Editing Atmospheric entry (section)

===Sphere or spherical section===
[[File:Apollo cm.jpg|thumb|[[Apollo command module]] flying with the blunt end of the [[heat shield]] at a non-zero [[angle of attack]] in order to establish a lifting entry and control the landing site (artistic rendition)]]
The simplest axisymmetric shape is the sphere or spherical section.<ref>{{cite web|last1=Przadka|first1=W.|last2=Miedzik|first2=J.|last3=Goujon-Durand|first3=S.|last4=Wesfreid|first4=J.E.|title=The wake behind the sphere; analysis of vortices during transition from steadiness to unsteadiness.|url=http://sphere.meil.pw.edu.pl/publi/AoM_60_2008_6.pdf|website=Polish french cooperation in fluid research.|publisher=Archive of Mechanics., 60, 6, pp. 467–474, Warszawa 2008. Received May 29, 2008; revised version November 13, 2008.|access-date=3 April 2015|archive-date=December 21, 2016|archive-url=https://web.archive.org/web/20161221044217/http://sphere.meil.pw.edu.pl/publi/AoM_60_2008_6.pdf|url-status=live}}</ref> This can either be a complete sphere or a spherical section forebody with a converging conical afterbody. The aerodynamics of a sphere or spherical section are easy to model analytically using Newtonian impact theory. Likewise, the spherical section's heat flux can be accurately modeled with the [[Fay-Riddell equation|Fay–Riddell equation]].<ref name="Fay-Riddell">{{cite journal|last1=Fay |first1=J. A. |last2=Riddell |first2=F. R. |title=Theory of Stagnation Point Heat Transfer in Dissociated Air |journal=Journal of the Aeronautical Sciences |volume=25 |pages=73–85 |date=February 1958 |url=http://pdf.aiaa.org/downloads/TOCPDFs/36_373-386.pdf |format=PDF Reprint |access-date=2009-06-29 |issue=2 |doi=10.2514/8.7517 |url-status=dead |archive-url=https://web.archive.org/web/20050107202757/http://pdf.aiaa.org/downloads/TOCPDFs/36_373-386.pdf |archive-date=2005-01-07 }}</ref> The static stability of a spherical section is assured if the vehicle's center of mass is upstream from the center of curvature (dynamic stability is more problematic). Pure spheres have no lift. However, by flying at an [[angle of attack]], a spherical section has modest aerodynamic lift thus providing some cross-range capability and widening its entry corridor. In the late 1950s and early 1960s, high-speed computers were not yet available and [[computational fluid dynamics]] was still embryonic. Because the spherical section was amenable to closed-form analysis, that geometry became the default for conservative design. Consequently, crewed capsules of that era were based upon the spherical section.

Pure spherical entry vehicles were used in the early Soviet [[Vostok programme|Vostok]] and [[Voskhod programme|Voskhod]] [[space capsule|capsule]]s and in Soviet Mars and [[Venera]] descent vehicles. The [[Apollo command module]] used a spherical section forebody heat shield with a converging conical afterbody. It flew a lifting entry with a hypersonic trim angle of attack of −27° (0° is blunt-end first) to yield an average L/D (lift-to-drag ratio) of 0.368.<ref>{{Cite web |url=https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19690029435_1969029435.pdf |title=Hillje, Ernest R., "Entry Aerodynamics at Lunar Return Conditions Obtained from the Flight of Apollo 4 (AS-501)," NASA TN D-5399, (1969). |access-date=July 7, 2017 |archive-date=September 16, 2020 |archive-url=https://web.archive.org/web/20200916020329/https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19690029435_1969029435.pdf |url-status=live }}</ref> The resultant lift achieved a measure of cross-range control by offsetting the vehicle's center of mass from its axis of symmetry, allowing the lift force to be directed left or right by rolling the capsule on its [[Flight control surfaces#Longitudinal axis|longitudinal axis]]. Other examples of the spherical section geometry in crewed capsules are [[Soyuz programme|Soyuz]]/[[Zond program|Zond]], [[Project Gemini|Gemini]], and [[Project Mercury|Mercury]]. Even these small amounts of lift allow trajectories that have very significant effects on peak [[g-force]], reducing it from 8–9&nbsp;g for a purely ballistic (slowed only by drag) trajectory to 4–5&nbsp;g, as well as greatly reducing the peak reentry heat.<ref>{{cite report|last1=Whittington|first1=Kurt Thomas|title=A Tool to Extrapolate Thermal Reentry Atmosphere Parameters Along a Body in Trajectory Space|url=http://repository.lib.ncsu.edu/ir/bitstream/1840.16/6817/1/etd.pdf|website=NCSU Libraries Technical Reports Repository|date=April 11, 2011|publisher=A thesis submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the degree of Master of Science Aerospace Engineering Raleigh, North Carolina 2011, pp.5|access-date=5 April 2015|archive-date=April 11, 2015|archive-url=https://web.archive.org/web/20150411064311/http://repository.lib.ncsu.edu/ir/bitstream/1840.16/6817/1/etd.pdf|url-status=live}}</ref>