Semi-major axis, energy and hyperbolic excess velocity

The semi major axis (${\displaystyle a\,\!}$) is not immediately visible with an hyperbolic trajectory but can be constructed as it is the distance from periapsis to the point where the two asymptotes cross. Usually, by convention, it is negative, to keep various equations consistent with elliptical orbits.

The semi major axis is directly linked to the specific orbital energy (${\displaystyle \epsilon \,}$) or characteristic energy ${\displaystyle C_{3}}$ of the orbit, and to the velocity the body attains at as the distance tends to infinity, the hyperbolic excess velocity (${\displaystyle v_{\infty }\,\!}$).

${\displaystyle v_{\infty }^{2}=2\epsilon =C_{3}=-\mu /a}$ or ${\displaystyle a=-{\mu /{v_{\infty }^{2}}}}$

where: ${\displaystyle \mu =Gm\,\!}$ is the standard gravitational parameter and ${\displaystyle C_{3}}$ is characteristic energy, commonly used in planning interplanetary missions

Note that the total energy is positive in the case of a hyperbolic trajectory (whereas it is negative for an elliptical orbit).

Eccentricity and angle between approach and departure

With a hyperbolic trajectory the orbital eccentricity (${\displaystyle e\,}$) is greater than 1. The eccentricity is directly related to the angle between the asymptotes. With eccentricity just over 1 the hyperbola is a sharp "v" shape. At ${\displaystyle e={\sqrt {2}}}$ the asymptotes are at right angles. With ${\displaystyle e>2}$ the asymptotes are more than 120° apart, and the periapsis distance is greater than the semi major axis. As eccentricity increases further the motion approaches a straight line.

The angle between the direction of periapsis and an asymptote from the central body is the true anomaly as distance tends to infinity (${\displaystyle \theta _{\infty }\,}$), so ${\displaystyle 2\theta _{\infty }\,}$ is the external angle between approach and departure directions (between asymptotes). Then

${\displaystyle \theta {_{\infty }}=\cos ^{-1}(-1/e)\,}$ or ${\displaystyle e=-1/\cos \theta {_{\infty }}\,}$

Impact parameter and the distance of closest approach

Hyperbolic trajectories followed by objects approaching central object (small dot) with same hyperbolic excess velocity (and semi-major axis (=1)) and from same direction but with different impact parameters and eccentricities. The yellow line indeed passes around the central dot, approaching it closely.

The impact parameter is the distance by which a body, if it continued on an unperturbed path, would miss the central body at its closest approach. With bodies experiencing gravitational forces and following hyperbolic trajectories it is equal to the semi-minor axis of the hyperbola.

In the situation of a spacecraft or comet approaching a planet, the impact parameter and excess velocity will be known accurately. If the central body is known the trajectory can now be found, including how close the approaching body will be at periapsis. If this is less than planet's radius an impact should be expected. The distance of closest approach, or periapsis distance, is given by:

${\displaystyle r_{p}=-a(e-1)={\frac {\mu }{v_{\infty }^{2}}}\left({\sqrt {1+\left(b{\frac {v_{\infty }^{2}}{\mu }}\right)^{2}}}-1\right)}$

So if a comet approaching Earth (effective radius ~6400 km) with a velocity of 12.5 km/s (the approximate minimum approach speed of a body coming from the outer Solar System) is to avoid a collision with Earth, the impact parameter will need to be at least 8600 km, or 34% more than the Earth's radius. A body approaching Jupiter (radius 70000 km) from the outer Solar System with a speed of 5.5 km/s, will need the impact parameter to be at least 770,000 km or 11 times Jupiter radius to avoid collision.

If the mass of the central body is not known, its standard gravitational parameter, and hence its mass, can be determined by the deflection of the smaller body together with the impact parameter and approach speed. Because typically all these variables can be determined accurately, a spacecraft flyby will provide a good estimate of a body's mass.

${\displaystyle \mu =bv_{\infty }^{2}\tan \delta /2}$ where ${\displaystyle \delta =2\theta _{\infty }-\pi }$ is the angle the smaller body is deflected from a straight line in its course.

Position

In a hyperbolic trajectory the true anomaly ${\displaystyle \theta }$ is linked to the distance between the orbiting bodies (${\displaystyle r\,}$) by the orbit equation:

${\displaystyle r={\frac {\ell }{1+e\cdot \cos \theta }}}$

The relation between the true anomaly θ and the eccentric anomaly E (alternatively the hyperbolic anomaly H) is:^[3]

${\displaystyle \cosh {E}={{\cos {\theta }+e} \over {1+e\cdot \cos {\theta }}}}$     or     ${\displaystyle \tan {\frac {\theta }{2}}={\sqrt {\frac {e+1}{e-1}}}\cdot \tanh {\frac {E}{2}}}$     or   ${\displaystyle \tanh {\frac {E}{2}}={\sqrt {\frac {e-1}{e+1}}}\cdot \tan {\frac {\theta }{2}}}$

The eccentric anomaly E is related to the mean anomaly M by Kepler's equation:

${\displaystyle M=e\sinh E-E}$

The mean anomaly is proportional to time

${\displaystyle M={\sqrt {\frac {\mu }{-a^{3}}}}.(t-\tau ),}$ where μ is a gravitational parameter and a is the semi-major axis of the orbit.

Flight path angle

The flight path angle (φ) is the angle between the direction of velocity and the perpendicular to the radial direction, so it is zero at periapsis and tends to 90 degrees at infinity.

${\displaystyle \tan(\phi )={\frac {e\cdot \sin \theta }{1+e\cdot \cos \theta }}}$

Velocity

Under standard assumptions the orbital speed (${\displaystyle v\,}$) of a body traveling along a hyperbolic trajectory can be computed from the vis-viva equation as:

${\displaystyle v={\sqrt {\mu \left({2 \over {r}}-{1 \over {a}}\right)}}}$^[4]

where:

• ${\displaystyle \mu \,}$ is standard gravitational parameter,
• ${\displaystyle r\,}$ is radial distance of orbiting body from central body,
• ${\displaystyle a\,\!}$ is the (negative) semi-major axis.

Under standard assumptions, at any position in the orbit the following relation holds for orbital velocity (${\displaystyle v\,}$), local escape velocity (${\displaystyle {v_{esc}}\,}$) and hyperbolic excess velocity (${\displaystyle v_{\infty }\,\!}$):

${\displaystyle v^{2}={v_{esc}}^{2}+{v_{\infty }}^{2}}$

Note that this means that a relatively small extra delta-v above that needed to accelerate to the escape speed results in a relatively large speed at infinity. For example, at a place where escape speed is 11.2 km/s, the addition of 0.4 km/s yields a hyperbolic excess speed of 3.02 km/s.

${\displaystyle {\sqrt {11.6^{2}-11.2^{2}}}=3.02}$

This is an example of the Oberth effect. The converse is also true - a body does not need to be slowed by much compared to its hyperbolic excess speed (e.g. by atmospheric drag near periapsis) for velocity to fall below escape velocity and so for the body to be captured.