Study of the payload extraction trajectory heavy class carrier rocket

The obtained equation (23) is solved by the method of successive approximations. In the first approximation only the first two summands are taken into consideration; the last two summands are neglected. Let us integrate the equation (23):

$\underset{V_0}{\overset{V}{\int }}dV=-W_h\cdot \underset{{\mu }_0}{\overset{\mu }{\int }}\frac{d\mu }{\mu }+T\cdot \underset{{\mu }_0}{\overset{\mu }{\int }}g\cdot \sin \theta \cdot d\mu ,$

$V^I-V_0=-W_h\cdot \ln \mu -T\cdot \underset{\mu }{\overset{1}{\int }}g\cdot \sin \theta \cdot d\mu ,$

$V^I=V_0-W_h\cdot \ln \mu -T\cdot \underset{\mu }{\overset{1}{\int }}g\cdot \sin \theta \cdot d\mu$ is the first Korolev integral.

In the first approximation we determine only the flight altitude. For this purpose, let us write the equation (2) taking into account (22):

$\frac{dY}{dt}=V\cdot \sin {\theta }_{\text{programs}}\underset{}{}\to \underset{}{}dY=V\cdot \sin {\theta }_{\text{programs}}\cdot dt,$

$dY=-T\cdot V\cdot \sin {\theta }_{\text{programs}}\cdot d\mu ,$

$\underset{y_0}{\overset{y^I}{\int }}dY=-T\cdot \underset{{\mu }_0}{\overset{\mu }{\int }}{V}^I\cdot \sin {\theta }_{{}_{\text{programs}}}\cdot d\mu ,$

$Y^I=y_0+T\cdot \underset{\mu }{\overset{1}{\int }}\sin {\theta }_{\text{programs}}\cdot d\mu$ is flight altitude in the first approximation.

Thus, the rocket velocity in the first approximation is equal to the ideal velocity minus velocity loss to overcome gravity.

When calculating the velocity in the second approximation, it is necessary to take into account the influence of the atmosphere and the change in pressure at the nozzle exit of the engine:

$d{V}^{II}=-\frac{W_h}{\mu }+T\cdot g\cdot \sin {\theta }_{\text{programs}}\cdot d\mu +\frac{T}{P_m}\cdot \frac{q^I\cdot {C_x}^I}{\mu }\cdot d\mu +(W_h-W_E)\cdot \frac{P_h^I}{P_E}\cdot \frac{d\mu }{\mu }$.   (24)

After the integration of the equation (24) we obtain:

$V^{II}=V_0-W_h\cdot \ln \mu -T\cdot J_1-\frac{T}{P_m}\cdot \underset{\mu }{\overset{1}{\int }}\frac{q^I\cdot {C_x}^I}{\mu }\cdot d\mu +(W_h-W_E)\cdot \underset{\mu }{\overset{1}{\int }}\frac{P_h^I}{P_E}\cdot \frac{d\mu }{\mu },$

where

$q^I=\frac{{\rho }^I\cdot {\left(V^I\right)}^2}{2},$

${\rho }_h^I={\rho }_0\cdot H\left(Y^I\right).$

The calculated velocity head q is close to the true q on the rocket flight path, since it is determined by the overestimated velocity and underestimated density. The value of the drag coefficient  depending on velocity is taken according to the equation (1).

$\frac{P_h^I}{P_E}=f\left(Y^I\right)$: this value is generally underestimated because it is determined by an overestimated altitude.

It is customary to denote:

$J_2=\underset{\mu }{\overset{1}{\int }}\frac{q^I\cdot {C_x}^I}{\mu }\cdot d\mu$ is the second Korolev integral;

$J_3=\underset{\mu }{\overset{1}{\int }}\frac{P_h^I}{P_E}\cdot \frac{d\mu }{\mu }$ is the third Korolev integral.

Thus, taking into account the losses, we obtain:

But the value of the third integral itself is insignificant; therefore this inaccuracy does not significantly affect the final value of the rocket velocity.

$V^{II}=V_0-W_h\cdot \ln \mu -T\cdot J_1-\frac{T}{P_m}\cdot J_2+(W_h-W_E)\cdot J_E$ is the rocket velocity formula in the second and final approximation.

Knowing the velocity of the rocket, you can find the altitude and range

$\frac{dX}{dt}=V\cdot \cos \theta ,$

$\frac{dY}{dt}=V\cdot \sin \theta ,$

After all the transformations, we obtain the formulae for determining the altitude and range in the second approximation:

$X=X_0+\underset{\mu }{\overset{1}{\int }}{V}^{II}\cdot \cos \theta \left(\mu \right)\cdot d\mu ,$

$Y=Y_0+\underset{\mu }{\overset{1}{\int }}{V}^{II}\cdot \sin \theta \left(\mu \right)\cdot d\mu ,$

Injection programme

The spacecraft injection programme is understood as a set of control functions that regulate the procedure for changing the thrust vector in nominal motion, set the law for changing the propellant mass flow rate and the angular orientation of the rocket in space and time. The injection programme is autonomously executed by the control system in accordance with the functions defined at the rocket design stage. These functions are the law of change in propellant consumption and the law of change in pitch angle at the injection stage.

With the development of rocket technology and the expansion of the list of tasks to be solved, the approach to the selection of the payload orbital injection programme has also changed. The simplest injection programme is a set law of pitch angle variation. As a rule, the mass fuel consumption on the marching sections is not regulated and remains practically constant. If the launch vehicle is designed to launch a spacecraft or a cargo module, one injection programme is used, if the launch vehicle is designed for a manned launch, another programme is used.

In general, programme selection is a search for a combination of trajectory parameters satisfying the flight task with minimum energy costs. This search is subject to limitations, primarily related to the design and operational features of the launch vehicle itself, for example, its power capabilities. Since the propulsion system (RD-171MV) of the designed rocket is more powerful, a number of constraints have significantly changed, expanding the ballistic capabilities of the rocket. To investigate the capabilities of the launch vehicle equipped with the RD-171MV engine, let us perform a variational analysis of the spacecraft injection trajectories using the developed computer programme created in the MAPLE software package [18].

Let us consider some programmes of spacecraft injection by a launch vehicle (Fig. 5). As a variable parameter, we take the change of pitch angle during the flight (Fig. 6). Early start of the rocket turn (trajectory 4) to the required final angle of spacecraft injection (for a circular orbit it is equal to zero degrees) leads to a significant deviation in range from the launch point to a given point in the target orbit. In this case, the time of delivery of the spacecraft to the target orbit will be maximised. Late turn with prolonged vertical ascent (trajectory 1) provides minimum time of spacecraft delivery to the target orbit with minimum range deviation from the launch point. In addition, Figures 5 and 6 show intermediate trajectories. The points on the trajectory plots correspond to the impulse supply for pitch angle changes. The number and time of such activations can also be varied.

Рис. 5. Зависимость высоты орбиты (Y) от дальности полета (Х) для разных программ выведения космического аппарата

Fig. 5. Dependence of orbit altitude (Y) on flight range (X) for different spacecraft injection programmes

Рис. 6. Изменение угла тангажа в процессе полета для разных траекторий выведения космического аппарата

Fig. 6. Changing the pitch angle during flight for different launch trajectories of the spacecraft

Depending on the chosen trajectory, the final velocity of the rocket, the time of injection of the spacecraft into the target orbit and the overload will vary.

Thus, taking into account the developer's requirements, we can select the parameters of the spacecraft injection trajectory [18], varying the change in pitch angle for given values of final velocity and overload on the active section of the trajectory.

Calculations for the Soyuz-5 launch vehicle

Using the developed computer programme [18], let us determine the most rational parameters of the payload injection trajectory for the Soyuz-5 launch vehicle (Fig. 7).

The main characteristics of the Soyuz-5 launch vehicle are given in Table 2 [8].

Table 2. Technical characteristics of the Soyuz-5 launch vehicle

Table 2 shows the maximum capabilities of the rocket. Therefore, each task requires adjustment of fuel mass by stages.

Рис. 7. Эскиз ракеты «Союз-5»: Р – полезный груз; – разгонный блок;  – масса топлива второй ступени;  – масса двигателя второй ступени;  – масса топлива первой ступени; – масса двигателя первой ступени

Fig. 7. Sketch of the Soyuz 5 rocket: P – payload;  – booster unit;  – the mass of the second stage fuel;  – the mass of the second stage engine;  – the mass of the first stage fuel;  – the mass of the first stage engine

To determine the final parameters of a rocket, it is necessary to choose the type and mass of the payload. According to the available information, the Soyuz-5 [8] can launch a satellite with a mass up to 2500 kg into geostationary orbit. Consequently, the Soyuz-5 can launch the payload with a fully fuelled booster into a low Earth orbit (altitude 200 km). The mass of the Glonass-K satellite is 935 kg. The mass of the fueled Fregat booster equals 6280 kg. Let us assume payload values equal to 7215kg. To keep the payload stable in orbit, the velocity must be at least:

$V_{\text{orbit}}=\sqrt{\frac{g\cdot R_{\text{earth}}}{R_{\text{earth}}+{Н}_{\text{orbit}}}}=\sqrt{\frac{9.81\cdot 6 378.1}{6 378.1+200}}\approx 7.783\frac{km}{s}=7783\frac{m}{s}.$

For further calculations of the rocket flight trajectory we will use the developed computer programme [18]. In the programme we set the tactical and technical characteristics of the rocket and the pitch angle change programme. The engine operation time, total flight time on the active section of the trajectory, engine thrust and mass flow rate are determined by the tactical and technical characteristics of the rocket. According to the obtained data, the rocket flight trajectory is calculated by the method of successive approximations. The obtained calculated trajectory parameters (overload, altitude, velocity, acceleration, drag, etc.) are deduced in the form of graphs and numerical values.

Numerical experiment

Let us consider the programmes of spacecraft injection into a reference orbit by a two-stage rocket using the specifications of the Soyuz-5 launch vehicle (Table 2). Let us use the variants of the rocket turn programme considered earlier (Fig. 6.). The data of calculations are presented in Table 3.

Table 3. Parameters of spacecraft delivery trajectory to the reference orbit with different launch programmes

The final velocity for all trajectories is not sufficient for stable operation of the spacecraft on the selected low Earth orbit. Nevertheless, for all the trajectories, after reaching the required orbit altitude, there is still a fuel reserve that can be spent for booster's acceleration. Using the fuel reserve, let us perform calculations of the payload velocity correction on the target orbit. The results of calculations are given in Table 4.

Table 4. Spacecraft final velocity on the reference orbit after trajectory correction

As can be seen from Table 4, the velocity is still insufficient, so we will correct the payload mass, which in turn will affect the mass of the rocket (Table 5).

Thus, the analysis of possibilities of payload injection by the Soyuz-5 launch vehicle with RD-171MV engine (Tables 2-4) allows choosing a trajectory with minimum time of spacecraft delivery to orbit (tk = 205 s), launching mass of the rocket (m0 = 551.5 tonnes) and longitudinal load factor (n = 2.25) (trajectory 1). However, the spacecraft experiences the maximum lateral load factor (n = 6.7) and has a smaller payload mass (mpg = 4.5 tonnes).

Table 5. Mass Characteristics of the Soyuz-5 launch vehicle for payload injection into a reference orbit (H = 200

The launch programme with maximum delivery time (tk = 353 s) (trajectory 4) provides maximum mass of the payload (mpg = 17 t) and minimum side load (n = 1.79), despite some increase in the launching mass of the rocket (m0 = 564 t) and a significant increase in longitudinal load factor (to n = 6.84) compared to the trajectory 1. Nevertheless, this trajectory variant will be considered preferable, as there are design methods to reduce the impact of axial overloads on the spacecraft design and the possibility of reducing overloads for the designed spacecraft taking into account changes in the customer's requirements by optimising the launch programme of the customer by optimising the withdrawal programme.

Conclusion

The study analysed the use of the latest RD-171MV rocket engine with improved energy and mass-size characteristics on the new Soyuz-5 heavy-lift launch vehicle. On the basis of the system of differential ballistic equations, calculations were carried out using Maple software for modelling the trajectories of payload injection by a heavy launch vehicle into a reference orbit. Based on the results of the calculations, trajectories with different ballistic parameters were proposed, taking into account possible changes in the customer's requirements for the spacecraft being designed. The analysis of the family of launch vehicle trajectories allows ensuring an efficient and energetically favourable injection of payloads to the target orbit. The presented results can be used in the future to optimise the energy costs of launching the designed spacecraft into orbit.