Library / Aerospace / AER-28
Aerospace
Steady climb angle from excess thrust
Flight-path angle of a steady climb with thrust along the flight path, constant true airspeed and no tangential acceleration.
- Ref
- AER-28
- Rev
- v1
- Field
- Aerospace
- Result
- γc [°]
In a steady climb along a straight path, forces along the path balance: T − D = W sin γc. The thrust in excess of drag therefore sets the climb angle, γc = asin[(T − D)/W]. A positive angle is a climb. The excess thrust must be less than the weight, as a vertical climb at constant speed needs T = D + W. The rate of climb is V sin γc.
Flight-path angle of a steady climb with thrust along the flight path, constant true airspeed and no tangential acceleration. D is the drag at the actual climb condition. It needs |T − D| ≤ W; a positive angle is a climb and a negative one a descent. The page gives degrees.
Calculator
- Excess thrust, T − D1,200 N
- Climb gradient, tan γc10.05 %
- Excess thrust to weight, (T − D)/W0.1
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The aircraft climbs at 5.739°, a gradient of 10.05 %, or 5 m/s of climb rate at 50 m/s true airspeed.
- Each extra 1000 N of thrust would add 4.82° to the climb angle.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°) | Local step | Change in result |
|---|---|---|---|---|---|---|
| T Thrust | 0 to 10,000 N | −9.594 | 41.81 | 51.4 | +10% | +1.54 ° |
| D Drag | 0 to 10,000 N | −34.52 | 15.47 | 49.98 | +10% | −0.959 ° |
| W Weight | 5,000 to 100,000 N | 0.6876 | 13.89 | 13.2 | +10% | −0.523 ° |
Sweep of Thrust
| Thrust (N) | Climb angle (°) |
|---|---|
| 0 | −9.594 |
| 1,250 | −3.583 |
| 2,500 | 2.388 |
| 3,750 | 8.386 |
| 5,000 | 14.48 |
| 6,250 | 20.74 |
| 7,500 | 27.28 |
| 8,750 | 34.23 |
| 10,000 | 41.81 |
Sweep one variable
Variables and how each one contributes
Climb angle
What the equation gives you. Everything else feeds it.
Thrust
The thrust along the flight path, in N. Raises the climb angle as the excess thrust. Raising it by 10 % (to 3,520) moves the result by +26.8 %, from 5.739 to 7.277 °.
Drag
The drag at the actual climb condition, in N. Lowers the climb angle. Raising it by 10 % (to 2,200) moves the result by −16.7 %, from 5.739 to 4.78 °.
Weight
The aircraft weight, in N. A heavier aircraft climbs at a lower angle for the same excess thrust. Raising it by 10 % (to 13,200) moves the result by −9.12 %, from 5.739 to 5.216 °.
Worked example with the values above
- Excess thrust: T − D = 3,200 − 2,000 = 1,200 N.
- Ratio to weight: (T − D)/W = 0.1.
- γc = asin(0.1) = 5.73917°.
Assumptions and limits
- Steady flight along a straight path at constant speed, with L/D and specific fuel consumption held constant.
- Still air, and weights taken at the start and end of the cruise segment.
Common mistakes
- Using mass-based fuel consumption directly in a weight-flow equation, which needs g₀.
- Using range formulas for a changing L/D without averaging.
- Treating a still-air glide ratio as the ratio over the ground in a headwind or tailwind.
Statistics and uncertainty
Climb, glide and Breguet results are quasi-steady estimates. Range from the Breguet equation is typically good to 5 % to 10 % for long cruise legs, and worse for short missions where climb and descent are large, or in wind, which the formulas ignore.
Where it is used
- Planning range, fuel, and glide-back or engine-out decisions.
- Comparing the efficiency of aircraft designs.
Maths you need
- Lift-to-drag ratio: L/D, the glide ratio
- Specific fuel consumption: fuel flow per unit thrust or power
- Weight fraction: the ratio of final to initial weight