**6. Aeroheating**

During most of the entry phase, the vehicle flies at hypersonic speeds. For instance, the aeroshell which landed the NASA rover *Curiosity*, started its descent on Mars at 125 km of altitude, at speed of 5800 m/s [12]. Using the MCD model at 125 km, a M<sup>∞</sup> = 34 is obtained. At such high Mach numbers, shock waves and turbulent boundary layers determine very high aerothermal loading conditions (i.e., large radiative and convective heat fluxes, and pressure loads) for the vehicle.

As it is well known, vehicle aeroheating is strictly related to the kind of entry trajectory. Shallower entry leads to a smaller heat rate, but increases the integrated heat loads, thus requiring a thicker TPS. Conversely, steep entries determinate higher heat rates and lower total heat loads. Therefore, in order to adopt a lightweight (passive) and fully reusable TPS, AoA and the flight-path angle are key parameters to trade-off thermal heating and vehicle mass (terminal landing speed).

Generally speaking, the first law of Thermodynamics states that during the descent, the huge amount of kinetic energy (KE) and potential energy (PE) of an entry vehicle dissipate into heat energy that warm up both spacecraft and the atmosphere surrounding it.

Moreover, heat transfer is mutually exchanged between the vehicle surface and the surrounding flow through convection and radiation. The energy balance at spacecraft wall suggests that the convective heat flux, *q*\_*conv*, is given by the sum of heat conducted into the TPS material, *q*\_*cond*, minus the amount of heat reradiated, *q*\_*rad*. By neglecting the conduction inside the heatshield, we have:

$$
\dot{q}\_{conv} = \dot{q}\_{rad} \tag{4}
$$

being *<sup>q</sup>*\_*conv* <sup>¼</sup> <sup>7</sup>*:*207*x*10<sup>4</sup> *<sup>ρ</sup>*0*:*<sup>47</sup> ∞ *Rn* <sup>0</sup>*:*<sup>54</sup> *v*<sup>3</sup>*:*<sup>5</sup> (W/m2 ), given by the Sutton-Graves stagnation-point relationship, and *<sup>q</sup>*\_*rad* <sup>¼</sup> *σεT*<sup>4</sup> *<sup>w</sup>*, according to the

Stephan-Boltzmann law. Thus, for a safe landing, this huge heat energy must be transferred to the spacecraft shock-layer (i.e., surrounding gas instead of spacecraft) as much as possible.

The heat that goes to the vehicle during time *dt* is given by the product of heat flux rate, *q*\_, and the reference aerodynamic surface of the vehicle:

$$\frac{dQ}{dt} = \dot{q} \mathbf{S}\_{r\circ} \tag{5}$$

which can be re-written as:

$$\dot{q} = \overline{\text{St}} \rho\_{\text{os}} v\_{\text{os}} \left[ \frac{v\_{\text{os}}^2}{2} + c\_p \left( T\_{\text{os}} - \overline{T}\_w \right) \right] \tag{6}$$

where *St* is a mean Stanton number, *Tw* is a vehicle mean temperature and *T*<sup>∞</sup> is the freestream temperature. Neglecting the thermal contribution *cp(T*∞*- Tw)*, and substituting in (5) one has:

$$dQ = \overline{\text{St}} \rho\_{\text{os}} \frac{\upsilon\_{\text{os}}^{-3}}{2} \text{S}\_{\text{ref}} dt \tag{7}$$

Vehicle acceleration related to ballistic coefficient reads:

$$m\frac{dv}{dt} = -\frac{1}{2}\rho v^2 \mathbf{c}\_D \,\mathbf{S}\_{ref} \tag{8}$$

Therefore, by replacing *Sref* given by Eq. (8) in Eq. (7) and integrating over time between initial speed, *vi*, and the current one at time *t*, *v*, it follows:

$$
\Delta Q = \frac{\overline{\text{St}}}{2\text{C}\_D} m \left( v\_i^2 - v^2 \right) \tag{9}
$$

Eq. (9) expresses the integrated heat load absorbed by the vehicle during the descent in the time interval *ti* ½ � , *t* . Here, spacecraft energy is represented only by KE. Anyway, this approximation can be accepted considering that the PE of entry vehicle is negligible if compared with KE.

According to Eq. (9), the fraction of spacecraft energy that converts into vehicle heating (i.e., *ΔQ*) depends on the *St=cD* ratio. Therefore, entry flights with high drag aeroshape (e.g., capsule aeroshapes) are suggested. This design solution, however, is not suitable for manned missions.

High lift configurations must be exploited in order to limit inertial loads and aerothermal loads by flying as much as possible shallow trajectory at higher altitudes, i.e. lower density. However, shallow entries (*γ* ≪ 1 and *α* <30° Þ are characterized by large flight time and consequently large integrated heat loads (*ΔQ*Þ.

## **7. Entry dynamics**

Mars entry trajectory is computed in a non-rotating, inertial, Mars-Fixed Mars-Centered reference frame [13]. A three degree-of-freedom numeric simulation is assumed, and the spacecraft is described as a point mass, performing a non-planar unpowered descent trajectory with a constant bank angle (**Figure 4**) [13]:

*Lifting Entry Analysis for Manned Mars Exploration Missions DOI: http://dx.doi.org/10.5772/intechopen.101993*

#### **Figure 4.** *Reference frames for entry flight equations.*

$$\begin{aligned} \frac{dV}{dt} &= -\frac{D}{m} - g\sin\chi\\ \frac{d\chi}{dt} &= \frac{1}{V} \left(\frac{L}{m}\cos\mu\_d - g\cos\chi + \frac{V^2}{r}\cos\chi\right) \\\ \frac{d\chi}{dt} &= \frac{1}{V} \left(\frac{L}{m}\frac{\sin\mu\_d}{\cos\chi} - \frac{V^2}{r}\cos\chi\cos\chi\tan\varrho\right) \\\ \frac{dr}{dt} &= V\sin\chi \\\ \frac{d\theta}{dt} &= \frac{V\cos\chi\cos\chi}{r\cos\rho} \\\ \frac{d\rho}{dt} &= \frac{V\cos\chi\sin\chi}{r} \end{aligned} \tag{10}$$

where *θ*, φ are the longitude and latitude in a spherical frame of reference, *χ* is the heading angle and *μ<sup>a</sup>* is the bank angle.

Lift and Drag force are given by *L* ¼ <sup>1</sup>*=*2*ρV*<sup>2</sup> *SrefCL* and *D* ¼ <sup>1</sup>*=*2*ρV*<sup>2</sup> *SrefCD* respectively where aerodynamic coefficients *CL* and *CD* are taken using an aerodynamic database in hypersonic and low supersonic regimes [10].

The first order ODE system is integrated over time with a fourth-order explicit Runge–Kutta method.
