Checking ORBIX against published values

Standard atmosphere at 1,000 m

Function

calculateStandardAtmosphere

Inputs

• Altitude 1,000 m (geopotential)

Source

U.S. Standard Atmosphere, 1976, NOAA, NASA and U.S. Air Force, NASA-TM-X-74335 (NASA Technical Reports Server). Table I, geopotential altitude, metric units, page 52, row H = 1000 m.

• Density is just outside the table's rounding. ORBIX divides by the rounded gas constant 287.05 J/(kg·K); the standard's value is R*/M₀ = 8,314.32 / 28.9644 = 287.053 J/(kg·K) (pages 3 and 9). With that value and the same pressure and temperature, the density is 1.11164 kg/m³, which rounds to the printed 1.1116.

Standard atmosphere at 5,000 m

Function

calculateStandardAtmosphere

Inputs

• Altitude 5,000 m (geopotential)

Source

U.S. Standard Atmosphere, 1976, NOAA, NASA and U.S. Air Force, NASA-TM-X-74335 (NASA Technical Reports Server). Table I, geopotential altitude, metric units, page 54, row H = 5000 m.

• Pressure is 0.55 Pa above the table value, about 0.001 percent. The table itself sits low of its own equation here: the standard's pressure equation for this layer, with its constants R*/M₀ = 8,314.32 / 28.9644, g₀ = 9.80665 m/s² and P₀ = 101,325 Pa, gives 54,019.9 Pa at H = 5,000 m, while the table prints 5.4019 + 2 mb (54,019 Pa). The same happens at 1,000 m, where the equation gives 89,874.6 Pa and the table prints 8.9874 + 2 mb. ORBIX, at 54,019.55 Pa, lies between the table and the equation, so the gap is not an ORBIX error. Why the table's last digit falls low has not been confirmed here.

Standard atmosphere at 11,000 m

Function

calculateStandardAtmosphere

Inputs

• Altitude 11,000 m (geopotential)

Source

U.S. Standard Atmosphere, 1976, NOAA, NASA and U.S. Air Force, NASA-TM-X-74335 (NASA Technical Reports Server). Table I, geopotential altitude, metric units, page 58, row H = 11000 m.

• 11,000 m is the top of the constant-lapse-rate layer and the highest altitude the ORBIX calculator accepts. Above it the standard holds temperature at 216.65 K, which ORBIX does not model, so no higher altitude is compared.

Standard atmosphere at 11,000 m, read as geometric altitude

Function

calculateStandardAtmosphere

Inputs

• Altitude 11,000 m, compared as geometric altitude Z

Source

U.S. Standard Atmosphere, 1976, NOAA, NASA and U.S. Air Force, NASA-TM-X-74335 (NASA Technical Reports Server). Table I, geometric altitude, metric units, page 59, row Z = 11000 m (H = 10981 m).

• The ORBIX atmosphere formula is the geopotential form, so its altitude input is a geopotential altitude. A geometric altitude of 11,000 m is a geopotential altitude of 10,981 m, so reading the input as geometric gives the differences shown. This row records the size of that error rather than hiding it.

Isentropic flow at Mach 2

Function

calculateIsentropicFlow

Inputs

• Mach number 2.00
• γ = 1.4

Source

Equations, Tables, and Charts for Compressible Flow, Ames Research Staff, NACA Report 1135 (NASA Technical Reports Server). Table II, supersonic flow, γ = 7/5, page 634, row M = 2.00.

• ORBIX reports stagnation-to-static ratios (for example T₀/T = 1.8). The table prints static-to-stagnation ratios, so the ORBIX values are shown inverted.

Normal shock at Mach 2

Function

calculateNormalShock

Inputs

• Upstream Mach number 2.00
• γ = 1.4

Source

Equations, Tables, and Charts for Compressible Flow, Ames Research Staff, NACA Report 1135 (NASA Technical Reports Server). Table II, supersonic flow, γ = 7/5, page 634, row M₁ = 2.00.

Total pressure recovery across a normal shock at Mach 2

Function

calculateTotalPressureRecovery

Inputs

• Upstream Mach number 2.00
• γ = 1.4

Source

Equations, Tables, and Charts for Compressible Flow, Ames Research Staff, NACA Report 1135 (NASA Technical Reports Server). Table II, supersonic flow, γ = 7/5, page 634, row M₁ = 2.00.

Oblique shock at Mach 3

Function

calculateObliqueShock

Inputs

• Upstream Mach number 3.0
• Deflection angle 12.7°
• γ = 1.4, weak shock

Source

Oblique Shock Wave, An Internet Book on Fluid Dynamics, brennen.caltech.edu. Oblique shock reflection example (Figure 6): M₁ = 3.0, shock angle 30°, deflection 12.7°, M₂ = 2.36.

• The source takes β = 30° as its input and reads the 12.7° deflection from its Figure 4; the exact deflection for M₁ = 3.0 and β = 30° is 12.77°. With β = 30° and 12.7°, the source's M₂ works out to 2.358, printed as 2.36. ORBIX works from deflection to shock angle, so given 12.7° it returns β = 29.93° and M₂ = 2.371. With the exact pair (β = 30°, 12.77°) M₂ is 2.367. M₂ is sensitive to β minus the deflection, so the chart-read deflection alone accounts for the gap.

Two-stage rocket delta-v

Function

calculateRocketEquation

Inputs

• Stage 1: initial mass 165,000 kg, final mass 45,000 kg, specific impulse 260 s
• Stage 2: initial mass 36,000 kg, final mass 6,000 kg, specific impulse 320 s
• Standard gravity 9.80665 m/s²

Source

Example Problems, Robert A. Braeunig, Rocket and Space Technology. Problem 1.12, two-stage rocket.

• The total is the sum of the two ORBIX stage results; ORBIX's rocket equation calculator handles one stage at a time.

Hohmann transfer, low Earth orbit to geosynchronous orbit

Function

calculateHohmannTransfer

Inputs

• Initial circular orbit radius 6,578,140 m (200 km altitude)
• Final circular orbit radius 42,164,170 m (geosynchronous)
• GM = 3.986005 × 10¹⁴ m³/s²

Source

Example Problems, Robert A. Braeunig, Rocket and Space Technology. Problem 4.19, Hohmann transfer from a 200 km parking orbit to geosynchronous altitude.

• The source subtracts speeds it has already rounded to whole meters per second (3,075 − 1,597 m/s for the second burn), then adds the rounded burns. ORBIX keeps full precision, so its second burn and total differ from the printed figures by about 1 m/s.

Orbital plane change of 8°

Function

calculateOrbitalPlaneChange

Inputs

• Orbital velocity 7,558 m/s (600 km circular orbit)
• Inclination change 8° (28° to 20°)

Source

Example Problems, Robert A. Braeunig, Rocket and Space Technology. Problem 4.21, simple plane change.

• The orbital velocity is the source's own rounded value, entered as printed.

Escape velocity at 200 km altitude

Function

calculateEscapeVelocity

Inputs

• Orbital radius 6,578,140 m (200 km altitude)
• GM = 3.986005 × 10¹⁴ m³/s²

Source

Example Problems, Robert A. Braeunig, Rocket and Space Technology. Problem 4.25, escape velocity from a 200 km orbit.

Orbital speed from the vis-viva equation

Function

calculateVisViva

Inputs

• Circular orbit: radius and semi-major axis 6,578,140 m
• Ellipse: perigee radius 6,628,140 m, apogee radius 6,878,140 m, semi-major axis 6,753,140 m
• GM = 3.986005 × 10¹⁴ m³/s²

Source

Example Problems, Robert A. Braeunig, Rocket and Space Technology. Problem 4.1 (circular orbit at 200 km) and Problem 4.4 (250 km by 500 km ellipse).

• The source uses the perigee and apogee velocity equations; the semi-major axis given to ORBIX is the mean of the two radii the source states.

Position on an elliptical orbit from Kepler's equation

Function

calculateKeplerPosition

Inputs

• Semi-major axis 7,500,000 m, eccentricity 0.1
• Starting true anomaly 90°, time elapsed 1,200 s (20 minutes)
• GM = 3.986005 × 10¹⁴ m³/s²

Source

Example Problems, Robert A. Braeunig, Rocket and Space Technology. Problem 4.14, true anomaly 20 minutes later (orbit from problem 4.13), method 2.

• The source rounds the mean motion to 0.00097202 rad/s and the starting mean anomaly to 1.37113 rad before adding them, so its M is 2.537554 rad. ORBIX keeps full precision and gets 2.537559 rad, which puts M and E just outside the printed rounding. Solving Kepler's equation from the source's own M gives E = 2.58996 rad, the printed value. The true anomaly agrees within rounding either way.