2d fixed dv transfer

7/27/2019 2D Fixed DV Transfer

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Planar Circle to Circle Transfer with Two Fixed Impulse V BurnsConsider the situation in which a spacecraft is in a low circular orbit and

must transfer to a higher circular orbit using a two burn transfer. Assume thatthe propulsion units are solid rockets and that the amount ofv available for

each burn is in excess of that needed for the Hohmann transfer between the twoorbits. Let the initial orbit radius be r1

and the target orbit radius be r2. The

velocities in the two circular orbits are Vc1 and Vc 2 , respectively, where

Vc1 =r1

, and Vc 2 =r2

.

Reference Hohmann Transfer:

It is advantageous in our analysis to know the details of the Hohmann transferbetween the two orbits. The semi-major axis of the Hohmann transfer betweenthe two orbits is given by the equation aH = r1 + r2( )/2 . The velocities in the

Hohmann transfer orbit at r1

and at r2

are given by Vt1H =2

r1

1

aH

and

Vt2H =2

r2

1

aH

, respectively. The individual V maneuvers for the

Hohmann transfer are V1H =Vt1H Vc1, and V2H =Vc2 Vt2 H . The total velocity

change required for the Hohmann transver, VHtotal , is given byVHtotal = V1H + V2 H.

Knowing the Hohmann values for V1H and V2H , we have a basis for

investigating the range of values of excess V over which a transfer between thetwo orbits is possible.

Planar Circle to Circle Transfers with Excess v and Fixed Burn Magnitudes

Let us assume that the amounts ofv available for the first and secondburns are v

1 avail= f

1v

1H and v2 avail = f2v2H , where f1> 1, and f2 > 1 . Inpractice, the factors f

1and f

2will usually be no larger than 105% of the

corresponding Hohmann velocity impulse value. Anything larger would bevery uneconomical. It should be obvious that f

1and f

2cannot be smaller than

1.0. For the current analysis, we will assume that the transfer and the v s take

place in the plane of the two orbits.

The eccentricity and semi-major axis of the transfer orbit will not be thesame as for the Hohmann transfer. Let us denote these variables by aTran andeTran . We will present the equations and will then discuss how they might besolved.


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In terms of the semi-major axis of the non-Hohmann transfer orbit, themagnitudes of the velocity in this orbit at r

1and r

2, respectively, are given by

vTran1 =2

r1

1

aTran

and vTran2 =

2

r21

1

aTran

.

Note that these relations give only the magnitudes of these velocities and nottheir directions. The angles between the velocity vectors and the localcircumferential direction (the flight path angles), denoted by the symbol withappropriate subscripts, must be known before we can find the directions of thevelocities. The angles

1and

2can be written in terms of the radii, the

velocities, and semi-latus rectum of the transfer orbit, pTran . The equations are

cos1

( )=pTran

r1vTran1( )

and cos2

( )=pTran

r2vTran2( )

.

The angles 1 and 2 are also related, through the law of cosines, to the velocitiesinvolved in the burns at the ends of the transfer trajectory. Specifically, theequations are

v1avail

2 = vTran12 + v c1

2 2vTran1vc1 cos 1( ) and v2avail2 = vTran2

2 + vc 22 2vTran2vc2 cos 2( ) .

We have eight nonlinear equations in eight unknowns. We can guess avariable, perhaps aTran , to begin an iterative solution to the problem.

After the equations above have been solved, the eccentricity, the perigeeradius, and the apogee radius of the transfer orbit can be determined

from the equations pTran = aTran 1 eTran2

( ) , rPTran = aTran 1 eTran( ), andrATran = aTran 1+ eTran( ), respectively.


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Algorithm Summary

Hohmann Equations

vc1 = r1

vc2 = r2

aH = r1 + r2( )/2

vt1H =2

r1

1

aH

vt2H =

2

r2

1

aH

v1H = v t1H vc1 v2H = vc2 vt2H

vHtotal = v1H + v2 H.

For the Hohmann transfer, we have 8 equations involving 11 parameters.The three known parameters are , r

1, and r

2. The other 8 parameters, vc1 ,

vc2 , aH , vt1H , vt2H , v1H , v2H , and vHtotal , can be determined from the 8equations. We will now assume that the Hohmann variables are now knownand will use them as reference values.

Fixed v Analysis: For the analysis of transfers with fixed v , it isconvenient to specify the two propulsion multipliers, f

1and f

2. There are 8

equations which govern the transfer . These are

v1 avail

= f1v

1H v2 avail = f2v2H

vTran1 =2

r1

1

aTran

vTran2 =

2

r21

1

aTran

cos1

( )=pTran

r1vTran1( )

cos2

( )=pTran

r2vTran2( )

v1avail

2 = vTran12 + v c1

2 2vTran1vc1 cos 1( ) v2avail2 = vTran2

2 + vc 22 2vTran2vc2 cos 2( ) .

In addition to the known quantities , r1, r

2, f

1and f

2, and the derived

quantities v1H , and v2H , these eight equations involve the eight unknowns

v1 avail

, v2 avail

, pTran , aTran , vTran1 , vTran2 , 1 , and 2 .


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Solution: We have 8 nonlinear equations in 8 unknowns. It is mostconvenient to use an interative procedure to solve the equations. We canguess one of the unknowns and use this value to start the solution of theentire set of equations. A very logical variable to guess is aTran , the semi-major axis of the transfer orbit. Note that it might be necessary to guessseveral starting values for a

Tran

in order to get a starting value that will resultin convergence. A first consideration might be that or guess for aTran belarger than the value found for the Hohmann transfer, and that aTran not betoo large.

Auxiliary Equations : These equations can be used to determine theeccentricity, perigee radius, and apogee radius of the transfer orbit after thecharacteristics of the transfer orbit are determined.

pTran = aTran 1 eTran2( )

rPTran = aTran 1 eTran( )

rATran = aTran 1+ eTran( )


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TK Solver Plus Model

TK Solver Rule Sheet

" Hohmann Transfer Reference Information (8 Equations in 8 unknowns)

vc1 = sqrt(mu/r1)vc2 = sqrt(mu/r2)aH = (r1 + r2)/2vt1H = sqrt(mu*(2/r1 - 1/aH))vt2H = sqrt(mu*(2/r2 - 1/aH))dv1H = vt1H - vc1dv2H = vc2 - vt2HdvHtot = dv1H + dv2H

" Fixed DV Transfer Calculations (8 Equations in 8 unknowns)

dv1avail = f1*dv1Hdv2avail = f2*dv2HvTran1 = sqrt(mu*(2/r1 - 1/aTran))

vTran2 = sqrt(mu*(2/r2 - 1/aTran))cos(gam1) = sqrt(mu*pTran)/(r1*vTran1)cos(gam2) = sqrt(mu*pTran)/(r2*vTran2)dv1avail^2 = vc1^2 + vTran1^2 - 2*vc1*vTran1*cos(gam1)dv2avail^2 = vc2^2 + vTran2^2 - 2*vc2*vTran2*cos(gam2)

" Auxiliary Equations

pTran = aTran*(1 - eTran^2) " Gives eccentricity of transferrpTran = aTran*(1 - eTran) " Gives perigee radius of transferraTran = aTran*(1 + eTran) " Gives apogee radius of transfer


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TK Solver Variable Sheet 1

St Input Name Output Unit Comment

39860 mu km^3/s^2 Gravitational constant6672.756 r1 km Low orbit radius6750 r2 km High orbit radius

aH 6711.378 km Hohmann semi-major axisvc1 2.4440834 km/s Low orbit circular velocityvc2 2.4300587 km/s High orbit circular velocityvt1H 2.4511058 km/s Perigee velocity in Hohmannvt2H 2.4230565 km/s Apogee velocity in Hohmanndv1H 7.022401 m/s Perigee V in Hohmanndv2H 7.0022238 m/s Apogee V in HohmanndvHtot 14.024625 m/s Total V in Hohmann

Fixed V Transfer Calculations

1.03 f1 Multiplier for Hohmann V11.04 f2 Multiplier for Hohmann V2aTran 6711.5421 km Semimajor axis of Transfer orbitpTran 6711.3151 km Parameter of Transfer orbitdv1avai 7.233073 m/s Available V1dv2avai 7.2823127 m/s Available V2vTran1 2.4511354 km/s Transfer Velocity at r1vTran2 2.4230864 km/s Transfer Velocity at r2gam1 .0376462 deg Flight Path Angle at r1gam2 .04963932 deg Flight Path Angle at r2

Auxiliary values

L eTran .00581625

L rpTran 6672.5061 kmL raTran 3645.0206 nmi


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TK Solver Variable Sheet 2

St Input Name Output Unit Comment

39860 mu km^3/s^2 Gravitational constant6672.756 r1 km Low orbit radius42162.632 r2 km High orbit radius

aH 24417.694 km Hohmann semi-major axisvc1 2.4440834 km/s Low orbit circular velocityvc2 .97231008 km/s High orbit circular velocityvt1H 3.2116452 km/s Perigee velocity in Hohmannvt2H .50828242 km/s Apogee velocity in Hohmanndv1H .76756176 km/s Perigee V in Hohmanndv2H .46402766 km/s Apogee V in HohmanndvHtot 1.2315894 km/s Total V in Hohmann

Fixed V Transfer Calculations

1.5 f1 Multiplier for Hohmann V12 f2 Multiplier for Hohmann V2

aTran 40159.722 km Semimajor axis of Transfer orbitpTran 11380.722 km Parameter of Transfer orbitdv1avai 1.1513426 km/s Available V1dv2avai .92805533 km/s Available V2vTran1 3.309766 km/s Transfer Velocity at r1vTran2 .94775369 km/s Transfer Velocity at r2gam1 15.33712 deg Flight Path Angle at r1gam2 57.791361 deg Flight Path Angle at r2

Auxiliary values

L eTran .84653029 Eccentricity of transfer orbitL rpTran 6163.3007 km Perigee radius of transfer orbitL raTran 40041.114 nmi Apogee radius of transfer orbit

2d fixed dv transfer

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