Analytic Considerations for Lifting Ascent Launch Vehicle Trajectories

Martin Bayer

Motivation

Is There Another Way?

???

Coarse analytical solution vs. Complex numerical approachCoarse analytical solution vs. Complex numerical approach

Tsiolkovsky Revisited (I)

The thrust of a rocket equals the negative propellant mass flow times the exhaust velocity:

(1) esp cmgIdtdmF 0

Note: The mass flow must be taken as negative in order to obtain a positive thrust value, since from the perspective of the vehicle it constitutes a mass loss due to propellant consumption

According to Newton’s Second Law, the vehicle acceleration equals the thrust divided by the mass:

(2) mF

dtdva

Inserting (1) in (2) yields:

(3) mc

dtdm

dtdv e

Tsiolkovsky Revisited (II)

Multiplication with dt and subsequent integration under the assumption of a constant ec leads to:

(4) e ev

v

m

me m

dmcdv0 0

Note: The integration from a larger 0m to a smaller em implies that dm is negative, as outlined above

Solving the integrals finally leads to the well known Tsiolkowsy rocket equation:

(5) e

ee mm

cvvv 00 ln

An analogous approach can be taken for the performance determination of propulsion systems with variable specific impulse

Specific Impulses (Isp) of Various Propulsion Systems

Isp as a Linear Function of Flight Velocity

Derivation of a Propulsive Equation for Variable Isp (I)

Airbreathing engines generally have specific impulses that vary considerably with the flight velocity and typically decrease for higher velocities

Often the specific impulse can at least for certain segments be approximated with reasonable accuracy as a linear function of velocity:

(6) )( '0

vIII spspsp

Analogous to the derivation of the rocket equation, the acceleration can be written as:

(7) m

vIIgdtdm

dtdv spsp )( '

0 0

Multiplication with dt , separation of the variables and subsequent integration yields:

(8)

e ev

v

m

mspsp mdm

vIIgdv

0 00)( '

0

Derivation of a Propulsive Equation for Variable Isp (II)

Solving the integrals leads to:

(9) 00

'0

'0

'0

ln)(

)(ln1

0

0

mm

vIIg

vIIg

Ige

spsp

espsp

sp

Some rearrangement yields:

(10) '

0

0

0

00'

' spIge

spsp

espsp

mm

vIIvII

The final result is:

(11)

'0

0

00'0 1

spIge

sp

spe m

mv

I

Ivvv

For a value of 'spI approaching zero, this equation transforms into the Tsiolkowsy

equation

Derivation of a Propulsive Equation for Variable Isp (III)

The maximum achievable ev (for 0m

me approaching zero) is:

(12) 'max0

sp

spe I

Iv

This boundary value is the velocity, for which the specific impulse becomes zero

Apart from fundamental evaluations, equation (11) enables quantitative analyses, if an effective specific impulse, which includes all loss terms, is known as a function of the flight velocity:

(13)

F

gmFDII speffsp

sin1

This requires however advance knowledge of the main trajectory parameters; if these are not known yet, the influence of drag and gravity losses has to be included explicitly

Typical Ascent Trajectory Constraints for HTHL Vehicles

Ascent Trajectory of Airbreathing HTHL TSTO Booster

Ascent Trajectory of Airbreathing HTHL SSTO

Ascent Trajectory of Airbreathing VTHL SSTO

Modeling of Ascent Trajectories

It is assumed, that for ascent flight segments of vehicles with horizontal takeoff a relationship between flight velocity and air density of the following form is valid:

(14) .constlvk

Some typical values for k are:

:1k Constant air mass flow (subsonic flight)

:0k Constant altitude (transonic flight)

:2k Constant dynamic pressure (supersonic flight)

:8.2k Constant total pressure/total temperature ratio aft of airbreather intake

:5.4...9.3k Airbreathing engine pressure and temperature limit (hypersonic flight)

:3.6...0.6k Constant aerothermodynamic stagnation point heat flux

Transonic dives like the SR-71 ‘dipsy doodle’ maneuver can also be described by a negative value of k

Ascent Gravity Loss Relationships

Determination of Ascent Gravity Losses (I)

The momentary gravity loss occuring during ascent can be expressed as:

(15) dtdh

vg

dtdh

dsdtg

dsdhgg sin

Equating the hydrostatic equation:

(16) dhgdp

With the differential formulation of the polytropic equation of state:

(17)

dpndp

Leads after division with dt to:

(18) dtg

dpndtdh

2

Determination of Ascent Gravity Losses (II)

From equation (14) follows:

(19) dtdv

vkl

dtd

k

1

Inserting equation (19) into equation (18) and using equation (14) leads to:

(20) dtdv

vgpkn

dtdh

Using the ideal gas equation:

(21) TRp

Finally leads to the expression:

(22) dtdv

vTRkng

2sin

Determination of Ascent Gravity Losses (III)

Using the Mach number relationship:

(23) 221Mv

TR

Yields the alternative formulation:

(24) dtdv

Mkng

2sin

The total gravity loss during a flight segment following the relationship defined in equation (14) is:

(25) dvv

TRkndtgve et

t

v

vg

0 02sin

Note: The integral gravity loss is independent of flight duration and acceleration for trajectory segments that follow the relationship defined in equation (14), since lower acceleration and associated longer flight duration lead to shallower flight path angles

Integration of Equation of Motion along Flight Path (I)

The simplified equation of motion along an ascent trajectory is:

(26) sin gmD

mF

dtdv

Using equation (22) leads to:

(27) m

DFdtdv

vTRkn

21

Inserting the relationship:

(28) mdmdt

With both dm and m once again being negative yields:

(29) mdmdv

DFTRknv

vm

2

2

Integration of Equation of Motion along Flight Path (II)

This can be rewritten as:

(30) mdmdv

DFTRknv

vgIF

sp

2

20

If all factors and variables on the left side of the equation are given for example either as constants or as linear functions of v , this equation can be integrated

Alternatively, m , F and D can also be expressed as the following functions:

(31) 0)()()( gvIvmvF sp

(32) 2

)()(

2 krefD vlAvc

vD

(33) kvlAvAA

vvm 10

0

)()()(

Equations (32) and (33) require however k to be an integer in order to lead to an integrable solution

Integration of Equation of Motion along Flight Path (III)

Solving the integrals:

(34)

ee m

m

v

v mdmdv

DFTRknv

vm

00

2

2

respectively the transformation obtained using equation (1):

(35)

ee m

m

v

v sp mdmdv

FDTRknv

vgIF

00

2

20

allows to determine the mass ratio em

m0 , and with that the propellant consumption,

which is required to achieve a given v

Integration of Equation of Motion along Flight Path (IV)

Integration of Equation of Motion along Flight Path (V)

A derivation of equation (29) can also be used to determine the theoretical optimum switching point in flight from one propulsion system to another, such as from airbreather to rocket:

(36)

DFTRknv

vmm

dvdm 2

2

As soon as the value of dvdm , which denotes the propellant mass increment necessary

for achieving a given velocity increment, for the airbreather exceeds the one for the rocket for the respective values of k , switching will lead to lowering the propellant flow required for vehicle acceleration and hence the propellant mass for the total mission

Analytically Recalculated Air Launch/HTHL Examples

Boundary Conditions and Results of Calculated Examples

Advantages and Limitations

Symbols and Abbreviations (I)

Symbols and Abbreviations (II)

Literature

Martin Bayer: Analytic performance considerations for lifting ascent trajectories of winged launch vehicles, Acta Astronautica 54, 2004, pp. 713 – 721