Editing Solar sail (section)

==Physical principles for reflective sails==

===Solar radiation pressure===
{{Main|Radiation pressure}}
The force imparted to a solar sail arises from the momentum of photons. The momentum of a [[photon]] or an entire flux is given by [[Energy–momentum relation|Einstein's relation]]:<ref>{{cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/relmom.html |title=Relativistic Momentum |publisher=Hyperphysics.phy-astr.gsu.edu |access-date=2015-02-02}}</ref><ref name="wright-appendixA">Wright, Appendix A</ref>

:<math>p = E/c</math>

where p is the momentum, E is the energy (of the photon or flux), and c is the [[speed of light]]. Specifically, the momentum of a photon depends on its wavelength {{math|''p'' {{=}} ''h/λ''}}

Solar radiation pressure can be related to the irradiance ([[solar constant]]) value of 1361&nbsp;W/m<sup>2</sup> at 1&nbsp;[[Astronomical unit|AU]] (Earth-Sun distance), as revised in 2011:<ref>{{cite journal|author1=Kopp, G. |author2=Lean, J. L. |author2-link=Judith Lean |title=A new, lower value of total solar irradiance: Evidence and climate significance|journal=Geophysical Research Letters |volume=38 |issue=1 |year=2011|pages=n/a |doi=10.1029/2010GL045777|bibcode=2011GeoRL..38.1706K |doi-access=free }}</ref><!--DO NOT CHANGE FROM SIG FIGURES: LEADS TO MISLEADING VALUES & INCONSISTENCIES-->

*perfect absorbance: F = 4.54&nbsp;μN per square metre (4.54&nbsp;μ[[Pascal (unit)|Pa]]) in the direction of the incident beam (a [[Inelastic collision|perfectly inelastic collision]])
*perfect reflectance: F = 9.08&nbsp;μN per square metre (9.08&nbsp;μPa) in the direction normal to surface (an [[elastic collision]])

An ideal sail is flat and has 100% [[specular reflection]]. An actual sail will have an overall efficiency of about 90%, about 8.17&nbsp;μN/m<sup>2</sup>,<ref name="wright-appendixA"/> due to curvature (billow), wrinkles, absorbance, re-radiation from front and back, non-specular effects, and other factors.
[[File:Sail-Force1.gif|thumb|Force on a sail results from reflecting the photon flux]]
The force on a sail and the actual acceleration of the craft vary by the inverse square of distance from the Sun (unless extremely close to the Sun<ref>McInnes, C. R. and Brown, J. C. (1989) ''Solar Sail Dynamics with an Extended Source of Radiation Pressure'', [[International Astronautical Federation]], IAF-89-350, October.</ref>), and by the square of the cosine of the angle between the sail force vector and the radial from the Sun, so

:<math>F = F_0 \cos^2(\theta) / R^2</math> (for an ideal sail)

where R is distance from the Sun in AU. An actual square sail can be modelled as:

:<math>F = F_0 (0.349 + 0.662 \cos( 2\theta) - 0.011 \cos( 4\theta)) / R^2</math>

Note that the force and acceleration approach zero generally around θ = 60° rather than 90° as one might expect with an ideal sail.<ref>Wright, Appendix B.</ref>

If some of the energy is absorbed, the absorbed energy will heat the sail, which re-radiates that energy from the front and rear surfaces, depending on the [[emissivity]] of those two surfaces.

[[Solar wind]], the flux of charged particles blown out from the Sun, exerts a nominal dynamic pressure of about 3 to 4 [[Pascal (unit)|nPa]], three orders of magnitude less than solar radiation pressure on a reflective sail.<ref>{{Cite web|url=http://www.swpc.noaa.gov/SWN/index.html|archive-url=https://web.archive.org/web/20141127142813/http://www.swpc.noaa.gov/SWN/index.html|url-status=dead|title=NOAA / Space Weather Prediction Center|archive-date=November 27, 2014}}</ref>

===Sail parameters===
Sail loading (areal density) is an important parameter, which is the total mass divided by the sail area, expressed in g/m<sup>2</sup>. It is represented by the Greek letter σ (sigma).

A sail craft has a characteristic acceleration, a<sub>c</sub>, which it would experience at 1&nbsp;AU when facing the Sun. Note this value accounts for both the incident and reflected momentums. Using the value from above of 9.08 μN per square metre of radiation pressure at 1 AU, a<sub>c</sub> is related to areal density by:

:a<sub>c</sub> = 9.08(efficiency) / σ mm/s<sup>2</sup>

Assuming 90% efficiency, a<sub>c</sub> =  8.17 / σ mm/s<sup>2</sup>

The lightness number, λ, is the dimensionless ratio of maximum vehicle acceleration divided by the Sun's local gravity.  Using the values at 1&nbsp;AU:

:λ = a<sub>c</sub> / 5.93

The lightness number is also independent of distance from the Sun because both gravity and light pressure fall off as the inverse square of the distance from the Sun. Therefore, this number defines the types of orbit maneuvers that are possible for a given vessel.

The table presents some example values. Payloads are not included. The first two are from the detailed design effort at JPL in the 1970s. The third, the lattice sailer, might represent about the best possible performance level.<ref name="Wright " />  The dimensions for square and lattice sails are edges.  The dimension for heliogyro is blade tip to blade tip.

{| class="wikitable" style="text-align: center;"
|-
!Type
!σ (g/m<sup>2</sup>)
!a<sub>c</sub> (mm/s<sup>2</sup>)
!λ
!Size (km<sup>2</sup>)
|-
|Square sail
|5.27
|1.56
|0.26
|0.820
|-
|Heliogyro
|6.39
|1.29
|0.22
|15
|-
|Lattice&nbsp;sailer
|0.07
|117
|20
|0.840
|}

===Attitude control===
An active [[Spacecraft attitude control|attitude control]] system (ACS) is essential for a sail craft to achieve and maintain a desired orientation. The required sail orientation changes slowly (often less than 1 degree per day) in interplanetary space, but much more rapidly in a planetary orbit. The ACS must be capable of meeting these orientation requirements. Attitude control is achieved by a relative shift between the craft's [[Center of pressure (fluid mechanics)|center of pressure]] and its [[center of mass]]. This can be achieved with control vanes, movement of individual sails, movement of a control mass, or altering reflectivity.

Holding a constant attitude requires that the ACS maintain a net torque of zero on the craft. The total force and torque on a sail, or set of sails, is not constant along a trajectory. The force changes with solar distance and sail angle, which changes the billow in the sail and deflects some elements of the supporting structure, resulting in changes in the sail force and torque.

Sail temperature also changes with solar distance and sail angle, which changes sail dimensions. The radiant heat from the sail changes the temperature of the supporting structure. Both factors affect total force and torque.

To hold the desired attitude the ACS must compensate for all of these changes.<ref name="wright 6+B">Wright, ibid., Ch 6 and Appendix B.</ref>

===Constraints===
In Earth orbit, solar pressure and drag pressure are typically equal at an altitude of about 800&nbsp;km, which means that a sail craft would have to operate above that altitude. Sail craft must operate in orbits where their turn rates are compatible with the orbits, which is generally a concern only for spinning disk configurations.

Sail operating temperatures are a function of solar distance, sail angle, reflectivity, and front and back emissivities. A sail can be used only where its temperature is kept within its material limits. Generally, a sail can be used rather close to the Sun, around 0.25&nbsp;AU, or even closer if carefully designed for those conditions.<ref name="Wright " />