Editing Sub-orbital spaceflight (section)

==Speed, range, and altitude==
To minimize the required [[delta-v]] (an [[astrodynamics|astrodynamical]] measure which strongly determines the required [[fuel]]), the high-altitude part of the flight is made with the [[rocket]]s off (this is technically called free-fall even for the upward part of the trajectory). (Compare with [[Oberth effect]].) The maximum [[speed]] in a flight is attained at the lowest altitude of this free-fall trajectory, both at the start and at the end of it.{{citation needed|date=May 2024}}

If one's goal is simply to "reach space", for example in competing for the [[Ansari X Prize]], horizontal motion is not needed. In this case the lowest required delta-v, to reach 100&nbsp;km altitude, is about 1.4&nbsp;[[km/s]]. Moving slower, with less free-fall, would require more delta-v.{{citation needed|date=May 2024}}

Compare this with orbital spaceflights: a low Earth orbit (LEO), with an altitude of about 300&nbsp;km, needs a speed around 7.7&nbsp;km/s, requiring a delta-v of about 9.2&nbsp;km/s. (If there were no atmospheric drag the theoretical minimum delta-v would be 8.1&nbsp;km/s to put a craft into a 300-kilometer high orbit starting from a stationary point like the South Pole. The theoretical minimum can be up to 0.46&nbsp;km/s less if launching eastward from near the equator.){{citation needed|date=May 2024}}

For sub-orbital spaceflights covering a horizontal distance the maximum speed and required delta-v are in between those of a vertical flight and a LEO. The maximum speed at the lower ends of the trajectory are now composed of a horizontal and a vertical component. The higher the horizontal [[distance]] covered, the greater the horizontal speed will be. (The vertical velocity will increase with distance for short distances but will decrease with distance at longer distances.) For the [[V-2 rocket]], just reaching space but with a range of about 330&nbsp;km, the maximum speed was 1.6&nbsp;km/s. [[Scaled Composites SpaceShipTwo]] which is under development will have a similar free-fall orbit but the announced maximum speed is 1.1&nbsp;km/s (perhaps because of engine shut-off at a higher altitude).{{citation needed|date=May 2024}}{{update inline|date=June 2024}}

For larger ranges, due to the elliptic orbit the maximum altitude can be much more than for a LEO. On a 10,000-kilometer intercontinental flight, such as that of an intercontinental ballistic missile or possible future [[commercial spaceflight]], the maximum speed is about 7&nbsp;km/s, and the maximum altitude may be more than 1300&nbsp;km.
Any [[spaceflight]] that returns to the surface, including sub-orbital ones, will undergo [[atmospheric reentry]]. The speed at the start of the reentry is basically the maximum speed of the flight. The [[aerodynamic heating]] caused will vary accordingly: it is much less for a flight with a maximum speed of only 1&nbsp;km/s than for one with a maximum speed of 7 or 8&nbsp;km/s.{{citation needed|date=May 2024}}

The minimum delta-v and the corresponding maximum altitude for a given range can be calculated, ''d'', assuming a spherical Earth of circumference {{val|40000|u=km}} and neglecting the Earth's rotation and atmosphere. Let θ be half the angle that the projectile is to go around the Earth, so in degrees it is 45°×''d''/{{val|10000|u=km}}. The minimum-delta-v trajectory corresponds to an ellipse with one focus at the centre of the Earth and the other at the point halfway between the launch point and the destination point (somewhere inside the Earth). (This is the orbit that minimizes the semi-major axis, which is equal to the sum of the distances from a point on the orbit to the two foci. Minimizing the semi-major axis minimizes the [[specific orbital energy]] and thus the delta-v, which is the speed of launch.) Geometrical arguments lead then to the following (with ''R'' being the radius of the Earth, about 6370&nbsp;km):

<math display="block">\text{major axis} = (1 + \sin\theta)R</math>

<math display="block">\text{minor axis} = R\sqrt{2\left(\sin\theta + \sin^2\theta\right)} = \sqrt{R\sin(\theta)\text{semi-major axis}}</math>

<math display="block">\text{distance of apogee from centre of Earth} = \frac{R}{2}(1 + \sin\theta + \cos\theta)</math>

<math display="block">\text{altitude of apogee above surface} = \left(\frac{\sin\theta}{2} - \sin^2\frac{\theta}{2}\right)R = \left(\frac{1}{\sqrt{2}}\sin\left(\theta + \frac{\pi}{4}\right) - \frac{1}{2}\right)R</math>

The altitude of apogee is maximized (at about 1320&nbsp;km) for a trajectory going one quarter of the way around the Earth ({{val|10000|u=km}}). Longer ranges will have lower apogees in the minimal-delta-v solution.

<math display="block">\text{specific kinetic energy at launch} = \frac{\mu}{R} - \frac\mu\text{major axis} = \frac{\mu}{R}\frac{\sin\theta}{1 + \sin\theta}</math>

<math display="block">\Delta v = \text{speed at launch} = \sqrt{2\frac{\mu}{R}\frac{\sin\theta}{1 + \sin\theta}} = \sqrt{2gR\frac{\sin\theta}{1 + \sin\theta}}</math>

(where ''g'' is the acceleration of gravity at the Earth's surface). The Δ''v'' increases with range, leveling off at 7.9&nbsp;km/s as the range approaches {{val|20000|u=km}} (halfway around the world). The minimum-delta-v trajectory for going halfway around the world corresponds to a circular orbit just above the surface (of course in reality it would have to be above the atmosphere). See lower for the time of flight.

An [[intercontinental ballistic missile]] is defined as a missile that can hit a target at least 5500&nbsp;km away, and according to the above formula this requires an initial speed of 6.1&nbsp;km/s. Increasing the speed to 7.9&nbsp;km/s to attain any point on Earth requires a considerably larger missile because the amount of fuel needed goes up exponentially with delta-v (see [[Rocket equation]]).

The initial direction of a minimum-delta-v trajectory points halfway between straight up and straight toward the destination point (which is below the horizon). Again, this is the case if the Earth's rotation is ignored. It is not exactly true for a rotating planet unless the launch takes place at a pole.<ref name="tpticbm">{{cite journal |last1=Blanco |first1=Philip |title=Modeling ICBM Trajectories Around a Rotating Globe with Systems Tool Kit |journal=The Physics Teacher |date=September 2020 |volume=58 |issue=7 |pages=494–496 |doi=10.1119/10.0002070 |bibcode=2020PhTea..58..494B|s2cid=225017449 }}</ref>