@astro.cal
let GM=Gn*Msun

!Mars orbit:
set a 1.523679  e .0934

!Earth circular orbit velocity
let vEarth=sqrt(GM/au)
? vEarth
    29784.72616569657    


!actual Mars orbit:
let rmax=a*(1+e)
let rmin=a*(1-e)
let voMars=sqrt(GM/(a*au*(1-e^2)))
let vmin=(1-e)*voMars
* ? vmin
    21971.77754130247    
let vmax=(1+e)*voMars
* ? vmax
    26498.94282336215    

!join at aphelion:
let a2=(1+rmax)/2
let e2=1-1/a2          !because: 1=a2*(1-e2)
!    .2498098132654847    
let vo2=sqrt(GM/(a2*au*(1-e2^2)))
let v2max=(1+e2)*vo2
? v2max
    33297.80278960929    
? v2max-vEarth
    3513.076623912722    
! above delta V to start Hohmann transfer

!find spacecraft velocity when it meets Mars:
let v2min=(1-e2)*vo2
? v2min
    19986.78889175906    
? vmin
    21971.77754130247    
? vmin-v2min
   1984.988649543408    
! above to join Mars orbit from Hohmann
! total delta V=5498
! trip time in years:
? .5*sqrt(a2^3)
    .7695076400738805    !years

!join perihelion:

let a2=(1+rmin)/2
let e2=1-1/a2
!    .1601463866427006    
let vo2=sqrt(GM/(a2*au*(1-e2^2)))
let v2max=(1+e2)*vo2
? v2max
    32081.15588281668    
? v2max-vEarth
    2296.429717120114    
! above delta V to start hohmann transfer

let v2min=(1-e2)*vo2
? v2min
    23224.20256536158    
? vmax-v2min
    3274.740258000569    
! above to join Mars orbit from Hohmann
! total delta V =  5571.
! time of trip in years:
? .5*sqrt(a2^3)
    .6496278150325970    
*
! about 100 m/s better to go to aphelion, but takes longer

!Remark: the total delta V available on the OMS Space Suttle was about 300 m/s

Halley's comet:
set ecc=.96714 a=17.834
!r=a(1-e^2)/(1+e cos(phi)), if cos(phi)=-e then r=a
let phi=acos(-ecc)
let u=2*atan(sqrt((1-ecc)/(1+ecc))*tan(phi/2))
!Remark: expect u=90 deg as r=a(1-e cos(u))
!t is the time since perihelion to when it achieves r=a
!w t=2 pi(t/T) = u-e sin(u)=pi/2-e
!(t/T)=1/4-e/(2 pi)
? 1/4-ecc/(2*pi)
    .9607488833810435E-01
? (1-2*res)
    .8078502233237913    
!the fraction closer than a is 2t/T, so seek 1-2t/T

Oumuamua:
when far: E=KE=1/2 m v^2=GMm/(2 a)
Vinfinity=sqrt(GM/a)
set a=1.2798 ecc=1.19951
? sqrt(GM/(a*au))
    26328.28428699930    
let wt=sqrt(GM/(a*au)^3)*100*year
set xl=7 xr=5
root ecc*sinh(x)-x-wt
 Root between    6.59924316406250       and    6.59930419921875     
? a*(ecc*cosh(6.59924)-1)
    562.5269938127005    

Mercury:
The orbit plane is fixed in space; the Earth rotates
in a orbit period of 88.91 minutes, the earth rotates:
? 360*88.9/(24*60)
   22.2250
the nubmers I provided suggest: 35-10=25...close enough
launch parallel to the equator to get full benefit of spin velocity
? 2*pi*Rearth*cos(32.5*pi/180)/(24*3600)
    391.1912761970551    
!nearly 900 mpg
let a=(2*Rearth+156e3+285e3)/2
!6598637
! R+156=a*(1-e)
!e=1-(R+156)/a
? 1-(Rearth+156e3)/a
   .9774745905859028E-02
   
rendezvous Space Station
recall: with phasing we catch up by stepping on the brakes, producing
an orbit with a shorter period.  One orbit later we'd find ourselves
back at the altitude of ISS, but a bit before the ISS could complete its full orbit.

!period of space station orbit:

let T=2*pi/sqrt(Gn*Mearth/(400e3+Rearth)^3)
!we seek an orbit that is 10 minutes less
let T=T-10*60
let a=(Gn*Mearth/(2*pi/T)^2)^(1/3)
!as this a is smaller that our current R, we will
!be at apogee after we fire rockets to slow down:
!a*(1+e)=Rearth+400e3
let ecc=(Rearth+400e3)/a-1
! circular orbit velocity: sqrt(GM/R)
! apogee velocity: (1-e)*v0
let vSS=sqrt(Gn*Mearth/(Rearth+400e3))
let v0=sqrt(Gn*Mearth/(a*(1-ecc^2)))
? VSS-(1-ecc)*v0
   309.9436491056822
!by itself this is barely within the space shuttles abilities
!and of course we would need to do it again when, one orbit later,
!we re-join the Space Station's orbit...solution: makeup 5 minutes
!per orbit and orbit twice, etc

!This orbit in fact does not work as the perigee is inside the Earth:
? a*(1-ecc)-Rearth
   -594876.7953126021    
!so a better strategy would be, say, to catch up 1 minute per orbit
let T=T+9*60                      !remember we went back 10 minutes
let a=(Gn*Mearth/(2*pi/T)^2)^(1/3)
let ecc=(Rearth+400e3)/a-1
let v0=sqrt(Gn*Mearth/(a*(1-ecc^2)))
? VSS-(1-ecc)*v0
    27.91843817025710    
? a*(1-ecc)-Rearth
    302183.9395177895