Solar radiation pressure: Question bank — Physics deep dive

Before the traps, let us re-earn every symbol the parent note used, and justify each link of its derivation chain — because a trap only teaches you if you already hold the honest version.

Symbols and chain, re-earned from zero

Definition Every letter, in plain words

$E$ — the energy carried by one photon (a single packet of light), measured in joules.
$c$ — the speed of light, $\approx 3 \times 1 0^{8} m/s$ .
$p$ — momentum, the "push-capacity" a moving thing carries.
$L_{⊙}$ — the Sun's luminosity: total light power it pours out, in watts.
$r$ — the distance from the Sun to the spacecraft.
$S$ — the solar irradiance (also called intensity): light power per unit area arriving at distance $r$ , in $W/m^{2}$ . At 1 AU this is the solar constant $\approx 1361 W/m^{2}$ .
$P_{rad}$ — the radiation pressure: force per unit area the light exerts.
$A$ — the spacecraft's cross-sectional area facing the flow of light.
$η$ — the reflectivity coefficient: a dial from $0$ (surface swallows all light) to $1$ (perfect mirror bounces it all back). See figure below.
$θ$ — the tilt angle between the surface's normal (the arrow sticking straight out of the face) and the Sun direction.
$m$ — the spacecraft mass; $a$ — its acceleration.

Formula The chain, one honest step at a time

Link 1 — one photon carries momentum. $p = E / c$ . WHY: relativity says $E^{2} = (p c)^{2} + (m c^{2})^{2}$ ; a photon has mass $m = 0$ , so $E = p c \Rightarrow p = E / c$ . A massless thing still carries momentum.

Link 2 — from energy flux to pressure. $P_{rad} = S / c$ (perfect absorber). WHY: $S$ is energy landing per area per second. Each joule carries $1/ c$ of momentum, so momentum landing per area per second is $S / c$ — and momentum-per-area-per-second is pressure.

Link 3 — the plate force. $F = P_{rad} A (1 + η) cos θ$ . WHY the $cos θ$ : only the projected area $A cos θ$ actually intercepts the beam — a tilted plate is a smaller target (see step figure). WHY the $(1 + η)$ : an absorber takes momentum $p$ ; a mirror sends the photon back, changing its momentum by $2 p$ , i.e. $(1 + η)$ from $1$ up to $2$ (see step figure). CAUTION: this single- $cos θ$ form is only the component along the Sun–spacecraft line. The true force on a tilted mirror points along the surface normal and scales as $cos^{2} θ$ — the second cosine is the normal-projection of the incoming momentum. Both facts are made visual below and re-tested in the Spot the error section.

Link 4 — acceleration. $a = F / m$ . WHY: orbits respond to acceleration, and Newton's $F = ma$ divides the force by mass — hence the design-critical ratio $A / m$ .

The figure above shows Link 3's geometry: the projected area shrinks by $cos θ$ (fewer photons caught) and the recoil doubles the momentum change for a mirror.

The second figure shows why a tilted mirror obeys $cos^{2} θ$ : one cosine from the smaller target, a second from keeping only the momentum component along the normal.

True or false — justify

A photon has zero mass, therefore it cannot exert a force.

False. Momentum is not

m v

for light; it is

p = E / c

with

E

the photon energy. Massless does not mean momentumless, so a stream of photons pushes on any surface it hits.

At 1 AU the base radiation pressure is roughly the same order as atmospheric pressure at sea level.

False.

P_{rad} \approx 4.5 \times 1 0^{- 6} Pa

, about 11 orders of magnitude weaker than the

\sim 1 0^{5} Pa

of sea-level air. Its power is being continuous and everywhere in space, not being strong.

A perfect mirror feels exactly twice the force of a perfect black absorber of the same area, both face-on.

True. The absorber gains one photon-momentum

p

; the mirror reverses it, gaining

2 p

(recoil). That is the

(1 + η)

factor going from

η = 0

η = 1

SRP and the Sun's gravity both fall off as

1/ r^{2}

, so their ratio is the same for every object.

False. Both accelerations fall as

1/ r^{2}

so the ratio is distance-independent, but it still

\propto A / m

. A solar sail's ratio is huge; a dense probe's is tiny. This is the central idea of Orbital perturbations by SRP.

Doubling the distance from the Sun halves the SRP force.

False. Irradiance

S \propto 1/ r^{2}

(the Inverse-square law), so doubling

r

cuts the force to a quarter, not a half.

A spacecraft twice as massive but with the same cross-section feels half the SRP acceleration.

True.

a_{SRP} \propto A / m

; double

m

at fixed

A

halves the acceleration, even though the force

F

is unchanged.

SRP always pushes a spacecraft directly away from the Sun.

False. For a flat plate that is not face-on, reflected photons leave along the surface normal, so the reaction force points partly sideways — that is exactly how Solar sails tack to raise or lower an orbit.

A transparent (fully transmitting) object at 1 AU feels essentially no SRP.

True. If photons pass straight through, no momentum is transferred, so

F \to 0

. SRP needs absorption or reflection to act.

Spot the error

"Reflection gives a single factor of

cos θ

because only the projected area

A cos θ

intercepts light."

The projection is one cosine, but a perfect mirror's normal-force also carries a second cosine (only the normal component of momentum reverses), giving

\propto cos^{2} θ

— exactly what figure s02 shows. The simple

(1 + η) cos θ

is only the along-Sun component approximation.

P_{rad} = S / c

works for any surface, so a mirror at 1 AU feels

4.5 \times 1 0^{- 6} Pa

P_{rad} = S / c

is the absorber base value. A face-on mirror gets

(1 + η) P_{rad} = 2 P_{rad} \approx 9 \times 1 0^{- 6} Pa

; the reflection factor

(1 + η)

was dropped.

"Since

p = E / c

, a brighter (higher-frequency) photon carries the same momentum as a dim one."

Momentum scales with energy,

p = E / c

. A higher-energy photon carries more momentum; only equal-energy photons carry equal momentum. (Foundation of Photon momentum and relativity.)

"SRP is negligible for real missions — it's a textbook curiosity."

For high-

A / m

objects (sails, balloons, defunct panels) SRP rivals or beats Atmospheric drag and perturbs orbits over months. Ignoring it wrecks long-term ephemerides.

"To get the irradiance we divide the Sun's power by the surface area of the Sun."

We divide

L_{⊙}

4 π r^{2}

, the area of the sphere at the spacecraft's distance

r

, not the Sun's own surface — that's what makes

S

fall with distance.

"Because the momentum change per photon is tiny, the total force fluctuates wildly."

The photon flux is astronomically large and steady, so by the law of large numbers the force is smooth and constant — that steadiness is precisely why it matters over long times.

"Painting a satellite black minimizes SRP because black absorbs instead of reflecting."

Absorbing (

η = 0

) gives the minimum factor

(1 + η) = 1

, so black does feel less than a mirror — but it still feels the full absorbing pressure. Zero SRP would need transparency, not blackness.

Why questions

Why does the same power pass through every sphere centred on the Sun?

Energy is conserved and the space between Sun and spacecraft neither creates nor absorbs light, so the total power

L_{⊙}

crossing each concentric sphere is fixed — only the area

4 π r^{2}

it spreads over grows, which is why

S = L_{⊙} /4 π r^{2}

Why do we convert energy flux to momentum flux with a factor

1/ c

Because pressure is momentum delivered per area per second, and each joule of light carries

1/ c

of momentum (

p = E / c

). Multiplying

S

(energy flux) by

1/ c

turns it into momentum flux = pressure

P_{rad}

Why is the area-to-mass ratio

A / m

, not area or mass alone, the key design number?

Orbits respond to acceleration

a = F / m

, and

F \propto A

, so

a \propto A / m

. Only the combination survives — a big light sail and a small dense probe with equal

A / m

accelerate the same.

Why is a solar sail made huge and thin rather than small and massive?

To maximize

A / m

: large area catches more photons while low mass means each unit of force produces more acceleration, giving the biggest useful thrust.

Why can a tiny

\sim 9 mN

force still produce a large

Δ v

Because it acts continuously for months or years in the frictionless vacuum of space; impulse

= F \times t

accumulates without any air to cancel it.

Why does the

E = p c

relation come from relativity and not Newtonian mechanics?

Newton's

p = m v

gives zero for a massless particle. The relativistic

E^{2} = (p c)^{2} + (m c^{2})^{2}

with

m = 0

yields

E = p c

, the only self-consistent statement (see Photon momentum and relativity).

Edge cases

What happens to SRP force as

θ \to 9 0^{\circ}

(sunlight grazing the plate edge-on)?

cos θ \to 0

, so the intercepted light — and hence the force — goes to zero: an edge-on plate catches essentially no photons.

What is the SRP force on a spacecraft in Earth's shadow (umbra)?

Zero direct sunlight means

S \to 0

there, so SRP vanishes — this on/off cycling per orbit is itself a perturbation (see Orbital perturbations) mission planners must model.

r \to \infty

(deep space, far from the Sun), what happens to SRP?

S \propto 1/ r^{2} \to 0

, so SRP fades away; far from any star, radiation pressure becomes negligible.

For a perfectly reflecting surface tilted at

θ

, why doesn't the force simply vanish like a mirror "letting light slide off"?

The normal component of incoming momentum still reverses on reflection, giving a force

\propto cos^{2} θ

along the normal — it weakens with tilt but only reaches zero at true edge-on (

θ = 9 0^{\circ}

What is the limiting reflectivity behaviour: how does force change from

η = 0

η = 1

The factor

(1 + η)

runs from

1

(pure absorber) to

2

(perfect mirror), so force at most doubles; there is no way to exceed

2 P_{rad} A cos θ

with a passive flat surface.

What happens if a surface re-emits absorbed light as heat unevenly (thermal recoil)?

You get an extra thrust from anisotropic thermal photons — the Yarkovsky effect, SRP's cousin, which is why absorbing bodies can still feel a directional push after the sunlight is gone.

Recall One-line self-audit (hide answers)

Massless ⇒ no push? ::: No — $p = E / c$ , momentum without mass.
Mirror vs black force ratio, face-on? ::: 2 to 1 (recoil, the $(1 + η)$ factor).
SRP/gravity ratio depends on? ::: $A / m$ , not distance.
Force in shadow or edge-on? ::: Zero in both cases.
What is $S$ ? ::: Solar irradiance, power per area $= L_{⊙} /4 π r^{2}$ .

8. Connections

Solar radiation pressure — parent note whose symbols ( $E, S, η, P_{rad}$ ) and chain these traps drill.
Solar sails — the tilt/reflection subtleties (Links 3) that let SRP become steerable propulsion.
Atmospheric drag — the competing non-conservative force the true/false items contrast against.
Orbital perturbations — where the $A / m$ dependence and shadow cycling matter long-term.
Inverse-square law — behind the $S \propto 1/ r^{2}$ distance edge cases.
Photon momentum and relativity — source of the $E = p c$ used in Link 1.
Yarkovsky effect — the thermal-recoil edge case, SRP's absorbing-body cousin.