 Good evening to everybody. As you might imagine, I am a sister of Samson. And I have the longest history knowing him among all of you. Because I'm the elder sister, one-year elder. So this is our... You cannot guess. This is me and this is you. And this is our father and this is our mother. So we have exactly one year and two weeks difference and I probably know him just right a few days after he was born. And so our parents were father physicists. My mother is still alive, mathematician. And it's Samson in maybe one and a half or two. So he was very active, very sweet, very kind. And always playing active things and already from that time he's driving a car, you can see. And it's maybe our... I'm ten, he's nine or something around. And he was climbing, you know, it was difficult to stop him. But he was not spoiling anything in fact. He was observing and studying and trying to invent something. And maybe it is around ten also. He was very much interested in chess. Our father was parallel to being physicist. Good chess player, Grossmeister. And by that time he gave up playing, but he was chief referee of Tbilisi. So he was very often going to chess palace and Samson would follow. And here are very famous Georgian chess players are here and some of them trainers, even this one is trainer I guess. And so here is my brother. And then you can recognize some of you are sitting here. So this is the time when he already went to Leningrad as a PhD student. You know, in between we studied always together in the same place. Like in kindergarten, the school, only three years he went to physical, mathematical, special school. Otherwise then I went to physics faculty. Next year he came to the same faculty. So we were always together. But at some stage he decoupled, let's say he went to Leningrad. And after that I'm seeing him rarely and more rarely and I'm thankful to organizers to arrange this very dense meeting with my brother. So here is Karepin. You can see this is Atya. This is you. Do you have this photo? Yeah. So this is Samson. He was always slim lately. He became not slim unfortunately. So he is Atya. This one I don't know, maybe one of you remember? Antonov. Antonov, yeah. Yeah. So and here is all the group. That was Michael Atya. So Takhtay Jankulish, Novikov, Padev, Samson, Karepin, Venkov, and Bolivanov. I don't remember all of them. But those years I would frequently visit Leningrad so I was very close to his colleagues and friends and all. Leon Tatyajan and the right one. Yes, of course. Here is the one. Chairman. So at that time my study was closer to what you were doing because I was working on a very, very particular interaction that would be governed by non-linear, non-local, non-linear Schrodinger equations and I had a chance to solve it with some perturbation method but so doing this similar thing like Zaharov and I was using generalized Zaharov equations. But late after those years, gradually I kind of shifted my direction to more like applicable realistic objects because all the referees would write, where do we find such objects and things like that? So I started to learn some astrophysical things. I started from easiest, I thought easiest solar physics and then I moved to general stellar things and then at some stage I got interested in disk jet formation problem and since it is related to some analytical approaches that we proposed on the Beltrami conditions, Bernoulli theory and things like that, I decided it was difficult for me to choose the subject for my talk here so at the end I decided I better concentrate on this problem. Maybe it's also a little bit spectacular and maybe somehow easier to be a presenter here among you that it's maybe very far from your interest. So initially we made two papers with Professor Yoshida. He's from the University of Tokyo, the same place our previous lecture was. It's just next building sits from this Kavel Institute and I often get there so it's a fantastic place and another co-author is Alexander Tezade from my, I'm chair of astrophysics so he works with me there. So I want to give outline, okay no need of outline, I will just start from the very beginning. So what is disk jet structure? I mean you maybe all have seen the jets coming out from the nearby areas of compact object, massive objects or whatever but disks are not easy to see. But it's already well proven that accretion disks, not all disks are accretion but accretion disks often are combined with spindle like jets of ejecting gas and it becomes a typical structure that accompanies the massive objects of various scales ranging from young stars to AGM and mechanism that rules each part of different system can be not universal. Here I give the neutron star jet and disk and here is the elliptical galaxy jet and disks shall be here somewhere. So I mean not always seen in the same frequencies and this very macroscopic structure. So a macroscopic disk jet geometry independent from all these scaling have a very marked similarity and maybe starting from the early 70s or previous century when the radio galaxies and quasars were discovered it was evident that the jets in different classes of physical systems have direct association with accretion disks but opposite is not true. In some objects you may have accretion disks without collimated jets because viscose transport or disk winds play the similar role in energy balance. So winds are far from central object from the disk coming out from the disk but jets are from the center. This is the difference in fact. And this macroscopic disk jet geometry has a marked similarity despite the huge variety of the scaling parameters like uranium number, longquist number, ionization fractions and many other parameters. And it is already known that in the disk region transport processes of mass, momentum and energy depends strongly on the scaling parameters. And classical processes like, you know, simple processes of viscosity are insufficient to account from the accretion rate and turbulence transports must be invoked. In addition, for powerful jets winds may also remove the angular momentum. And connection of the disk and the jet is more complicated. Mass and energy of jets are fed by the accreting flow of disks and the mechanism is how this transfer occurs is still not clear. I mean, we pretend that we know how it happens but generally it's not the final, let's say, clearance. The major constituent of the jets is the material of accretion disk surrounding the central object. This one, later at the end I will speak about the protostellar jet. This one is the protostellar jet and here is the accretion disk around the accreting young stellar object that is neutral, not charged. So what are the basic properties I continue? Here is, again, the pulsar jet. It has magnetic fields, strong magnetic fields. So for the fastest outflows contributions may come also from the outer regions of the compact object, outer regions of the disk as well as from the compact object. Livio suggested in 97 that powerful jets can be produced by system in which on top of an accretion disk threaded by a vertical filter exists an additional source of energy and wind associated with the central object. Launching of an outflow from accretion disk requires a hot corona, corona of disk and also can be assumed that extensive hot atmosphere around the compact object can also provide the additional acceleration. So I want to specify here some of the community members normally say the jet acceleration problem as if jet existed and then they are solving the acceleration of jet but we propose the different approach giving the idea of formation of disk jet structure. So it's one structure like disk jet and the acceleration is the next stage of the formed jet, let's say because if nothing existed in that space it cannot be accelerated. Maybe you need to eject some gas and then try to accelerate it. So this accretion ejection process gives you one structure and then additional effects of some other physical properties can give you extra acceleration. So it's also believed that magnetic fields are playing a very important role in defining the local accretion. This is disk, here is the compact star and these are magnetic fields threading the disk and due to the accretion the field lines are advected to the center, you can see. So plus its rotation also. So when magnetic field is advected inwards by the accretive material or generated locally by some mechanism jet along the magnetic field lines can become super alphanic. And agent jets, there is an alternative idea based on the electrodynamical processes extracting energy from rotating black holes. This is the very famous work of Blentford and Snarchek. And extra galactic radiogen, these two classes are very powerful and hot. Jets might be accelerated by highly disorganized magnetic fields and here are the launch factors and many other parameters play a role. But I will be speaking about the universal mechanism where I don't need all these additional effects. So in addition to the energetics to account for the acceleration of this ejecting flow we have to explain how these stream lines or magnetic field lines change the topology through the disk jet connections. So correct theory must explain also this connection. It's not that you just give a model. So our proposition is that disk jet system is a generalized Beltramel vortex. So this was one of the pioneering paper of Blentford and Pine also next one by Begenwan. It's still, all these theories are based on having strong magnetic fields. But we saw that the collimatic structure of jet is a natural consequence of the alignment of the flow velocity and the vorticity. So called Beltramel condition the term is the structure and there is no requirement of having magnetic fields. That's the universal. So on a clarion disk the vorticity becomes a vertical vector with magnitude proportional to this value. I'm covering it. Sorry. Myself I hear. But it's not for the recording. Sorry. So alignment is the unique solution for avoiding the singularity of Coriolis force near the center. But speaking about the vorticity we need to explicitly define which type of vorticity. So in our model vorticity is generalized appropriately by dissipation in the disk. So the dissipation that causes the aggression. So simplest theory, the MHD model it's not even MHD because my disk is practically very having low ionization. So in fact it is neutral here. The equations are neutral here. So my title is wrong. I'm sorry. So this momentum, this momentum and here is the equation of motion. So this mass, this flow velocity is effective viscosity tensor, weak gravity potential and here are given the normalization parameters and V dot and rho dot are the representative flow velocities of the disk. So remember I mentioned that the jet is fed by the matter and energy of disk. So we are normalizing to the disk background parameters. So that we could track what would happen later. So some manipulation with the equations gives this equation where omega 2 is the so-called generalized vorticity and it brings the impact of the dissipation. Dispation is here as you guys might see. We shall show the generalized bentrami condition demanding omega dot to be parallel to p2 is a unique condition to avoid the singular energy density. I don't remember if I introduced what was rho 1 and rho 2. Okay. So rho 1 is the, I split rho here. Sorry, I missed here. I introduced two parameters. So density is splitting two. So in conventional you have rho 1. And rho 2 is one. So with no dissipation rho 2 goes to one. But with dissipation, this one carries the information about the dissipation. So with no viscosity, rho 1 is one. So and correspondingly you have p2 and p1. With rho 2 and rho 1. Can be. If you have slow acceleration and slow density variation it can be. And for such system it's okay. It's not charge. I mean I try to speak about the universal property. The charge is the next step. Yeah. So p2 is defined by rho 2 and p1 is by rho 2. V stays the same. Velocity is the same. So the requirement of having this left-hand side zero that is beltrami condition is the possibility of avoiding the singularity. So since we are speaking about acceleration, so the mean velocity in the disk is the toroidal one. The V-teta. A cylindrical geometry. And if you introduce the viscosity then you have VR also appearing with it's still smaller than V-teta but the acceleration is due to the VR. So it goes to center. So it's like slowly you have material coming close to center. But due to the singularity it has to go somewhere. It cannot go to the exact zero. So this is the singularity. So you need to break matter from disk to somewhere and the jet is a natural consequence of this breaking. So it's estimation here given if since our acceleration is small normally it is like that this dissipation is not so strong. The flow in the disk is primarily azimuthal and the viscosity force can be approximating then this can be shown that it's proportional. This is the proportionality and eta is some local viscosity coefficient representing the viscosity. So if one compares to this equation then we can change this term by this value. So where we put new to be positive. And this viscosity force is again primarily azimuthal and if one goes to this equation you see that we can find the solution when this term vanishes. So the viscosity defines the partial inertial term. This is part of the inertial term due to the dissipation p2 is defined by rho2. So if one assumes that this entire term vanishes so viscosity is balanced by the partial inertial term and also invoking the continuity equation. This is steady state solution. One can come to this kind of the relation and this is the final relation that defines this rho2 by local viscosity. So this is the determination of rho2 by local viscosity. And after some algebra you can also estimate rho1. So after these estimations we also speak about the poloidal components that becomes defining for jet. So the vorticity includes the conventional vorticity includes the singularity and to eliminate the divergence of this left-hand side of that equation these two vector fields must align each other. That is the Beltrami condition. And after this alignment we are left with Bernoulli condition that is again generalized because of the dissipation. Remember rho2 is now bringing the local dissipation information. So at the end we have determining equation some type of artificial rho2 for given local viscosity velocity field and equation for enthalpy. So these are three equations to be solved. These are all estimations I give and I want to give here some remark that if we would start from the conventional vorticity then that would mean that we don't have local aggression so that means that you could arrive to the adequate condition for Keplerian flow. So the conventional Beltrami condition would not give any jet solution. So we need to generalize this condition and our equation was exactly the generalized case because it was with P2 and omega2 so you have the information coming from dissipation. So that includes local viscosity effects. So then this type of generalized Beltrami condition give you the consistency with this jet structure formation. So here is briefly shown that if one our momentum field is divergence free because we are thinking about steady state solutions. So if we split this momentum field in let's say this is the toroidal file and this is the poloidal file in the stream function then one can solve the equations and here you see the momentum field distribution here the density and here is the total density and here we have introduced the orthogonal variables like tau that defines mostly the jet structure and sigma that is working mostly in the disk. They are orthogonal and we could find this let's say universal solution for very very simple model of type I showed about. I mean we just estimated the local viscosity. But you know this was very mathematical so all the astrophysicists want to compare with realistic object. So they want to show them that this type of solution coincides with realistic concrete object. So where? So double double I say no way sorry misspelled sorry no no it's me yeah but so this was kind of a universal minimal model where we could show that narrowly collimated jet is a unique structure that is amenable to the singularity of the capillary and vorticity and the Beltrami condition alignment of the flow and generalized vorticity characterizes the geometry of the disk structure and we could find the analytic solution very simple in the similarity solution and yeah so the important here is that instead of introducing magnetic field we worked with vorticity so vorticity plays here the same role as magnetic field in all those blend world Begelman and other theories and in reality many disks do not have magnetic field so but the model has to be some minimal model has to show the universal character of this type of structure but as I said this was not enough you know most of mathematical journals accept jet papers but astrophysical journals do not accept if you don't show the exact parameters of jet or something so we had to yeah you know what the generalized to have a generalized description of the Beltrami condition you have to invent this generalized part is and the omega 2 is this one omega 2 is this one yes omega 2 is this one oh no this is omega 2 yeah index is missing yes so omega 2 is this one so this one has to be parallel to this one so this is the generalized condition in fact it is reduced because in a way it's not related to the total momentum or total density you see you have the reduced part of the density only this one to have the total density and total momentum you must have both so it's kind of a reduced that carries only the information about the dissipation but row one follows I showed you the condition that they are linked to each other yeah so the vorticity plays the role of the magnetic field and if you have charged disks then you may have some additional effects and those would work for next step as I said those would work for next stage acceleration and not deformation I mean for formation area you don't need magnetic field so the other additional effects will work for acceleration so this is what is the calculation of this simple model that these are the streamlines and we could show that the generalized Beltrami flow this is generalized Beltrami flow so P2 so this system can be models like this but to convince the astrophysical community you need to go to the realistic object so for that you have to invoke some physics that would bring physics mostly to the central the area to the central object like whole term or turbulent viscosity or many other effects that are very physical yes so and you need to also connect to the coupled systems like jet and disk and jet and for this you need to invoke concrete object reality let's say but up to now I was not speaking about the contribution of central object I will not speak up to the end because central object is even more next step because the main thing is disk and jet is material and energy comes from disk and not the central object these are only pulsar and agn jets where you can imagine that central object can contribute otherwise not so and this was the idea with the Gitev Zadze joined us and we thought about the concrete herbiharo young stellar objects many many types of young stellar objects that are characterized by outflows bipolar outflows and these collimated bipolar outflows from young stellar objects are known to be parsec scale, non-relativistic and supersonic and their velocities are within these values and these jets are launched in the vicinity of protostar and protostar is the star that continues aggression so in the evolution of protostar you may observe the jet launching so aggression goes on and also star evolution and parallel you may observe the jet formation also the molecular clouds exhibit such type of collimated jets and today we know exactly that morphology sizes and velocities of the jet outflows can be used to estimate the mass, luminosity or age of these also it's even opposite so if you can observe and detect such type of jets their parameters can show you the parameters of this star a critical star or disk so it's interesting to solve this problem so what else this I don't hold so yeah there were several parameters of papers following these observations that use self-similar solutions to model these type of structures or specifically to model the jet velocity and there are some studies that we are using some instability theories to construct the bipolar jet outflows but instability mechanics cannot explain long-lived steady structure they can explain some bursts that are short-lived so we wanted to show that we can explain more long-lived steady structure like disk jets are and Shakura and Sulyav in these early years after the observations suggested some special disk-co-sacration-disk models and community would think that it was natural to assume that jets are driven magnetically from the agration-disk so when this magnetic field is directed by the acrylic material or some generated locally magnetic field the centrifugal force due to rotation may boost the jet along the magnetic field lines even after supra-alphanic speed the magnetic field is necessary to help later on Blendford and Pines studied the magnetosentrifugal acceleration again having the magnetic field and they were the first to show theoretically breaking of the metal in the azimuthal direction decided this and outflow acceleration and showed the conditions for the collimation of the flow in this problem it is not necessary to solve the formation problem you need to also show the collimation of the jet jets are very collimated long structures and hot and fast and this collimation stands for long sometimes some solutions showed the formation but jets would break immediately you need to also have it for a long time and collimate it so collimation mechanism can differ from the formation mechanism so there are some other models Blendford and Pines and these other models and angular momentum energy and mass can be removed from the and they were all showing how the angular momentum, mass and energy were removed from the disk and the jet was fitted by that so collimation can be provided by stratified thermal pressure from the external medium and acceleration efficiency depends on the pressure gradient of the medium in all these theories so we started all this material for concrete objects and we started to invoke our this simple universal 2011 model to the realistic object of ESOS invoking the turbulent viscosity of Shakurite's Uniaev that could work as the main reason for the aggression and this was assumed un-magnetized so no pre-existed global magnetic field so the theory is the same I mean we have the same question but for concrete objects it's again biotropic H now H is the enthalpy normalization is similar so we split our density in here we started to call to use more like physical notations like row one that was conventional is row ideal and row two now becomes row reduced so then you introduce P ideal and P reduced and you have similar equation for motion but now important is to define determine correctly the viscosity term so again invoking this peltramid condition and trying to make this term zero we can define the a local viscosity so and arrive to the Bernoulli condition so we employ the viscosity model when the small scale turbulence creates the anomalous dissipation that can be described by using the alpha viscosity model of Shakur and Sindhyaev but we went a little bit further so we assumed that this velocity field is mostly azimuthal so in viscosity term you have only this components left and then we introduce background constant pressure that will be directly linked to Shakur and Sindhyaev and we add here the perturbation sorry possibly I don't know you know this came to us gradually we first started without P dot thinking that everything will be inside but strangely analysis gave not realistic P was coming to be negative but at the end also physically you need to have something background to have the Keplerian disc not always you have the dissipation so the Keplerian this solution must be there too so the P dot serves for that and P dot but P dot now here also is a little bit generalized by Shakur and Sindhyaev and adding the P you have generalization due to turbulent viscosity so then so this alpha dot and P dot give you this Shakur and Sindhyaev solution and then assuming flow to be axisimetric rotating locally with Keplerian angular velocity you know locally it's Keplerian but due to the viscosity it slowly accretes to the compact object so at every ring you have Keplerian but this flow shouldn't be like you know in the sun differential so if you slice this those flows that along the lines usually like in the sun the people I think assume them differential dependence differential rotation in sun is working in the equator lines yeah so in this case if you start from very small alpha and the observation shows that this Shakur and Sindhyaev coefficient alpha is very small 10 to the minus 3 if it's smaller so if it is very slow accretion you can assume that locally you have Keplerian that I'm telling so locally you can but it's still varying so after introducing this constraint you can estimate the tensor of viscosity this is Shakur and Sindhyaev this is the new term and then you find this components and as I said slowly accretion flow minimal model gives you p dot and again invoking this factorization by colloidal and colloidal components of the velocity field and by the way this stream function exactly matches the actual momentum field and this condition of reality remember I said that viscosity term has to be balanced by the partial inertia term this is the condition so it's dependent on beta the beta now plays the role of generalized alpha let's say and if you take beta 0 that means dissipation 0 you don't have any condition for that so also PR is constant, rho r becomes constant, not locally defined rho r becomes 1 so it's 0 to 0 so generalized Bernoulli condition is written like this for sonic energy is written here and generalized Beltrami condition is here extra terms are here and you have all the equations to define the flow so this is 1, this is 2 and this is 3 with this epsilon em and to try to solve these equations we go to similarity variables again orthogonal, the similar way like in the minimal model and it's obvious that gravity depends only on sigma on the disc parameter and also the Psi is only tau dependent in this model because of slow accretion and then one can separate the variables in this solution so this condition balancing condition is this one and Beltrami conditions can be written in these two equations and the Bernoulli conditions in these two and all of these equations are the full set and we had to work to solve on that and so in the similarity variables we can split pressure and two type of densities and this is Beltrami parameter and azimuthal velocity as I said locally Keplerian this is the one and our estimations give this type of dependencies of sigma for sigma parts of these parameters and introducing this function this was a guess I mean there were several suggestions and finally this type of function worked we arrived to the defining equation so this is the equation solving of each gives you the final structure and remember our row two yeah this is what it is row two is now the tau dependent only sigma dependent ones are one read index so solving this equation you arrive to the structure solution and apparently there were three solutions of this equation the one that was beta equals to zero gives you one solution that corresponds to the background dissipation model and two separate solutions for the dissipated dissipated flow in the disc and the jet so here are the estimations for small beta small so this is w plus and w minus and combining this sigma dependent and tau dependent parts you arrive to the defining parameters of the entire so this set is the final solution but we need to we need to find w w functions so the appropriate choice of w will give you the final structure and by the way here we show the character of the solution so you have one solution that is a radial vertical accretion flow that is represented by w plus with a v r negative and v z negative so flow is creating and also squeezed to the central plane and another solution is radial vertical ejection so the accretion flow and ejection flow this corresponds to disc and disc corresponds to jet so with w minus so we r and v z are both negative and then we could manage to calculate the dependence of Beltrami parameter on all the disc depending on the disc parameters and interestingly both solutions of lambda grow with beta this is the disc solution and this is the jet solution for disc you see that for a long period of time it stays very small but jet solution grows so and here is the final solution represented so this is disc flow and it's proportional to beta you see and this is a v r and v z so disc has v r and v z v theta is we know so and for jet it has v r and v z inversely proportional to beta so your small beta gives you very strong jet flow and sigma dependence is the same so to get the disc but we need to have one structure so it must be continuous because it is for velocity field remember you cannot have jumps in the velocity field if it is a physical solution so for physical solution we need to connect all these three solutions so here is the picture so this is disc this is jet and as you can see because there was a solution with beta equals zero so any w is okay so in this region flow goes ballistically to the jet so this is disc region and because for beta equals zero your equations are valid for any w so it goes ballistically to the jet solution so this is how it looks like so but this is now more realistic picture of generalized vortex yes and sorry no no it is still neutral yeah it is still neutral so you can see that you do not need any magnetic field if you properly invoke the local accretion you will have this vortex playing the role of magnetic field I mean they are one can in some way imagine that B plus V is a generalized vortex or generalized magnetic field so you make one of these zero another part works so you do not need to have B if you have Karloff V whatever type so this is the accretion velocity estimated again proportional to beta and tau dependent so this is these two fields are combined drawn here and this is the ejection velocity inversely proportional to beta and both proportional to Keplerian velocity so in low beta limit the derived solution gives you this jet structure that corresponds to the slowly accretion flow in the disk with fast outflow in the jet area and this solution matches the properties of astrophysical accretion ejection flows and it is important that this solution does not depend on the explicit profile of the tau dependent part of density so any function that would properly show the disk jet structure density could work so this was this freedom that made us let's say happy so also decreasing beta is to increase of ratio between vertical and disk part but the transition region is put by hand? no, it's defined by the viscosity it's everywhere the beta it's not by hand so there were three solutions of that equation quadratic equation I mean beta equals 0 beta equals 0 is a solution and we thought these are only two solutions but at the end we realized redefining the function when we introduced this function remember I told you there were several this one this one was introduced in a different way before it included beta and we lost that solution but then we made it free from beta and then you came up to this equation beta equals 0 is a solution when beta equals 0 there is no equation on W so that was an interesting point we missed one solution in the beginning I found this representation for W so then if beta equals 0 there is no equation on W so you don't have an equation but you won't so any W can work for this connection for one value of parameter of equation which is beta equals 0 any W is a solution this was the main guess here when was the definition of beta it was not alpha 0 no alpha 0 extended generalized alpha 0 with all the R and Z dependent it's proportional no alpha 0 is an extra next term there are two proportional but it is R and Z dependent beta is a local parameter so that was the interesting finding by the way so that is why because any W can work in this area so because the requirement that the continuity velocity field must be provided velocity field is a physical field so you must have it continuous so this is like ballistic flow and where did I here so as I told that picture is independent of raw tau so you can invoke any row that will be realistic for this gadget so invoking our let's say 10 years before finding of row this is this is the idea but we added also tau 0 to the divergence it was for avoiding the divergence that tau equals 0 tau was Z over R remember so tau can be 0 it's Z 0 but it never happens in fact because there is a compact object there so flow avoids compact object so it never happens but for simulation it was necessary to invoke this so and these are two sets of these powers and as you can see the second set gives you very similar picture to this gadget for density so if you invoke this row in those definitions self-similar definitions for physical parameters you can arrive to the final solution so and this those final definitions we are by four equations if you remember 49 and 45 is this for density and this is the self-similar background flow that represents the disk jet so we need to also show the ratio if it is a real fast flow to show how the accretion and ejecting flow magnitudes are dependent and those equations exactly give you you see remember it or one was directly proportional to beta another was inversely so if you estimate this ratio you arrive that beta square also defines this ratio between accretion velocity and so you reduce the beta and then you have a faster outflow so the local is strange but it's like that so if you go to the realistic objects when beta is smaller than alpha beta was local I remember so then you can estimate that the ejection flow can be 10 to the 4 power faster than accretion flow and I give below since you are not familiar with this community result I give below some papers that are coinciding these results so to also illustrate yeah major property was estimated outflow long and accelerated on this one dimension so we had to give this vertical flow distribution I think I am saving the time I am close to the end so so the outflow velocities on this picture we show that outflow velocity is maximal near the center object it's still big but at some height it starts to decline let's say so it's clear now that there must be some other physical processes added to show that it constantly accelerates and keeps its structure similar to observed structures so conclusion is that our solutions can describe the formation of this jet structure and not the further acceleration for further acceleration of this form jet and further collimation you need to add the physics related to concrete objects like if you have magnetized disks and the properties from charged disks add and also if you have relativistic compact object you may have some extra sources coming from hot atmosphere or whatever corona but here as you can see this is not fully showing the long the one dimension of jet structure so also this picture I show because the astrophysical community likes to speak in the Mach number languages they don't like only pictures they want to also track how the Mach number behaves so this is the estimated Mach number and interestingly in this test that we did in this small narrow area the Mach number is greater than 1 so it shows you that our solution also explains the supersonic flow behavior closer to the axis so it was a good finding as well so this I don't want to talk about here I give the papers the results of with Mach our theoretical findings so these are big observations of ALMA heavy heart objects and also this protostellar system was well studied in this paper recently and all this data match our results if you take those boundary parameters from those so as you can see the realistic disk jet system jet flow streaming away from the central object is likely to undergo cooling in reality we didn't have any energy equation if you remember so this was the simplest model other transport equations the first one is energy equation then you may have cooling that will reduce local sound speed so then it will explain further supersonic acceleration or expansion or whatever many many thinking the expansion words so I have two slides more I guess but few words I will say yeah so I showed the decrease of the beta leads to the increase of the ratio of vertical and horizontal velocities and our structure depends on the thermal properties of the disk flow this is local alpha and in case of ESOS our solution shows weaker jets at the later stage of the evolution of the protostellar in the temperature of the central object corresponding this matter increases and this is the summary so we constructed the analytic configuration of the hydrodynamic jet from young stellar objects using the Beltrami Bernoulli flow model for this disk jet structure formation and we used the extended turbulent viscosity model extended I mean when I say extended it was not exactly Shakura signal but it was extended Shakura signal model and derived solution describes the disk structure flow with jet properties linked to the properties of the aggression this flow so all the solution carries the information from beta that is local property of the disk so one can use these derived solutions to analyze the astrophysical jets from ESOS and link the properties of outlaws with the local conditions at the inner edge of the aggression flows this is our result I think I don't need to speak about yeah these are about additional effects so what if you want to move to relativistic jets or other objects of course mathematics will be more difficult because relativity effects and others we started with some students and I can tell you that this new effects mostly work on collimation the disk jet is there but collimation can be provided more strongly by the additional effects and of course everybody thanks my brother but I want to thank first of all organizers because they allowed me to be part of you this time because all my life I was part of my brother's life you know it somehow it happened that we are part from each other and now I am with you with friends thank you I want to thank you I think he is the kindest person in the world so he couldn't stand me here thank you no it's very easy for you I think yeah more about this transition so there was the region of the transition and was it totally fixed it was you had three solutions I'll come to this solution one to an intermediate solution this is the equation yes it's forced the place where you go from this solution to the intermediate solution is forced on you by the equations yeah it's forced by the equations now it's forced by equations but freedom is in the choice so what if equation gives you concrete W at this place and equation gives you concrete W here flow must go ballistic to that point and here those regions where you make the transition the place yes it's fixed we are not fixing it the problem fix system problem fix system and then when they are they are dynamically in a way depending on the parameters so whatever the parameters of the physical parameters are given and these points are found then flow must go ballistic to this place so this was the idea other questions please just a generic question that you can match this to the observable observed it's like you know some of the parameters you pick up from the disk of those observations and then you define the beta and invoke here and then you get whatever you get and yeah in the jet so it matched jet so from jet you can go back and then figure out what was there so because the disks as you can see are not so well seen but for disks are many theories you know but in most of them I mean it was the minimal set of equations of course energy equation has to be added and many others but we wanted to show I mean with Yoshida, Yoshida is partly mathematician partly physical experimentalist great person so with Yoshida we discussed a lot for one year spoke with all these you know people that do simulation all this community mostly are based on simulation so they have big codes and play with codes and whatever they get from code it is for them the reality so we spoke with them and we thought that it is not possible that code must give you realistic solution so we first estimated the viscosity and we imagine that this has to be linked to the minimal so in fact the minimal model does not require all these complexities so all the complex things give you extra acceleration extra collimation or whatever I already gave to my diploma students relativistic or too fluid effects you know charged fluid too fluid whatever those add to the collimation it is very more difficult so you need a lot of assumptions to do but it is possible because of observations again so you know observational features and then you do then you use the assumptions so anyway the alpha parameter is better you know they fit into the data but you have to add more features yes that is what I want to know ok if there is no other questions thank you