All,
A while back I wanted to model a solar sail so I used the special
relativy equation and integrated over a distance to get the
approximate velocity vs time. Basicly, I did a force ballance to
calculated the accelleration and then integrated the accelleration to
get the velocity and the distance (two integrations). I think this
is how I did it.

p = gamma*m*u

dp/dt = d gamma/dt * (m*U) + d(m*u)/dt * gamma
dpt/dt = d gamma
d gamma/dt = (u/c^2)*gamma^3* du/dt
after some manipulation (I love when they do this in books )
dp/dt = ma*[gamma^3 * (u/c^2) + gamma]

I obtained a final equation for the accelleration:

a = (dEphotons/dt)/ m*c[gamma^3 * (u/c)^2 + gamma]

m = mass
u = accelleration
c = speed of light
E = energy of the light
p = momentum
(Note: all of these values are in the stationary reference frame)

I integrated (numerically) this eqation with respect to time to get
the velocity and again to get the distance.

Now, I am an engineer and not a physicist so I used Euclidean space to
integrate over - I'm not concerned about small changes in cordinates
due to small variations in the the coordinated system due to gravity.

Does my rational have any inherent flaws? The results seemed
reasonable.

Eric

(Eric Malroy) wrote in message ...
All,
A while back I wanted to model a solar sail so I used the special
relativy equation and integrated over a distance to get the
approximate velocity vs time. Basicly, I did a force ballance to
calculated the accelleration and then integrated the accelleration to
get the velocity and the distance (two integrations). I think this
is how I did it.

p = gamma*m*u

dp/dt = d gamma/dt * (m*U) + d(m*u)/dt * gamma
dpt/dt = d gamma
d gamma/dt = (u/c^2)*gamma^3* du/dt
after some manipulation (I love when they do this in books )
dp/dt = ma*[gamma^3 * (u/c^2) + gamma]

I obtained a final equation for the accelleration:

a = (dEphotons/dt)/ m*c[gamma^3 * (u/c)^2 + gamma]

m = mass
u = accelleration
c = speed of light
E = energy of the light
p = momentum
(Note: all of these values are in the stationary reference frame)

I integrated (numerically) this eqation with respect to time to get
the velocity and again to get the distance.

Now, I am an engineer and not a physicist so I used Euclidean space to
integrate over - I'm not concerned about small changes in cordinates
due to small variations in the the coordinated system due to gravity.

Does my rational have any inherent flaws? The results seemed
reasonable.

Eric

Yes. p = gamma*m*u is wrong, if you are taking p to be momentum and u
to be acceleration. It really should be dp/dt = gamma*m*u, or F =
gamma*m*u.

(...Starblade Riven Darksquall...)

Eric Malroy wrote:

All,
A while back I wanted to model a solar sail so I used the special
relativy equation and integrated over a distance to get the
approximate velocity vs time. Basicly, I did a force ballance to
calculated the accelleration and then integrated the accelleration to
get the velocity and the distance (two integrations). I think this
is how I did it.

p = gamma*m*u

dp/dt = d gamma/dt * (m*U) + d(m*u)/dt * gamma
dpt/dt = d gamma
d gamma/dt = (u/c^2)*gamma^3* du/dt
after some manipulation (I love when they do this in books )
dp/dt = ma*[gamma^3 * (u/c^2) + gamma]

I obtained a final equation for the accelleration:

a = (dEphotons/dt)/ m*c[gamma^3 * (u/c)^2 + gamma]

m = mass
u = accelleration
c = speed of light
E = energy of the light
p = momentum
(Note: all of these values are in the stationary reference frame)

I integrated (numerically) this eqation with respect to time to get
the velocity and again to get the distance.

Now, I am an engineer and not a physicist so I used Euclidean space to
integrate over - I'm not concerned about small changes in cordinates
due to small variations in the the coordinated system due to gravity.

Does my rational have any inherent flaws? The results seemed
reasonable.

Eric

I haven't checked the calculations, but I have number
of questions.

First, what is the probability a solar sail will reach
relativistic speeds? I would argue it's nil. Hence, you
can take gamma to be 1.

Second, how do you intend on determining dEphotons/dt?
Precisely how does the energy of the photon change with
time?

Third, do you understand the concept of sailing? If you do,
why is the calculation independent of the surface area of
the sail?

Eric Malroy wrote:
[snip]
p = gamma*m*u

dp/dt = d gamma/dt * (m*U) + d(m*u)/dt * gamma
dpt/dt = d gamma
d gamma/dt = (u/c^2)*gamma^3* du/dt
after some manipulation (I love when they do this in books )
dp/dt = ma*[gamma^3 * (u/c^2) + gamma]

I obtained a final equation for the accelleration:

a = (dEphotons/dt)/ m*c[gamma^3 * (u/c)^2 + gamma]
[snip]
Does my rational have any inherent flaws? The results seemed
reasonable.

Actually, there are two major flaws, namely,

(1) you applied special relativity to the
sail
(2) your calculation violates the conservation
of momemtum


Let

Flux = solar radiation flux in Watts/meter^2
E = energy of the photons in Joules
t = time in seconds
A = area in meters^2 of the sales
c = speed of light in meters/sec.
m = mass of payload and sail in kilograms

Flux = E/(t*A)

hence

E = Flux*t*A = pc

or

p = Flux*t*A/c

Assuming you're bouncing light off a mirror and
perpendicular to the mirror, then the momentum
change is twice the incident momentum, and

dp/dt = 2*Flux*A/c

ma = 2*Flux*A/c

or

a = 2*Flux*A/(m*c)

At the earth the solar radiation flux is

Flux = 1.4 x 10^3 Watts/meter^2.
Let

r = distance from the sail to center of the sun
r_earth = distance from center of sun to center of sail


Then

Flux(r) = Flux*r_earth^2/r^2

and

a = 2*Flux(r)*A/(m*c)