinite set of cylindrically symmetric Fourier coefficients, by applying an azimuthal Fourier expansion of the angle-dependent PRD function. Such a decomposition method allows one to construct an infinite set of integral equations for the Fourier coefficients of the irreducible source vector components, the solution of which is possible only if we truncate this infinite set. Although the truncated integral equations can be numerically solved, this solution is still computationally demanding (at least 54 integral equations need to be solved simultaneously). Therefore, to simplify the problem, here we have presented an approximate method of solution, based on the single scattering approximation. An approximate solution can be obtained by keeping only the contribution of the zeroth order Fourier coefficient, namely, in the source terms for the K = 2 components (see Eq. (16)), which are responsible for the generation of Stokes Q and U. As a result, the number of Fourier coefficients reduces to 25 for K = 2 components. The Stokes I in this approximation is assumed to be cylindrically symmetric and decoupled from Stokes Q and U, so that it is computed through a solution of the scalar non-LTE transfer equation with an angle-dependent PRD function. An ALI method associated with a core-wing technique or frequency-angle by frequency-angle technique is used to solve this scalar transfer problem (see Sampoorna et al. 2011).

We have presented the single scattering approximation for a Hanle scattering redistribution matrix with (i) 1D cut-off assumption (see Sect. 3) and (ii) the more realistic frequency domains (see Sect. 4) introduced by Bommier (1997b). We show that the single scattered solution provides a reasonable approximation to the emergent solutions computed by solving the full polarized line transfer equation with angle-dependent PRD and the Hanle effect. To compute the full solution, we use a perturbation method developed by Nagendra et al. (2002). While the perturbation method used by Nagendra et al. (2002) is computationally expensive, the single scattering approximation presented here provides a rapid method of solution. For example, for T = 104 with 41 depth points, seven μ points (eight χ points required only in the case of perturbation method), and 39 frequency points, the perturbation method requires 24 min with 2.7 G Bytes of memory, while single scattered solution can be computed in 20 s with 22 M Bytes of memory, on a Sun Fire V20z Server, 2385 MHz, with a Single-core AMD Opteron processor.

The differences between single scattered and multiple scattered solutions for angle-dependent, as well as angle-averaged PRD are somewhat similar to those for CRD in the line core region. In the case of an angle-dependent and an angle-averaged type-II PRD function, the differences are particularly large at the near wing maximum in Q/I (see Fig. 1c). This difference is reduced in the presence of elastic collisions (see Fig. 2b). Thus, elastic collisions seem to expand the domain of applicability of single scattering approximation. We have also presented the sensitivity of U/I profiles to the choice of different frequency domains (see Fig. 3).

 Fig. 3The effect of three types of frequency domains on Q/I and U/I profiles computed using angle-dependent PRD functions. Solid lines correspond to the 1D cut-off assumption (xc = 3), dotted lines to 2D square domains (), and dashed lines to the approximation-II of Bommier (1997b). The model parameters are the same as in Fig. 2a. Panel a) corresponds to single scattered solutions, and panel b) to the multiple scattered solutions. Open with DEXTER

The single scattered solution presented in this paper can be improved by including higher orders of scattering. Such a method was presented in Frisch et al. (2009) for the Hanle effect with CRD, and in Sampoorna et al. (2011)

for Rayleigh scattering with angle-dependent PRD, where it was referred to as “scattering expansion method”. It is also possible to construct such a scattering expansion method for the case of Hanle effect with an angle-dependent PRD discussed in this paper. The single scattered solution should be seen as an “approximate solution” to the exact solution, and can be used in many practical applications that do not require highly accurate solution of this rather difficult problem, involving angle-dependent PRD matrices.

## Acknowledgments

I am very grateful to Drs. H. Frisch and K. N. Nagendra for stimulating discussions and critical comments on the manuscript. Thanks are also due to L. S. Anusha for useful comments.

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## Online material

### Appendix A: The azimuthal Fourier coefficients

The azimuthal Fourier coefficients are defined in Eq. (12). It is easy to verify that they satisfy the properties (A.1)(A.2)In this Appendix, we give only those Fourier coefficients that are of relevance to the problem at hand, namely those corresponding to K′ = Q′ = 0. Owing to the above-mentioned symmetry relations, there are only four independent coefficients, which are given by (A.3)If we chose the reference angle γ = 0 (see Landi Degl’Innocenti & Landolfi 2004), then from Eq. (A.6) of Frisch (2010), it is clear that are real and hence that are also real. Using Eq. (A.6) of Frisch (2010) in our Eq. (A.3), we obtain the explicit form for these coefficients (A.4)

### Appendix B: The explicit form of the real components of

In this Appendix, we list the explicit analytic forms of the 25 non-zero real components of , defined in Eqs. (19), (20), and (25) of the text. For K = 2, we define (B.1)We can express all the 25 non-zero real components of in terms of as follows : (B.2)The above equations are valid for x < xc in the case of a 1D cut-off assumption. For x > xc, we have to set . In the case of approximation-II, the various appearing in the above equations should be taken inside the frequency integral appearing in Eq. (B.1), and Eqs. (106)–(113) of Bommier (1997b) have to be used.

The various appearing in the above set of equations are given by Eq. (6) in the case of the 1D cut-off assumption and Eqs. (A11)–(A18) of Anusha et al. (in press) in the case of the 2D frequency domains of Bommier (1997b). Below we give the explicit forms of appearing in those equations. We introduce the abbreviations, CB = cosθB, SB = sinθB, c1 = cosχB, s1 = sinχB, c2 = cos2χB, and s2 = sin2χB. In terms of the elements of the matrix given below (see also Appendix C in Frisch 2007), we have (B.3)where (B.4)

## All Figures

 Fig. 1Comparison of single scattered (dashed lines) and multiple scattered (solid lines) solutions computed with the angle-dependent PRD. A 1D cut-off assumption is used with xc = 3. The model parameters are (a,   ϵ,   r,   ΓE/ΓR) = (10-3,   10-3,   0,   0) and the magnetic field parameters are (Γ,   θB,   χB) = (1,   30°,   0°). The line of sight is represented by μ = 0.11 and χ = 0°. Panel a) corresponds to T = 1, panel b) to T = 100, and panel c) to T = 104. Open with DEXTER In the text
 Fig. 2Comparison of single scattered (dashed lines) and multiple scattered (solid lines) solutions computed with approximation-II of Bommier (1997b). The model parameters are (T,   a,   ϵ,   r) = (104,   10-3,   10-3,   0) and the magnetic field parameters are (Γ,   θB,   χB) = (1,   30°,   0°). Panel a) corresponds to collisionless case (ΓE/ΓR = 0) and panel b) to an equal mix of type-II and III scattering (ΓE/ΓR = 1). Open with DEXTER In the text
 Fig. 3The effect of three types of frequency domains on Q/I and U/I profiles computed using angle-dependent PRD functions. Solid lines correspond to the 1D cut-off assumption (xc = 3), dotted lines to 2D square domains (), and dashed lines to the approximation-II of Bommier (1997b). The model parameters are the same as in Fig. 2a. Panel a) corresponds to single scattered solutions, and panel b) to the multiple scattered solutions. Open with DEXTER In the text

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In his TASI notes Oliver DeWolfe starts with the KG equation on the Poincaré patch metric $$ds^2=\frac{r^2}{L^2}(-dt^2+dx^2)+\frac{L^2}{r^2}dr^2.$$ When we use the ansatz$$\phi(r\rightarrow\infty,x,t)=\frac{\alpha(x,t)L^{2\Delta_-}}{r^{\Delta_-}}+\dots+\frac{\beta(x,t)L^{2\Delta_+}}{r^{\Delta_+}}+\dots$$ on the KG equation $$\left(-\frac{1}{\sqrt{-g}}\partial_\mu\sqrt{-g}g^{\mu\nu}\partial_\nu+m^2\right)\phi=0$$ we end up with an equation which is an ODE in r since the $$\partial_t^2$$,$$\partial_x^2$$ terms have a factor $${1}/{r^2}$$ which we can ignore.
The equation I end up with is $$-\frac{3r}{L^2}\partial_r\phi-\frac{r^2}{L^2}\partial_r^2\phi+m^2\phi=0.$$
When I substitute the ansatz I don't confirm that the equation is satisfied since I have a lot of terms with different powers of $$r$$. How can I show that this is indeed satisfied?
To solve equations of the form $$ar^2 \partial^2_{r}\phi + b r \partial_r\phi +c \phi=0$$ with constants $$a$$, $$b$$, $$c$$ (which is what you have) one sets $$\phi(r) =r^\lambda$$ so the equation becomes $$(a\lambda(\lambda-1) +b\lambda +c)r^\lambda=0.$$ Thus one solves the quadradic equation $$a\lambda(\lambda-1) +b\lambda +c=0$$ to get the two values of $$\lambda$$.
• The ansatz given is identical to mine! The $\Delta_{\pm}$ in the ansatz are just my $-\lambda$.I don't see what your problem is. When one includes the $1/r^2$ terms it is more complicated, but it's still the usual Frobenius indicial equation at a regular singular point. May 9 '20 at 12:56

format: plain


text:
# How to calcute a determinant of this matrix? [closed]

$$\begin{vmatrix} -t & 1 & 0 & 0& \ldots & 0 & 0 &0\\ n & -t & 2 & 0& \ldots & 0 & 0 &0\\ 0 & n-1& -t & 3&\ldots & 0 & 0 &0\\ 0 & 0 & n-2& -t& \ldots & 0& 0 &0\\ \vdots&\vdots&\vdots&\vdots& \ddots&\vdots&\vdots&\vdots\\ 0 & 0 & 0 & 0&\ldots & -t& n-1& 0 \\ 0 & 0 & 0 & 0&\ldots & 2& -t & n \\ 0 & 0 & 0 & 0&\ldots & 0&1 & -t \end{vmatrix}$$

It is needed to find an eigenvalue. I just know that for $$n=m$$ there are Fibonacci number $$F_{m+2}$$ summands in the determinant formula.

## closed as off-topic by Eevee Trainer, Cesareo, GNUSupporter 8964民主女神 地下教會, Rhys Steele, Alex ProvostMar 23 at 1:12

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Eevee Trainer, Cesareo, GNUSupporter 8964民主女神 地下教會, Rhys Steele, Alex Provost
If this question can be reworded to fit the rules in the help center, please edit the question.

• You appear to have a tridiagonal matrix. One thing you can do is see the determinant section of that page for a recursive formula for the determinant. – Minus One-Twelfth Mar 22 at 7:10
• @Andrey Komisarov I don't understand your Fibonacci number comment. – Max Mar 22 at 7:27
• @Max for n=1 there are t^2-1 (2 addend), for n=2 there are (3 addend), for n=3 there are (5 addend)... – Andrey Komisarov Mar 22 at 7:30
• I think you mean "summand". Edit: Google says "addend" is valid (though I still think "summand" is more common). – Max Mar 22 at 7:35
• @Max Sorry, I mixed. – Andrey Komisarov Mar 22 at 7:37

Your matrix is $$-tId$$ plus the matrix that appears in Proposition 2 on page 16 of https://arxiv.org/pdf/1210.8062.pdf
The eigenvalues are thus $$n-t, n-2-t, \ldots, 2-n-t, -n-t$$.
Same solution in more concrete language: Consider the operator $$w\frac{\partial}{\partial z}+z\frac{\partial}{\partial w}$$ acting on the space of homogeneous degree $$n$$ polynomials in $$z$$ and $$w$$. A natural basis for that space is $$z^n, z^{n-1}w, \ldots, w^n$$; in that basis the matrix of this operator is like the one in the problem with $$t=0$$. However, in the basis $$(z+w)^{n-k} (z-w)^{k}$$ the matrix is diagonal with eigenvalues $$n-2k$$.


text:
HERE Places (Search) API
In maintenance
Places (Search) API Developer's Guide

# Browse by Corridor Entrypoint

The Browse by Corridor Entrypoint represent sets of places along a route area sorted by distance from starting point of the route.

The Browse by Corridor Entrypoint is a Places (Search) API Deprecated entrypoint.

## Entrypoint URI

/browse/by-corridor

## Entrypoint Parameters

Parameter Type Description
at Position (format: latitude,longitude[;cgen=(map|gps|sgps)][;u=\d+]); required, unless one of the Geolocation or X-Map-Viewport headers or the in are set. Coordinates of search location expressed as latitude, longitude. Additional parameters can be passed which provide more context such as the uncertainty and how the coordinates were generated. For example, "52.5304417,13.4111201", "52.5304417,13.4111201;cgen=gps;u=100" or "52.5304417,13.4111201;u=100". For a full description, see the Location Contexts documentation.
route Route (format: $start(latitude,longitude|[width])|0..n(latitude,longitude|[width])|end(latitude,longitude[width])$[;w=\d+]); required

This parameter limits search results to the boundary of an area around a route. A route consists of a start coordinate, 0..n turning coordinates and an end coordinate. An optional width in meters can be passed to provide off the route max distance in meters, default: 1000m

The points should describe the geometric shape of the route with high precision. Especially, it’s usually not sufficient to only pass maneuvre points as they would be shown to the driver.

width specifier is CW or HW. It can follow any of the points to specify whether a route segment is located in a city or on a highway respectively, where segment is the point preceding specifier and following points till the next specifier. This will modify the search distance and possibly algorithm for the segment. DW will reset to default width (specified by w=\d+ ) and algorithm.

There are practical limitations to URL lengths. Thus, route parameters with more than 4700 characters will be rejected.

A way to reduce this parameter's length is to use a more space-efficient HERE polyline encoding and supply that as the compressedRoute parameter.

compressedRoute HereEncodedRoute (format: [a-zA-Z0-9._-]+(;w=\d+)?); required

This parameter is the new preferred way to specify a route. It contains the coordinates in HERE polyline encoding, which is basically a URL-safe version of Google's polyline encoding format.

Additionally, it may contain width attributes for individual parts of the route, i.e. one specify segments to have city width (.C), highway width (.H), or default width (.D).

By supplying the suffix ';w=\d+', one can define the actual width of the default width segments in meters.

cat Comma-separated list; optional A comma-separated list of category ids defining an OR-filter that all places reachable through the resource must match. For a list of supported categories, see the Categories documentation.
name String; optional

Plain-text name of a place used to filter out the results. For example, "Brandenburger Tor"

The name parameter is mutually exclusive with the 'q' parameter.

urgency Double; optional; default: 1.0.

This parameter is useful when search of interesting places along the route is performed and route parameter is specified.

There are two different ways of calculating distance along a route. The first way is to compute distance from where you are, along the route, and then perpendicular to the route. The second is to compute distance as the amount of extra travel needed to include the point in the route. The "urgency" parameter allows you to choose between the two, or choose a blending of the two. A value of 1.0 selects for distance along the route, and a value of 0.0 selects the amount of extra travel needed. Values in between weight each method correspondingly.

cs Comma-separated list; optional A comma-separated ordered list of category systems defining which type of category systems should be returned in the response. Valid category systems are 'places', 'cuisines', and 'pds'. For example cs=places,cuisines

## GET Method

The GET method provides access to places for the given route and matching the category filter criteria.

## Representation Modifiers

The following options are available in this context:

Parameter Type Description
size Number (non-negative integer); optional The maximum number of result items in each collection.
tf String; optional; default: html. Text format. Determines how rich text properties such as location.address.text should be rendered. Note: plain text can still be multiline. In this case newline symbol ("\n") is used to separate lines.
Supported values are:
• html
• plain
show_refs Comma-separated list; optional A list of one or more external system names or reference types. This parameter exposes place related external references in response. For a full description see Representation Modifiers documentation.
show_content Comma-separated list; optional A list of one or more available content types you can add to the response. For a full description see Representation Modifiers documentation.

For additional information and examples, see Changing Responses with Representation Modifiers.

## Response Media Type

Responses to requests to this endpoint will have the urn:nlp-types:search media type. See the urn:nlp-types:search media type documentation for details about the structure and content of the response.

## Request Example

Find petrol stations within a route corridor area between Berlin and Hamburg with 1 km off route distance.

In the following request example, URL encoding is omitted for better readability. An actual request would need to be URL encoded.

https://places.sit.ls.hereapi.com/places/v1/browse/by-corridor
?app_id={YOUR_APP_ID}
&app_code={YOUR_APP_CODE}
&route=[52.5160,13.3771|52.5111,13.3712|52.5355,13.3634|52.5400,13.3704|52.5626,13.3307|52.5665,13.3076|52.6007,13.2806|52.6135,13.2484|52.6303,13.2406|52.6651,13.2410|52.7074,13.1926|52.7045,13.0661|52.7191,12.9621|52.7636,12.8263|52.7861,12.8000|52.8335,12.7919|52.9002,12.7451|52.9708,12.6311|53.0526,12.5392|53.0867,12.5169|53.1146,12.4687|53.1334,12.4644|53.1415,12.4225|53.1666,12.3722|53.1785,12.3050|53.2570,12.1618|53.2893,12.0618|53.3000,11.9373|53.3316,11.8724|53.3463,11.8190|53.3669,11.7328|53.3725,11.6427|53.4154,11.5505|53.4309,11.4906|53.4342,11.4000|53.4655,11.3370|53.4873,11.2631|53.4860,11.2011|53.5110,10.9647|53.5128,10.8414|53.5495,10.6892|53.5692,10.5155|53.5596,10.4259|53.5682,10.2999|53.5571,10.2020|53.5672,10.1279|53.5534,9.9924];w=1000
&cat=petrol-station
&pretty

## Response Example

{
"results": {
"items": [
{
"position": [52.53723, 13.37472],
"distance": 277,
"title": "Total",
"averageRating": 0.0,
"category": {
"id": "petrol-station",
"title": "Petrol station",
"href": "http://...",
"type": "urn:nlp-types:category"
},
"icon": "http://...",
"vicinity": "Chausseestraße 62
10115 Berlin",
"having": [],
"type": "urn:nlp-types:place",
"href": "http://...",
"id": "276u33db-cc5a1377043842ba8fc3c6acd0b36e7d"
},
{
"position": [52.542679, 13.37774],
"distance": 822,
"title": "Esso Station Mathias Fuhrmann",
"averageRating": 0.0,
"category": {
"id": "petrol-station",
"title": "Petrol station",
"href": "http://...",
"type": "urn:nlp-types:category"
},
"icon": "http://...",
"vicinity": "Gerichtstr. 4
13347 Berlin",
"having": [],
"type": "urn:nlp-types:place",
"href": "http://...",
},
{
"position": [52.53391, 13.36515],
"distance": 839,
"title": "Total",
"averageRating": 0.0,
"category": {
"id": "petrol-station",
"title": "Petrol station",
"href": "http://...",
"type": "urn:nlp-types:category"
},
"vicinity": "Heidestraße 19
10557 Berlin",
"having": [],
"type": "urn:nlp-types:place",
"href": "http://...",
"id": "276u33db-6bcb7b2e4fec41e9a295bba5709b00e4"
},
...
],
"search": {
"context": {
"location": {
"position": [52.5353, 13.3773]
}
}
}
}
}


text:
# uniqueness of $f$-localization

The $f$-localization I mean is the one described and studied in detail in the book by E. D. Farjoun; $L_f$ is a homotopy idempotent functor which associates to each space $X$ an $f$-equivalence $X\to L_f(X)$ where $L_f(X)$ is $f$-local.

$f$-localization has a kind of uniqueness: if $F$ is some other coaugmented functor with the property that $F(X)$ is $f$-local for every $X$ (I'm happy to assume that $F = L_g$ for some map $g$), then there is a commutative square of functors and natural transformations, which I don't know how to draw here. The square would show that the composites $$id \xrightarrow{\iota} L_f \xrightarrow{L_f(j)} L_f\circ F \qquad \mathrm{and} \qquad id \xrightarrow{j} F \xrightarrow{\iota_F} L_f \circ F$$ are equal. And $\iota_F$ is a weak equivalence for every space $X$; thus $F$ factors through $L_f$ up to weak equivalence'.

My Question: Suppose $X\to Y$ is an $f$-equivalence; then $L_f(X) \to L_f(Y)$ is a weak equivalence; does it follow that $(L_f\circ F)(X) \to (L_f\circ F)(Y)$ is a weak equivalence?

-

If you allow $F$ to be a continuous functor, which it appears you do since localization functors are continuous, then I believe this is true. Corollary 2.1.1 in the following paper seems to be useful:

E.Dror Farjoun, Higher homotopies of natural constructions, Journal of Pure and Applied Algebra, Volume 108, Issue 1, 8 April 1996, Pages 23-34, ISSN 0022-4049,

It says that if $\phi: X \rightarrow Y$ is a map and $G$ is a continuous, coaugmented, idempotent functor such that for all spaces $A$, the induced map $map(Y,GA) \rightarrow map(X,GA)$ is a weak equivalence, then $G(\phi): G(X) \rightarrow G(Y)$ is a weak equivalence. Letting $G = L_f \circ F$ should give the result you want. The induced map will be an equivalence since $\phi$ is an $f$-equivalence and $L_f\circ F(A)$ is $f$-local for all spaces $A$.

-

I think that in general the answer is no', since your functor $F$ need not preserve $f$-equivalences. Take $f\colon \ast \to \ast$ to be the identity map or any other weak equivalence, then the $f$-local objects are the fibrant objects and $L_f$ is equivalent to the fibrant replacement is simplicial sets. Now take any functor $F$ which does not preserve weak equivalences and takes fibrant values. For example, $F(-) = \mathrm{fib}(\mathrm{hom}(A,-))$, where $\mathrm{fib}$ is a fibrant replacement functor and $A$ a non-trivial space. Then $L_f\circ F$ does not preserve weak equivalences (which are also $f$-equivalences) of non-fibrant simplicial sets.

I think that the condition that you need to assume about $F$ is that $F$ preserves $f$-equivalences.

-
I really like this example. I never considered this $F$ before, but it does exactly what you want. I also agree that in situations like this the condition you suggest sounds right –  David White May 1 '13 at 15:46

An $f$-equivalence is a map $\alpha:X\to Y$ which induces a weak equivalence $\alpha^\ast:map_\ast(Y,Q)\to map_\ast (X,Q)$ for all $f$-local spaces $Q$; equivalently, $L_f(\alpha)$ is a weak equivalence. Therefore the assumption implies that every $g$-local space is $f$-local, so an $f$-equivalence $\alpha$ is also a $g$-equivalence; so $L_g(\alpha)$ is a weak equivalence.

-
So this answers it for $F=L_g$, and Matthew Sartwell's answer is slightly more general, but Boris shows you can't go more general than that. Interesting. +1 to everyone –  David White May 1 '13 at 15:36


text:
# How many moles of each element are present in 1.000L of a 0.295 M solution of sodium chloride?

Feb 25, 2018

$0.295$.

#### Explanation:

You know that molarity is defined as the number of moles of solute present in exactly $\text{1 L}$ of a solution.

Basically, what this means is that if you know the molarity of a solution, you know how many moles of solute are present for every $\text{1 L}$ of that solution.

In your case, you're dealing with a $\text{0.295-M}$ solution of sodium chloride, so right from the start, you can say that this solution will contain $0.295$ moles of sodium chloride, the solute, for every $\text{1 L}$ of the solution.

So in your $\text{1.000-L}$ sample of this solution, you will have $0.295$ moles of sodium chloride.

Now, sodium chloride is soluble in water, which implies that it exists as ions in aqueous solution.

${\text{NaCl"_ ((aq)) -> "Na"_ ((aq))^(+) + "Cl}}_{\left(a q\right)}^{-}$

So for every mole of sodium chloride that you dissolve in water to make your solution, you will end up with $1$ mole of sodium cations, ${\text{Na}}^{+}$, and $1$ mole of chloride anions, ${\text{Cl}}^{-}$.

This means that your $\text{1.000-L}$ sample contains $0.295$ moles of sodium cations and $0.295$ moles of chloride anions.

If you want the number of moles of each element, you can say that the $0.295$ moles of sodium cations correspond to $0.295$ moles of sodium, the parent element of the cation and that the $0.295$ moles of chloride anions correspond to $0.295$ moles of chlorine, the parent element of the anion.


text:
ï»¿ The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative:(d) work done by an applied force on a body moving on a rough horizontal plane with uniform velocity from Physics Work, Energy and Power Class 11 Manipur Board

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Class 10 Class 12

The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative:

(d) work done by an applied force on a body moving on a rough horizontal plane with uniform velocity

Positive quantity.

In thisÂ case, the body is moving on a rough horizontal plane. Frictional force opposes the motion of the body. Therefore, in order to maintain a uniform velocity, a uniform force must be applied to the body. Since the applied force acts in the direction of motion of the body, the work done is positive.
288 Views

What is watt?

SI unit of power is Watt.Â
773 Views

Define the unit joule.

Work done is said to be one joule if one newton of Â force displaces the body through a distance of one meter in the direction of applied Â forceÂ .
799 Views

What are different units of energy?

The different units of energy are :

(i) Joule Â  Â  Â  Â  Â  Â
(ii) Erg
(iii) eV Â  Â  Â  Â  Â  Â
(iv) KWh
(v) Calorie.

804 Views

Is work a scalar or a vector quantity?

Work is dot product of two vectors. i.e.Â .

And dot product is a scalar quantity. Therefore, work is a scalar quantity.Â
1450 Views

Define watt.

Power is said to be one watt if one joule of work is done in one second.Â
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text:
# How to put a tick mark above a letter? [duplicate]

I am looking for some way of putting a tick and cross marks above a word or a letter. Something like this

For math mode, I'd recommend using \overset and the symbols from the pifont package.

\documentclass{article}

\usepackage{amsmath}
\usepackage{pifont}

\begin{document}

$\overset{\text{\ding{51}}}{A}$
$\overset{\text{\ding{55}}}{B}$

\end{document}

Use stackengine (I've not searched for the symbols, you'll find them on your own):

\documentclass{article}
\usepackage{stackengine}

\begin{document}
\stackon{A}{cm}
\stackon{B}{x}
\end{document}

• Thanks @TeXnician. But it is showing error stackengine.sty not found. – Seeker Oct 11 '17 at 7:08
• @Seeker Which TeX distro are you using (TeX Live, MikTeX, ...)? Try using its package manager. – TeXnician Oct 11 '17 at 7:13

By simple, can use \mathop and \limits commands, example shown below:

\documentclass{book}

\begin{document}

$\mathop{\mathrm{A}}\limits^\surd$\quad
$\mathop{\mathrm{B}}\limits^\times$
\end{document}


text:
# very simple question, plz I need the answer

Discussion in 'General Electronics Chat' started by fankoushat, Mar 28, 2010.

1. ### fankoushat Thread Starter New Member

Mar 8, 2010
17
0
Hi guys,

Consider the following circuit:

http://www.postimage.org/image.php?v=gxFeffA

V source , connected in series with resistor R , and open circuit.

what is the voltage dropped between the open circuit terminals?

equal to V source, or equal to V source - V dropped on R?

Thanks alot

Last edited: Mar 28, 2010
2. ### dsp_redux Active Member

Apr 11, 2009
182
5
I don't see any circuits...?

3. ### beenthere Retired Moderator

Apr 20, 2004
15,815
283
There is no current in the circuit, so there is no drop across the resistor.

4. ### dsp_redux Active Member

Apr 11, 2009
182
5
$V=R*I$
If $I=0A$ from the open circuit, $V=0V$. Same thing beenthere said.

5. ### Jony130 AAC Fanatic!

Feb 17, 2009
4,019
1,123
But voltage between the open circuit terminals is equal Vsource

6. ### fankoushat Thread Starter New Member

Mar 8, 2010
17
0
thanks johny, but why people say " the inductor at the beginning of connecting it with a dc source, will have voltage difference equals to V source - V dropped on R, because it is considered as open circuit".

So voltage difference measured between the two open circuit terminal will be the same as V source.

7. ### dsp_redux Active Member

Apr 11, 2009
182
5
When you do a DC analysis, you can consider capacitors as "open circuit" and inductors as "short circuit". There is no voltage drop in a short circuit. DC voltage is considered as a "0 frequency" or 0Hz. Impedance of a capacitor is $X_C = \frac{1}{2\pi f C}$. If you take $f\rightarrow 0$, $X_C=\infty$ hense the capacitor considered as an open circuit. For the inductor $X_L = 2\pi f L$. $X_L = 0$ when $f=0$. It is considered as a short circuit.

8. ### redlight000 Member

Feb 26, 2010
66
2
Hi dsp,
Just asking how do you get those electronic equations on the post? are they built in this actual forum?? or you got a special program to do it??


text:
## College Algebra (10th Edition)

$\color{blue}{T=d^2\sqrt[3]{x}}$
RECALL: (1) If $y$ varies directly as $x$, then $y=kx$ where $k$ is the constant of proportionality. (2) If $y$ varies inversely as $x$, then $y=\dfrac{k}{x}$ where $k$ is the constant of proportionality. (3) If $y$ varies jointly as $x$ and $z$, then $y = kxz$ where $k$ is the constant of proportionality. Notice that when the variation is direct or joint, the variable/ are on the numerator while if the variation is inverse, the variable is in the denominator. $T$ varies jointly with the cube root of $x$ and the square of $d$. Using the formula in (3) above, the equation of the joint variation is: $T=k\cdot \sqrt[3]{x} \cdot d^2 \\T=kd^2\sqrt[3]{x}$ Since $T=18$ when $x=8$ and $d=3$, substituting these into the tentative equation above gives: $\require{cancel} T=kd^2\sqrt[3]{x} \\18=k(3^2)\sqrt[3]{8} \\18=k(9)(2) \\18=k(18) \\\frac{18}{18}=k \\1=k$ Thus, the equation of the joint variation is: $T=1\cdot d^2\sqrt[3]{x} \\\color{blue}{T=d^2\sqrt[3]{x}}$


text:
# Super Mario World Physics (2012-07-29)

Super Mario!

Arguably one of the greatest video game series of all time, I spent countless hours playing these games as a kid. One thing I always particularly loved about the games is that the kinematics / physics of Mario’s motion were just so.. fun! Not necessarily that they felt real, but when you ran, you ran fast. When you jumped, you jumped high, etc. I think this is a problem that I have with numerous other platforming games.. yea, sure, the gameplay can be fun, but moving around just seems so… boring in comparison to Mario.

Since I’ve started dabbling around with my own demos and games lately, I figured it would be necessary for me to capture that sense of kinematic joy. Since I couldn’t find any definitive break-downs of Super Mario physics online (though I’ll be honest, I didn’t look too far), I decided it would be simple enough for me to do my own investigation. Thankfully, the process turned out to be a lot simpler than I was expecting thanks to the tools I used - ZSNES, MPlayer, PhysMo, and Matlab (though you could just as easily use GNU Octave instead of Matlab). The process I used went like this:

1. Record a video of Mario going through his various motions using ZSNES and mencoder
1. Import the video into PhysMo, and analyse the motion
1. Calibrate the analysis by defining Mario’s height as 1 m

2. Pick a non-deforming location on Mario’s sprite, then follow that point around in time and space, recording where it goes as Mario moves around

3. Export the results to be analysed elsewhere

1. Import the motion capture results into Matlab and perform regression analyses on them to figure out the constants of motion

If you want to try this yourself, it’s certainly easy enough to do and figure out, but more likely you’re just interested in seeing some of the results I found. Here are the plots of the 5 basic motions I examined (walking, running, jumping, high-jumping, and following off a ledge):

In each of the plots above I’ve included a simple linear regression analysis based on the physical model that makes sense - basically, for vertical movement a quadratic equation indicating constant gravity / acceleration, and a linear equation for horizontal movement indicating constant horizontal movement speed. For the relatively simple motion that we see here, this is all we really need, once we know a thing or two about Newton’s laws of motion and calculus - which I’ll quickly go over so you know what I’m talking about.

Basically, when thinking about motion, there are a couple key relationships that exist - the velocity of the motion is the derivative with respect to time (rate of change) of the position of the object, and the acceleration of the motion is the derivative of the velocity with respect to time:

Then, assuming we have a constant acceleration due to gravity (in the real world this acceleration is 9.81 m/s^2), we can figure out what the regression coefficients in the above analyses mean:

Applying this math to the regression coefficients that I calculated, we find that the coefficient in front of the term is half of gravity, while the coefficient in front of the t term is the “jump” velocity (we can ignore the y_0 parameter, as it really doesn’t mean much here). Using this, I found three separate gravities and push-off velocities, depending on the scenario (assuming short-Mario is 1m tall):

• When jumping regularly, Mario experienced a gravity of 67.82 m/s² (6.9 times Earth gravity). Here he had a push-off velocity of 17.36 m/s.
• When high-jumping, Mario experienced a gravity of 34.79 m/s² (3.5 times Earth gravity). Here he had a push-off velocity of 15.21 m/s.
• When falling off a ledge, Mario experienced a gravity of 55.88 m/s² (5.7 times Earth gravity).

As for running, Mario seemed to follow a perfectly constant run velocity pattern (as is expected, really). When walking, he moved at a speed of about 3.7 m/s, and when running, moved at a speed of about 9.1 m/s.

Now, it is a good idea to take these results with a grain of salt - I did the whole experiment and analysis in only one shot, with one trial for each type of motion, and made a few assumptions about the nature of the motion as well. That said, these values should give you a good starting point to produce “fun” motion in your games! Also, of course, Mario is the full property of Nintendo (or whoever the owner of it is), and I do not claim to possess ownership or even inside knowledge of the Mario name, game, physics, programming, etc. I do own a copy of game, which I still play occasionally (no emulator will ever be able to replace the feel of holding that plastic controller in your hand, sitting in your couch in front of the TV!), but that’s about it.


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# Use Lorentz Force to Find Magnetic Field Components

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So I am constructing an analogy between the self replicating fracturing effect on thin films and the path of a charged particle. (Qualitatively, several cracks have similar shapes to charged particle motion) I won't go into the details of the fracture mechanics, so I will only use E+M nomenclature.

My problem: Start with a parameterized path function, s, of a charged particle from which the velocity and acceleration functions can be easily derived. (a and R are constants)
$$\vec{s} = ( tcos(at) , tsin(at),(x^{2} + y^{2}) R )$$
I want to use this in a modified Lorentz Force equation where the mass-charge ratio is set to one (m/q=1)In the equation, I ultimately want to find a way to solve for all x, y, z components of E and B without resorting to Maxwell's equations.
$$dv/dt = E + v × B$$

Current attempt: In order to reduce the amount of unknowns from 6 to 3, (i.e. from Ex, Ey, Ez, Bx, By, Bz to just Bx, By, Bz) I thought I can use the relation between kinetic energy and the electric field. I am not sure if this is the right thing to do, so feel free to suggest a different method for minimizing unknowns. I used the following relation for this (m/q=1 again):
$$Work = qEs = \frac{1}{2}mv^{2}$$

So now s, v, dv/dt, and each component of E is known thus leaving the components of B left to be solved. The following vector equation is the result of taking the cross product:
$$(a_{x},a_y,a_z )= (E_x,E_y,E_z)+(v_yB_z-v_zB_y,v_zB_x-v_xB_z, v_xB_y-v_yB_x)$$

I think this then simplifies to:
$$\begin{pmatrix} 0& -v_{z} &v_{y} \\ v_{z}&0 &-v_{x} \\ -v_{y}& v_{x}& 0 \end{pmatrix}\begin{pmatrix} B_{x}\\ B_{y}\\ B_{z} \end{pmatrix} =\begin{pmatrix} a_{x}-E_{x}\\a_{y}-E_{y} \\ a_{z}-E_{z} \end{pmatrix}$$
Which doesn't seem to have a solution. What identities can I implement to yield solutions for this equation? Do I need to set one of the B components to 0? Is my equation for E useful in this case? Can I solve for Ex, Ey, Ez, Bx, By, Bz numerically- what numerical method would be useful?

tldr: I just want to know how to take a particle path and derive the electric and magnetic field components

Feel free to ask any questions for clarification!

Twigg
Gold Member
If your interest is in thin film cracking, can't you reduce the problem to 2 spatial dimensions? This would be the same as adding the constraints ##v_{z} = 0## and ##\frac{dv_{z}}{dt} = 0##, which means ##E_{z} + (v_{x}B_{y} - v_{y}B_{x}) = 0##. That would take you down to five unknowns.

To simplify the problem (whether you work in 2D or 3D), you will have to introduce the constraint of a steady-state field: ##\nabla \times \vec{E} = 0##, and of course ##\nabla \cdot \vec{B} = 0##. This will get incredibly messy, because it couples the equations of motion of nearby points.

One thing you could try: write an implicit finite-difference algorithm for solving the field equations over a grid of points, then further constrain this by adding the equations of motion given the velocity and acceleration at each point, and see if that gives you a fully defined system of equations. It's not exactly easy or guaranteed to work (the equations of motion might give you an under-defined system, it's hard to say).

Example in 2D:

In 2D, you only have ##E_{x}, E_{y}##, and ##B_{z}## to worry about, and ##\nabla \cdot \vec{B} = 0## doesn't matter. Suppose I take a grid of 4 points with spacing a, where (1,1) is the bottom left, (1,2) is the bottom right, (2,1) is the top left, (2,2) is the top right.

The field equations at (1,1) say ##\frac{E_{y}(1,2) - E_{y}(1,1)}{a} - \frac{E_{x}(2,1) - E_{x}(1,1)}{a} = 0##. This is one equation in twelve unknowns (3 unknown field variables per point, and 4 points).

The equations of motion at (1,1) say ##\vec{a} = \vec{E} + \vec{v} \times \vec{B}##, where ##\vec{a}## and ##\vec{v}## are known constants. This is 2 equations in 12 unknowns.

Taking both field equations and equations of motion together, the point (1,1) gives you 3 equations in 12 unknowns. If you repeat this process for the three remaining points in the grid, you obtain a total of 12 equations in 12 unknowns. Assuming the equations are all linearly independent, you will have a unique solution for the fields. You could check this by writing a 12x12 linear system for all these equations and taking its determinant. If it is zero, the method doesn't work. If it isn't zero, you can use any algorithm for solving linear systems to find the E and B fields.

Thanks, do you have any insight for the 3D example? Can it be done using the same algorithm you proposed for 2D? I am working in 3D because the delamination of the thin film cracks creates an upwards peeling effect along with the planar propagation of the crack.

So, the example I used for the path function was the Archimedian spiral (b), but it actually has a parabaloid shape. When I started this work, I was thinking in 2D, but the delamination peeling is actually really integral to these cracks. (The peeling is seen via the color warping between crack lines)

Twigg
Gold Member
A paraboloid shape simplifies things, because that means the force in the z-axis is a constant. Since ##z = Rt^{2}##, you know that ##a_{z} = 2R## and thus ##F_{z} = 2mR##. A constant ##F_{z}## requires that ##B_{x} = B_{y} = 0## and ##F_{z} = qE_{z}##, or else the z-component of acceleration would depend on velocity in the plane. ##\nabla \cdot \vec{B} = 0## implies that ##B_{z}## is constant. That makes the problem effectively 2D, and maybe the algorithm I suggested can solve it.

In general, I suppose you could extend the algorithm I suggested to 3D, in which case the points would have indices, and you would have to use all four of the static field equations at each point plus the three equations of motion per point. Try a 2x2x2 example by hand to see if you get enough equations to fully determine the fields.

If it's not too much to ask, do you think you can run through the 3D example?

Twigg
Gold Member
Sure thing! Sorry for not elucidating.

Consider a 2 by 2 by 2 cubic grid of points, where each point is given by three indices as shown in the attached image (sorry for the low quality!). You have 6 unknowns per point and 8 points, so 48 unknowns total.

First, we have to formulate the field equations over the discrete mesh. In general, if we use (i,j,k) to denote the indices of the points, then ##\nabla \times \vec{E}=0## turns into three finite-difference equations (using forward differences only):

##\frac{E_{z}(i,j+1,k) - E_{z}(i,j,k)}{\Delta y} - \frac{E_{y}(i,j,k+1)-E_{y}(i,j,k)}{\Delta z} = 0##
##\frac{E_{x}(i,j,k+1) - E_{x}(i,j,k)}{\Delta z} - \frac{E_{z}(i+1,j,k)-E_{z}(i,j,k)}{\Delta x} = 0##
##\frac{E_{y}(i+1,j,k) - E_{y}(i,j,k)}{\Delta x} - \frac{E_{x}(i,j+1,k)-E_{x}(i,j,k)}{\Delta y} = 0##

(It might work out better if you use the average of forward and backward differences for the discrete derivative, but those are practical details you'll only know once you implement. See https://en.wikipedia.org/wiki/Finite_difference for explanation of forward and backward differences.)

And the remaining field equation ##\nabla \cdot \vec{B} = 0## becomes

##\frac{B_{x}(i+1,j,k) - B_{x}(i,j,k)}{\Delta x} + \frac{B_{y}(i,j+1,k)-B_{y}(i,j,k)}{\Delta y} + \frac{B_{z}(i,j,k+1) - B_{z}(i,j,k)}{\Delta z} = 0 ##

This gives 4 field equations per point and 8 points, so 32 equations in 48 unknowns.

Then apply the equations of motion at each point, ##\vec{a}(i,j,k) = \vec{E}(i,j,k) + \vec{v}(i,j,k) \times \vec{B}(i,j,k)## where ##\vec{a}## and ##\vec{v}## are known for all (i,j,k). That gives 3 equations per points and 8 points, so 24 equations in 48 unknowns.

Taking the field equations and equations of motion together, you have 56 equations in 48 unknowns. My interpretation of this is that not every velocity/acceleration pattern is compatible with a static electromagnetic field. There is a catch though, because if you do this numerically the odds of getting a perfectly solvable system are slim, because all it takes is one slight deviation in the last decimal place of a floating point number to make the system unsolvable. With that in mind, you may need to resort to a least-squares solution if the direct approach fails.

Let me know if anything wasn't clear.

P.S.: As you can see, the linear systems get very large very fast (in this case, 56 equations in 48 variables for an 8 point grid). You will need to use sparse matrix encoding (supported in MATLAB, BLAS, Lapack, and Armadillo, that I know of, plus probably a lot of others I don't know of) to reduce the memory load.

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Ok, so I have attempted your method, but I just learned forward/backward difference methods last semester. I have reduced the problem significantly and felt an update was finally appropriate.

For the spiral-paraboloid shape $$(atcos(at),atsin(at),Rt^{2})$$
I can make 2 assumptions:
1.) The B and E-fields are parallel and are only in the z direction (pointing out the mouth of the paraboloid).
2.) Since the acceleration in the z direction is just 2R (constant), the E-field is also constant- denoted E0

So this really helps reduce the Lorentz Equation $$\vec{a} = \vec{v}_{y}\vec{B}_{z}\hat{x}+\vec{v}_{x}\vec{B}_{z}\hat{y}+E_{0}\hat{z}$$
Where the acceleration and velocity vector functions are known. Can Bz be solved in this case? I'm sorry if my questions seems rudimentary; I tried solving and got two different values for Bz. I'm not sure if I trust my methods though. I tried the following...
$$\vec{v}_{y}\vec{B}_{z}=\frac{\delta v_{x}}{\delta{t}} , \vec{v}_{x}\vec{B}_{z} =-\frac{\delta v_{y}}{\delta{t}}$$