# Formula for solar radiation pressure

Gold Member

I'm looking for a formula that gives the amount of force in Newtons that is produced from the pressure of solar radiation.

$$F_R = C_R \frac{I}{c}S$$
where I is the radiation intensity, c is the speed of light, and S is the cross-sectional (sun-facing) area of the object being pushed, and Cr is the solar radiation coefficient which equals 1.0. (Does this value have units? I'm assuming it doesn't. And why clutter up the formula by multiplying it by 1?)

I don't quite know what I (intensity) is. Googling it, I find that its units are the Candela (CD). But that it is also a measure of energy which is expressed in Watts. (aka Joules / second, or Nm/s)

So:

$$N = \frac{Nm/s} {m/s} m^2$$

$$N = N m^2$$ which makes no sense. Unless the solar radiation coefficient has units of /m^2. Maybe that's why it's thrown in?

Furthermore, Intensity should change with distance. So I'm guessing that for Intensity I would use:

Luminosity of the Sun (3.9e26 watts) * (area of a sphere of 1 solar radii / area of a sphere of radius = distance), which would give me Intensity in Watts.

Ultimately, I'm trying to figure out how much solar radiation pushes against small objects at any given instant, causing their orbits to spiral outward.

Any thoughts? And any idea which is stronger: pressure from solar radiation or pressure from solar wind? Am I correct in assuming that pressure from solar radiation is basically constant, while pressure from solar wind varies with solar activity? Which is more responsible for creating the tails on comets?

Astronuc
Staff Emeritus
In MKS (SI) force should be in Newtons, while pressure is in Pa, which is N/m2.

Radiation intensity might be in photons/unit area or energy/unit area. I would expect the former.

Radiation pressure is the pressure exerted upon any surface exposed to electromagnetic radiation. If absorbed, the pressure is the energy flux density divided by the speed of light. If the radiation is totally reflected, the radiation pressure is doubled. For example, the radiation of the Sun at the Earth has an energy flux density of 1370 W/m2, so the radiation pressure is 4.6 μPa (absorbed)
from Wikipedia

I checked an old textbook, which gives

Force - 1/c ( L r2/d2 ), where L is solar luminosity, and L/(4$\pi$d2 is the total radiant energy from the sun, and r is the radius of the object (round particle).

So L/(4c$\pi$d2 would be pressure.

Remember, photon momentum p = E/c.

The solar wind (http://en.wikipedia.org/wiki/Solar_Wind) has the stronger effect on a comets tail - but one should do a calculation to show the pressure of solar wind.

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Gold Member

Would I be correct to assume that pressure = force if my cross-section were exactly 1 square meter?

SpaceTiger
Staff Emeritus
Gold Member
Astronuc said:
Force - 1/c ( L r2/d2 ), where L is solar luminosity, and L/(4$\pi$d2 is the total radiant energy from the sun, and r is the radius of the object (round particle).

That's strange, I would expect it to be:

$$force = \frac{flux*area}{c}$$

or just the energy absorbed per unit time divided by the speed of light (dp/dt). This would be with

$$flux= \frac{L}{4\pi d^2}$$

and

$$area = \pi r^2$$

$$force = \frac{L}{c}(\frac{r^2}{4d^2})$$

That is, with an extra factor of four in the denominator. The equation for pressure would be the same, however.

Would I be correct to assume that pressure = force if my cross-section were exactly 1 square meter?

They would have different units, but would be numerically the same in mks units.

lightgrav
Homework Helper
Intensity is POWER/Area ; the "C_R" in your formula
is probably supposed to be the reflection coefficient
(1 for fully absorbed, 2 for fully reflected ...).

At a distance "a" from the Sun, in astronomical units,
the radiation pressure is C_R (4.57E-6 N/m^2)/a^2
as Astronuc's quote implies, for fully absorbed light.

This pressure does NOT double if the surface reflects!
A sphere reflects directly backward (180 degrees)
only light that hits the *center* of the sphere;
by impact parameter .7r , the deflection is 90deg.
The outer half of the cross-sectional Area "S"
is less effective for momentum than if it absorbed!
So "C_R" is shape-dependent and non-trivial.

Wikipedia claims (Solar wind) that the Sun emits
800 kg/s in the solar-wind protons & electrons;
at a speed of 450 km/s , at 1 a.u., that's
(mv/t)/(4 pi r^2) = 1.27E-15 N/m^2 = wimpy.
(Is 800 kg/s way too small? 1 m^2 at Earth gets)
( 1.2 Million protons, 0.2 Million He++, 1.4 M e- )
( per second, on avg. We only get +1 mol/day? )

Astronuc
Staff Emeritus
Space Tiger - thanks for the correction. I believe I = L / (4$\pi$d2), where d is distance from sun.

Tony - Sorry for the crappy post. I was in a rush then and didn't get back in time to edit it.

lightgrav makes a good point - what I posted is the incident energy and assumes it is absorbed. Photons could be reflected or scattered (depending on surface characteristics, e.g. composition, and particle shape) and so the momentum transfer could be different.

I'd have to check the solar wind numbers. Here is what Hyperphysics has about Solar Wind

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BobG
Homework Helper
tony873004 said:

I'm looking for a formula that gives the amount of force in Newtons that is produced from the pressure of solar radiation.

$$F_R = C_R \frac{I}{c}S$$
where I is the radiation intensity, c is the speed of light, and S is the cross-sectional (sun-facing) area of the object being pushed, and Cr is the solar radiation coefficient which equals 1.0. (Does this value have units? I'm assuming it doesn't. And why clutter up the formula by multiplying it by 1?)

I don't quite know what I (intensity) is. Googling it, I find that its units are the Candela (CD). But that it is also a measure of energy which is expressed in Watts. (aka Joules / second, or Nm/s)

So:

$$N = \frac{Nm/s} {m/s} m^2$$

$$N = N m^2$$ which makes no sense. Unless the solar radiation coefficient has units of /m^2. Maybe that's why it's thrown in?

Furthermore, Intensity should change with distance. So I'm guessing that for Intensity I would use:

Luminosity of the Sun (3.9e26 watts) * (area of a sphere of 1 solar radii / area of a sphere of radius = distance), which would give me Intensity in Watts.

Ultimately, I'm trying to figure out how much solar radiation pushes against small objects at any given instant, causing their orbits to spiral outward.

Any thoughts? And any idea which is stronger: pressure from solar radiation or pressure from solar wind? Am I correct in assuming that pressure from solar radiation is basically constant, while pressure from solar wind varies with solar activity? Which is more responsible for creating the tails on comets?
Yes, solar pressure is nearly constant if you're talking about objects orbiting the Earth. The variation in the object's distance from the Sun is insignicant when compared to the distance between the Sun and Earth. But solar pressure wouldn't cause the objects to spiral outward - it would just make the orbits more elliptical (in about 200 years, some of the GPS satellites will reach the geostationary belt at apogee).

If you're talking about objects that orbit the Sun, the pressure in a nearly circular orbit could be considered nearly constant. That wouldn't apply to comets.

The solar radiation coefficient is an indication of how reflective the object is. A transparent object has a reflectivity of 0, a totally absorbant object has a reflectivity of 1, and a perfect mirror has a reflectivity of 2. A certain amount of momentum is transferred with each photon absorbed. A certain amount of momentum is transferred with each photon emitted. If the object is transparent, every photon absorbed on one side is emitted out the opposite side giving a net of zero. If perfectly reflective, every photon absorbed is emitted back out the same side, doubling the amount of momentum transferred.

Gold Member
Thanks, ST, Astronuc LG and Bob. Playing around with these formulas in a simulation, I can see that it's a very weak force. But if I create 1 kg objects with the radius of Earth, they get blown away quickly :rofl:

I could probably just google this, but how is aldebo and reflectivity related? Or are they the same?

Astronuc
Staff Emeritus