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[Python] Simulating rocket trajectory to ISS

  1. Oct 19, 2013 #1
    background information is that I have been working on code for a rocket launch to the ISS and I have gotten it close. The problem is when calculating the Force of drag, the problem occurs with it. As entering space, and for this "model" were going to say that temperature is 0 K in space or basically just out of the atmosphere. Obviously absolute 0 isn't possible, but perhaps that would be a simple solution to the problem. When entering space, T= 0 K, and there isn't any air, well not in the "model". So since there isn't any air then there wouldn't be any drag force. I did the calculation for my formula and at y= 44307.69 m, the rocket leaves the atmosphere. the equations I use for temperature is T=Tz - L*y where Tz is the Temperature at time 0, L is temperature lapse rate (assuming linear as y increases) of value 0.0065 K/m, and y is height in m.

    Next, is the pressure, and as you know as y increases pressure should also go down. Now as leaving the atmosphere there should not be any pressure. This formula is P= Pnot * (T/Tz)^((g*M)/(R*L)); where Pnot is pressure @ y=0 or 101 kPa, T=current temperature, Tz= Temp @ y=0 (288K), g=gravity (which uses another formula but I have that one working correctly), M= molar mass of air value of 0.02896 kg/Mole, R=8.314 J/K*M, and L is the temperature lapse rate again.

    Anyways this is a lot of information obviously, and my question is how to get the force of drag to stop at that y= 44307.69 m, and then keep calculating in the while loop for the computer program. also using the Euler method, in the python language.

    Trying to get it to go to the ISS and then come back to earth.

    Any help would be appreciated. Thanks.


    COPY OF THE CODE ATTACHED BELOW

    Code (Text):

    from __future__ import division, print_function
    from pylab import *
    from scipy.interpolate import interp1d

    dt= 0.01 #seconds, the timestep
    t= 0.0 #seconds, initial time

    yfinal= 355000.0 #m, =355km how far up the ISS is traveling
    y= 0.0 #m at ground level
    v= 0.0 #m/s initial speed
    pay= 5000.0 #kg
    me= 10000.0 #kg, empty mass with NO fuel or payload
    gas= 10000.0 #kg of rocket fuel, 10,000 is max fuel
    m= 20000.0 #kg   ###+ pay #total mass, empty rocket, fuel, and payload
    Em= 10000.0
    Ev= 7000.0 #m/s exhaust velocity
    Br= 100.0 #kg/s  burn rate of the fuel
    A= 10.0 #m**2   Reference or cross sectional area
    Cd= 0.5  #drag coefficient

    gs=9.8 #m/s**2  initial gravity at the earths surface
    Vex= 7000.0 #m/s exit velocity of the fuel being ejected out
    Ft= Br * Vex #force of thrust, acceleration upward
    a=0.0
    nogas = gas/Br  #the point at which the rocket runs out of gas
    Re= 6378100.0 #meters radius of the earth
    g= 0.0
    rho= 0.0
    R= 8.314 #Joules/Kelvin*Mole
    M= 0.02896 #kg/mole        
    T= 288.0
    Pz=101000.0  #Pascal  Pressure at time 0
    Tz= 288.0 #Kelvin   Tempature at time 0. Aprox. 14.85 Celsius
    L= 0.0065 #Kelvin/meter  "Tempature lapse rate"
    P= 101000.0 #Pressure in Pascal
    Fd= 0.0
    Fg= 196000
    #initial array values
    tArray = array([t])
    yArray = array([y])
    vArray = array([v])
    aArray= array([a])
    gArray= array([g])  
    mArray= array([m])
    FtArray= array([Ft])
    FdArray = array([Fd])
    FgArray= array([Fg])    
    TArray = array([T])  
    rhoArray= array([rho])
    PArray = array([P])
    RArray= array([R])
    MArray= array([M])
    #calculations for arrays
    #if y<=44307.69231:
       # break(T)
    while (y<=yfinal):# and (T>=0):# and (P>=10100): # and (nogas>=t):
       
        t= t + dt  #increments TIME
       
        gas= gas - Br*dt
        m = Em + gas
        Ft= Br * Vex

        T= Tz - (L*y)#*dt
       
        g=  (gs/((1+y/Re)**2))
        P= Pz*((abs(T/Tz))**((g*M)/(R*L)))
        rho= (P*M)/(R*T) #*dt
        if(T < 0):
            Fd= 0
       
        Fd=  .5*rho*A*Cd*v*abs(v)
       
       
        Fg= (m)*g
        a=(-Fg-Fd+Ft)/m
       
        y= y + v*dt #increments HEIGHT
             
        v= v + a*dt #increments VELOCITY

           
       
        yArray = append(yArray, y) #fills in height array
        vArray = append(vArray, v)#fills in velocity array
        tArray = append(tArray, t)#fills in time array
        aArray = append(aArray, a)
        gArray = append(gArray, g)
        mArray = append(mArray, m)
        FtArray= append(FtArray, Ft)
        FgArray= append(FgArray, Fg)
        FdArray= append(FdArray, Fd)
        TArray = append(TArray, T)
        rhoArray= append(rhoArray, rho)
        PArray = append(PArray, P)
        RArray = append(RArray, R)
        MArray= append(MArray, M)
        print("time", t, "secs", '\t', "height", y, "m", '\t', 'velocity', v, "m/s")
        print(aArray)
     
    Last edited by a moderator: Oct 20, 2013
  2. jcsd
  3. Oct 20, 2013 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Your approach with if is good:
    Code (Text):
    if(condition)
       Fd= 0
    else
       Fd= .5*rho*A*Cd*v*abs(v)
    There are some other issues, however:
    This is true close to the ground, maybe for the first 10km, but it is completely wrong afterwards. Your atmospheric model is not useful - above the height where your atmosphere ends, the space shuttle performed its re-entry maneuvers, and it was glowing red from the drag there.

    Your model seems to be one-dimensional, looking only at the height. That is completely unrealistic - the ISS is orbiting earth with a speed of ~7-8km/s, and most of the rocket acceleration is needed to get this speed. The height is not so crucial, it is just needed to get out of the atmosphere.
     
  4. Oct 20, 2013 #3

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    The ISS is *not* out of the atmosphere. Its orbit decays due to atmospheric drag. Here's a plot of the ISS's altitude from 1999 to 2009.

    640px-Internationale_Raumstation_Bahnhöhe_%28dumb_version%29.png

    The discontinuities are where the ISS did a reboost to raise its altitude. The slopes of the continuous segments vary quite a bit. Notice how the ISS maintained a rather high altitude during 2001-2002. The periods between reboosts are rather steep lines. Notice how the altitude is much lower in 2006-2007, and yet even with that significant decrease in altitude, the drag losses in 2006-2007 are much lower than they were in 2001-2002.

    I wouldn't expect you to model this, but the reason is the solar cycle. Solar max puffs up the upper atmosphere. Solar cycle 23 had a maximum in 2000 and another in 2002. The ISS altitude was raised during that solar max so as to reduce the need for reboosts. (That sharp drop in altitude in April-May 2000: That's because of STS-101, whose launch was delayed a number of times. NASA didn't want to do a reboost during that interval unless they absolutely had to.)

    What about the flip side, the period encompassing 2006 and 2007? Launches are considerably cheaper if the launch vehicle doesn't have to go so high. Solar cycle 23 ended with a prolonged minimum, and the Earth's upper atmosphere contracted in response. NASA and Russia took advantage of this extended lull by keeping the ISS altitude low during 2006 and 2007.

    While NASA and Russia do need to model this behavior, you don't. That takes a fairly sophisticated atmosphere model. Having an atmosphere model that has the atmosphere just end is not a great idea. Having an atmosphere model with a constant lapse rate until the atmosphere ends is a very bad idea. Going from sea level on up, temperature in the Earth's atmosphere drops with increasing altitude in the troposphere, then rises in the stratosphere, then drops again in the mesosphere, and rises once again in the thermosphere. A fairly simple model that exhibits these behaviors is the US Standard Atmosphere model. There's even a python model that incorporates the US Standard Atmosphere, up to 84 km altitude: https://pypi.python.org/pypi/AeroCalc.
     
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