Equation of a conic section:
relating energy to geometry
relating energy to velocity and position:
Velocity found from geometry and position: using equatings (1a) (1b), we get
Velocity can be found knowing Energy and position: from (1b) we solve for
Relating angular momentum to geometry:
Which is valid for any orbit.
from geometry (1)
from geometry (2)
from geometry (3)
by geometry deﬁnition.
can also be found from physics as
so, given the angular momentum and the mechanical energy, we can ﬁnd
These can be derived from (1) and (2)
where where is the average angular velocity of the probe, and is the time we wish to ﬁnd the angle at which the probe is located. hence is the distance traveled (in radian angles) by the probe in the eccentric model. But (you see this have units as radians per unit time). so (4) is
Solve using Newton method. use
eﬀective oﬃcially is the total impulse per total mass expelled to generate this impulse.
i.e. eﬀective =
in units: so it has the units of speed!
To convert it to , notice that eﬀective is the same as
In units then i.e. seconds, which is what we use
The above is the average speciﬁc impulse.
To ﬁnd the instantaneous speciﬁc impulse
From the net
from the net
impulse = change in momentum
Speciﬁc impulse is deﬁned as , in any system of units you care to name.
One deﬁnition I saw of speciﬁc impulse is
is the momentum gained per unit weight of propellant used during this momentum change. The momentum gained results from the loss of the from the total mass.
Hence, assume we consume propellant mass, then
Final velocity =
payload ratio (Prussing def) this is class deﬁnition also
payload ratio (Wisel def)
Notice that can also be written as
Methods: Given masses, if asked to ﬁnd do
is the rate at which the propellant is consumed assumed constant. Also called engine mass ﬂow rate. in other words, it is the rate at which MASS is exiting the nozzle of the rocket engine. If we multiply this quantity by how fast this rate of mass is changing (i.e. ), we get an acceleration time mass, hence force, which is the engine thrust. This causes the rocket to go up.
For space shuttle, m/sec. kg/sec, hence engine generates a thrust (force) of Newton
Impulse = thrust * time thrust applied.
i.e. total impulse
So, Thrust is the rate of change of impulse. The faster the impulse changes, the larger the thrust.
so, if I can ﬁnd given , then I can ﬁnd the time it takes to reach burn out for some given
In other words, time it takes to burnout
Speciﬁc Impulse (ISP), or how much thrust you get from each pound of fuel is very important.
Generally, DELTA V = LN(MASS RATIO)* ISP*G That means that the Speciﬁc Impulse (ISP), or how much thrust you get from each pound of fuel is very important, and the Mass Ratio, or what percentage of your vehicle is propellant is less important. For each stage you can set an ISP to determine how much propellant you will use for the thrust you need. Then set a mass ratio to determine how much metal you wish to wrap around the propellant. The rule of Thumb is that higher stages get the better ISPs and Mass Ratios because they are smaller and they include the cost of the boosters. Boosters are the work horses, low ISP because of atmospheric back pressure, and heavy, but you can buy them by the pound cheap. Also, the ISP is set mostly by the propellant choice, the Mass Ratio on the other hand is determined by how much money you wish to spend on light weight materials. The lightest know material for construction is Unobtainium.
Solve for :