For hyperbolic orbits, the orbit curve approaches a straight line at great distance from the orbiting centre. The more eccentric the orbit, the greater the angle between these asymptote straight lines.
ϑ
The total thrust capacity of the rocket, as defined by the integral of the thrust by time. (thrust × time). The total deltaV of a single stage rocket is dependent on the fuel load mass of the rocket as a ratio of the total rocket mass, and the exhaust velocity of the rocket engine.
The remaining deltaV corresponds to the fuel left after the thrust schedule has completed.
This is non-editable here, as the direction of the initial thrust is determined by the latitude, day and hour of launch. The assumption being the rocket should launch vertically initially to escape Earth's gravity as quickly as possible. To establish an Earth orbit, a rocket will usually begin deviating slowly from this vertical course soon after lift off.

The azimuth direction is like a compass direction on the Solar plane. On day zero, the direction of the Sun from the Earth is 90°
Chemical Rocket
Uses a chemical reaction to produce thrust. Commonly kerosene or hydrogen reacting with oxygen. Exhaust velocity is typically about 2.5km/s. Which means 1kg of fuel expended in 1 second produces 2500N of force.

Deuterium Rocket
Such a rocket would use the nuclear reaction of deuterium (hydrogen with neutrons) into helium to produce thrust. Such a rocket has not been invented, and is at present unfeasible. If such a rocket existed, its exhaust velocity would likely be in the order of 500km/s, or 500,000N of force for every kg of fuel per second.

Antimatter Rocket
A rocket that might get you to the nearest star within a lifetime, but in all practicality impossible to create. Such a rocket would react electrons with positrons to produce high energy gamma radiation. Assuming this radiation can be directed perfectly in the thrust direction, then I kg of fuel would produce Mass×speed of light kgs.ms-1 of momentum. 1 kg of fuel in a second would give you 3×108N of force.
The amount of fuel as a ratio of the total rocket's mass determines how much total thrust can be maintained. In general, the total change in velocity a rocket can achieve in open space is given by

ΔV = Ve × logε ( Mi/ Mf )

Where:

ΔV = change in velocity
Mi = Rocket's intial mass
Mf = Rocket's final mass
Ve = Rocket's exhaust velocity
logε is the natural logarithm of base ε

Note, that this does not take into account thrust used to escape a planet's gravity.

The mass of the rocket is not important, as we are considering the fuel load as a ratio of the total mass, the thrust can be divided by the unknown rocket mass to get acceleration - the fuel usage rate as a ratio of total mass is independent of the actual mass and therefore not important to this simulation.
The number of days to complete a full orbit is calculated by dividing the current "Areal Velocity" (or sweep) into the area of the deduced ellipse:

Period (seconds) = π × a × b / Av

where a and b are the major and minor radii of the deduced ellipse in metres.
Av is the Areal Velocity in (metres2 second-1)

The period will be constant while the two body solution of planet and sun is an accurate approximation. In reality, effects from other planets disturb the elliptical orbit, hence why the period and areal velocity may be seen to change. A more accurate orbital period would need to take into account these effects.
The day of launch will define where the Earth is in regard to the solar system for the launch. Day zero is the day when the Earth is furtherest from the sun, which occurs usually on July 4th. Note that it is about 13 days after the Southern Hemipshere winter solstice, when the South Pole is tilted most away from the Sun.

On day zero, the Earth's position is defined by convention here as [-1, 0, 0], therefore a noon launch towards the sun will have a positive x component. The Earth's relative orbital velocity on day zero is defined as [0,0,0.99] (slghtly below 1 in the z-component of the orbital plane), so the z-component is initially increasing. Note that the Y-component is defined here as perpendicular to the plane of orbit.

The time of launch will affect the direction the rocket is pointing towards when it launches. A vertical launch is assumed. It will also affect how the rotational velocity of the Earth assists the rocket. In general, dawn launches give the rocket an assist towards the sun, and evening launches assist the rocket away from the sun.

The hour can be set from 0 to 24, with 12 representing a noon launch.
This will calculate an intersection trajectory to the target body.
This will populate the inclination, azimuth and burn-time input fields which will need to be applied (apply button below) to use.

A calculation can only be made if the rocket is in space (ie after launch) and if the target shares the same main orbit body. Wait for the rocket to escape Earths gravitational influence before trying to calculate orbit tranjectories to other planets

See Hohmann transfer orbit for the description of the ideal transfer orbit.

As two solutions are generally found, the one requiring the least amount of rocket fuel is chosen by default, unless you tick the option 'direct', in which case it will always choose the quickest.
If the target is the body the rocket orbits, then the default calculation gives you a simple linear collision calculation, but will give you a circular orbit if 'orbit' is ticked (what could possibly go wrong?)
Launch thrust can be entered as an acceleration. It will need to be initially greater than 9.8ms-2 for the rocket to lift against the gravity at the surface.

The thrust acceleration will be considered constant, with the burn rate adjusting as the rocket mass depletes.
The gravitational accelerations will however be constantly re-calculated.

A thrust schedule can be set up on the next interface to determine where the rocket accelerates towards after escaping the Earth's gravity.

Note that this simulator only considers the thrust and gravity, no consideration is made of air resistance during the initial launch.
The Launch latitude can be entered as between -90 and +90 degrees°. A latitude closer to zero gives the strongest rotational assistance. The launch direction and rotational assist vectors will be determined by the seasonal tilt angle (23.45°), the time of launch and the latitude.

At equinox dates (day 80 and day 259) the evening and dawn launches will result in the least rotational assist in the directions perpendicular to the orbital plane, while noon and midnight launches will have strong components in this direction.

Correspondingly, the solstice dates ( ie day 170) will have significant rotational assistance perpendicular to the orbital plane for evening and dawn launches.

As most rockets wish to explore the planets in the orbital plane, and want to use rotational assist, it is usually most useful to use low latitudes and dawn launches. (Dawn launches to slingshot around the sun)
The position is calculated from the centre of mass of the two objects, this is most often likely very close to the centre of the largest of the two objects, ie sun. The three numbers are the vector coordinates for the X,Y and Z axis directions. The X/Z plane is designated as the elliptical plane (the common orbital plane for the solar system). The directions of the X and Z axis are arbitary.

The units used are astronomical units (AU) for the primary and secondary bodies, ie sun and planets. One astronomical unit (AU) is defined as the average distance of the Earth from the Sun.

1 AU = 149597870700 metres

For moons we use a Lunar distance (LD), which is defined as the average distance of the Earth's moon (Luna) from the Earth.

1 LD = 384402000 metres = 0.002569569 AU

These Earth-centric units are used for the convenience of being able to conceptualise the distances.
The distance from the centre of mass of the two objects, this is most often likely very close to the centre of the largest of the two objects, ie sun.

The units used are astronomical units (AU) for the primary and secondary bodies, ie sun and planets. One astronomical unit (AU) is defined as the average distance of the Earth from the Sun.

1 AU = 149597870700 metres

For moons we use a Lunar distance (LD), which is defined as the average distance of the Earth's moon (Luna) from the Earth.

1 LD = 384402000 metres = 0.002569569 AU

These Earth-centric units are used for the convenience of being able to conceptualise the distances.
The position angle from the centre of the orbit ellipse - from the main axis (containing the centre of mass). The angle here is always positive between 0 and 180°

This is a dynamically calculated figure derived from the current orbital distance and velocity to deduce the ellipse properties.
The centre of mass of the two bodies is one of the ellipse focal points.

ϑ
The following equations relate the distance to focal point (orbital distance) to the angle and and ellipse radii.

r2 = a2sin2ϑ + b2cos2ϑ

where ϑ is angle from major axis and r is distance from centre

the cosine rule also gives us:

f2 + r2 - 2frcosϑ = d2

where f is the length of the triangle base from the ellpse centre to the focal point, and d is the current distance from focal point (distance form centre of mass) and has been solved from solving the closest approach using the orbital energy.
The closest and furtherest approach are calculated for common centre of mass of the two bodies. With a planet and sun, this is generally very close to the centre of the sun. For Jupiter and the sun, this centre just clears the surface of the sun.

To calculate the approaches, the centre of mass is treated as the foci of the orbit ellipse with a reduced mass of m13/(m1+m2)2 where m2 is the mass (kg) of the body being analyzed and m1 is the mass (kg) of the body it is most effected by gravitationally (ie, the sun).

The orbit where only two bodies are involved obeys two fundamental rules in Newtonian physics: conservation of angular momentum and conservation of energy.
This is represented by the following equations:

V × R × sinϑ = P
E = m2V2/2 - G×Mr×m1/2×R

ϑ
V⃗
R⃗

  • E is the constant energy (kg.m.s-2)
  • V is the velocity of the analyzed object with respect to the centre of mass
  • P is the Areal Velocity (metres2.second-1) - ie angular momentum divided by the mass
  • m1(kg) is the mass of the analyzed body
  • Mr is the reduced mass (kg) described above
  • ϑ is the angle between the velocity tangent and the position from centre of mass position
  • R (metres) is the distance from the centre of mass
  • G is the gravitational constant

At the closest and furtherest approaches, the velocity tangent is orthogonal and sinϑ becomes 1. The two equations reduce to a quadratic:

Er×R2/2 + G×Mr×R/2 - P2/2 = 0 (note: Er = E/m2)

The two solutions for R give the closest and furtherest approaches respectively assuming the body is in orbit.
Kepler's second law of planetary motion states:

A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time

The rate at which this line sweeps out area is called the Areal Velocity. This is related to the angular momentum of the orbiting body divided by it's mass. The Areal Velocity can be calculated by:

V × R × sinϑ / 2 = P

  • P is the Areal Velocity (metres2.second-1) - ie angular momentum divided by the mass (and halved)
  • R (metres) is the distance from the centre of mass
  • ϑ is the angle between the velocity tangent and the position from centre of mass position
ϑ
V⃗
R⃗
ΔA
Above, ΔA is the small segment of area swept over a small increment of time Δt.
ΔA/Δt is the Areal Velocity
The major and minor axii of the elipse (A & B), can be deduced from the focii distances derived from the closest and furtherest approach distances.
A
B
f

The foci distance f has the following properties:

f2 + a2 = B2

ie B shares the length of the hypotenuse of the above triangle with side A and length f.
The main axis direction projected onto the x-z plane expressed as an angle from the postive z direction. The main axis direction here is considered to be the vector from the ellipse centre, through the centre of mass to the point of closest approach.
ϑ
The main axis direction expressed as a unit vector (length 1) of the x,y,z axis directions. The main axis direction here is considered to be the vector from the ellipse centre, through the centre of mass to the point of closest approach.
The deviation of the orbital path plane from the x-z plane.

A vector direction orthogonal to the orbit plane is deduced from a vector "cross product" of the velocity and relative position vectors, represented:

V⃗×R⃗ = P⃗

P⃗ is the angular momentum and is considered to have a direction orthogonally towards the observer if the rotation is seen to be anti-clockwise. The position vector R⃗ is from the centre of mass, and the velocity is with respect to the inertial frame of the centre of mass.

The elleptical plane of the solar system can be defined in numerous ways, but here the inclination is simply the deviation from the x-z plane which approximates the ecliptic. The ecliptic is defined by the Earth's orbital plane, but even this varies.
V⃗
R⃗
P⃗
The eccentricity is a way to define the elongation of an ellipse. The smaller the number the more the ellipse is like a circle.

A
B
f

The eccentricity is defined by the ratio of the focus distance divided by the major radius:

E = f / B

  • E is the eccentricity
  • f is the distance from the centre to the focal point
  • B is the major radius
The lost time a clock on the rocket would appear to lose if compared to a clock in a stationery frame with respect to the Solar System but away from the gravity well.

The time delta is integrated from the time of launch and is an integration of the inverted lorentz factor minus one by the time interval. ie ∫ Δt/γ

where γ is the Lorentz Factor determined by relative speed and gravitational potential and
Δt is the infinitessimal increments of the base time.
The Lorent factor can be crudely be thought to be the time dilation factor between two clocks in different frames. In this case a clock on the rocket and one in a stationary frame with the Solar System but far enough away so that the gravity is negligible. An Earth clock comparison might be desirable, but the calculations would be more complex given that the Earth frame is affected by gravity and complex motion.

A time dilation however is only one aspect of the Lorentz factor. It also describes the the apparent distorion of space in the relative direction of motion.

While it might be tempting to think that the clock in the rocket is going slower, in reality the Lorentz factor is symmetrical. It describes an apparent distortion when we rotate between different frame coordinates. The rotation in this case using an imaginary number. Like rotating a line segment, the length of the line is the same in both frames, howevever the projection of the rotated line-segment onto the original frame changes the size of the time and lengths in the same way your shadow is not the same length as you. This apparent change in size is defined by the Lorentz factor.

The Lorentz factor here is a combination of the gravitational potential of the rocket's position in relation to the surrounding masses of the Solar System and the relative velocity of the rocket.