# Analytical theories for spacecraft entry into planetary atmospheres and design of planetary probes

#### Abstract

This dissertation deals with the development of analytical theories for spacecraft entry into planetary atmospheres and the design of entry spacecraft or probes for planetary science and human exploration missions. Poincaré’s method of small parameters is used to develop an improved approximate analytical solution for Yaroshevskii’s classical planetary entry equation for the ballistic entry of a spacecraft into planetary atmospheres. From this solution, other important expressions are developed including deceleration, stagnation-point heat rate, and stagnation-point integrated heat load. The accuracy of the solution is assessed via numerical integration of the exact equations of motion. The solution is also compared to the classical solutions of Yaroshevskii and Allen and Eggers. The new second-order analytical solution is more accurate than Yaroshevskii’s fifth-order solution for a range of shallow (-3 deg) to steep (up to -90 deg) entry flight path angles, thereby extending the range of applicability of the solution as compared to the classical Yaroshevskii solution, which is restricted to an entry flight path of approximately -40 deg. Universal planetary entry equations are used to develop a new analytical theory for ballistic entry of spacecraft for moderate to large initial flight path angles. Chapman’s altitude variable is used as the independent variable. Poincaré’s method of small parameters is used to develop an analytical solution for the velocity and the flight path angle. The new solution is used to formulate key expressions for range, time-of-flight, deceleration, and aerodynamic heating parameters (e.g., stagnation-point heat rate, total stagnation-point heat load, and average heat input). The classical approximate solution of Chapman’s entry equation appears as the zero-order term in the new solution. The new solution represents an order of magnitude enhancement in the accuracy compared to existing analytical solutions for moderate to large entry flight path angles. The analytical theory is very accurate for moderate to large entry angles and for any entry speed. A new analytical theory is developed for ballistic entry at circular speed for zero initial flight path angle and for ballistic entry at circular speed for very small to large initial flight path angles. Two separate solutions for zero and non-zero initial flight path angles are needed to avoid a singularity. The classical Yaroshevskii’s solution enters as the zero-order term in the solutions. Using the new solutions, other important expressions are developed such as time-of-flight, range, deceleration, and aerodynamic heating parameters (e.g. average heat input, stagnation-point heat rate, and total stagnation-point heat load). Large-scale human exploration of Mars and in situ exploration of Venus pose great challenges for entry, descent, and landing of spacecraft. The Adaptive Deployable Entry and Placement Technology (ADEPT), a mechanically deployable decelerator, presents an enabling alternative to the traditional rigid aeroshell technology. ADEPT helps in lowering the ballistic coefficient of an entry vehicle and also presents attractive options for lifting and guided entry. Optimal trajectory solutions which minimize peak deceleration and peak heat-flux are computed for four different control strategies. The deployable decelerator for human Mars missions (requiring a landed mass of 40 mt) presents an acceptable entry environment—peak heat-flux of < 80 W/cm2, and peak deceleration of less than 4 G (compared to 200 W/cm2 and 15 G for Mars Science Laboratory respectively). For lifting and guided entry for Venus in situ missions, ADEPT could lead to a two-order-of-magnitude decrease in peak deceleration and to a 50% decrease in peak heat-flux compared to conventional rigid aeroshell technology. There exist a number of attractive trajectory candidates for round-trip human missions to Mars and Venus. However, the speeds the spacecraft will encounter during Earth reentry are unprecedented. NASA’s Stardust entry robotic spacecraft which entered Earth (ballistic entry) at a speed of 12.9 km/s is the fastest reentry achieved by a manmade object to date. The goal is to assess the feasibility of Earth reentry for fast free returns Mars and Venus human missions within human tolerance limits and capabilities of current state-of-art vehicle and thermal protection system technologies.

#### Degree

Ph.D.

#### Advisors

Grant, Purdue University.

#### Subject Area

Aerospace engineering

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