# Reading — The Radar Range Equation

By the end of this lesson you should be able to:

1. Write the **radar range equation** and explain the physical origin of each term.
2. Derive the **maximum detection range** $R_\text{max}$ and show why it scales as the fourth root of every term.
3. Work a numerical example from transmit power and antenna gains to a detection range.
4. Quantify how an RCS reduction translates into a detection-range reduction.

## Building the equation

The range equation is just bookkeeping on power, applied twice — out to the target and back.

**Step 1 — transmit toward the target.** A transmitter radiates $P_t$ watts. An antenna of gain $G_t$ concentrates that into a beam, so the power density at range $R$ is

$$
S_\text{inc} = \frac{P_t G_t}{4\pi R^2}.
$$

The $4\pi R^2$ is the surface area of a sphere of radius $R$ — the one-way spreading loss from Lesson 2.

**Step 2 — the target re-radiates.** The target intercepts some power and scatters it back. Its **radar cross section** $\sigma$ (units: m²) is defined exactly so that the power "captured and re-radiated isotropically" is $S_\text{inc}\,\sigma$. The echo then spreads back over another sphere, giving a density at the radar of

$$
S_\text{echo} = \frac{P_t G_t}{4\pi R^2}\cdot\frac{\sigma}{4\pi R^2}.
$$

**Step 3 — the radar collects the echo.** A receive antenna with gain $G_r$ has effective aperture $A_e = G_r \lambda^2 / 4\pi$. Multiplying the echo density by $A_e$ gives the received power:

$$
\boxed{\,P_r = \frac{P_t\, G_t\, G_r\, \lambda^2\, \sigma}{(4\pi)^3\, R^4}\,}
$$

The headline feature is the $R^4$ in the denominator. The signal spreads out **on the way to the target and again on the way back** — two factors of $1/R^2$ multiply into a fourth-power law.

:::{admonition} Key Concept
:class: key-concept

The $1/R^4$ in the radar range equation comes from two-way spreading: $1/R^2$ out, $1/R^2$ back. This single fact is why radar detection is so range-limited, and why it is the most leverage-rich link in the kill chain.
:::

## Maximum detection range

A radar declares detection when received power meets or exceeds its minimum detectable signal $S_\text{min}$. Setting $P_r = S_\text{min}$ and solving for range:

$$
R_\text{max} = \left[\frac{P_t\, G_t\, G_r\, \lambda^2\, \sigma}{(4\pi)^3\, S_\text{min}}\right]^{1/4}.
$$

Every term sits under a fourth root. Bundling everything except $\sigma$ into a constant $K$ recovers the Lesson 1 form,

$$
R_\text{max} = K\,\sigma^{1/4}.
$$

The fourth root is brutal on the radar designer and merciful to the target. To **double** detection range you must raise $P_t G_t G_r$ by a factor of **16**. Conversely, the target gets the same leverage in reverse — a 16-fold RCS cut only halves the range, but the radar's billion-watt options have run out long before the target's shaping options do.

### The cost of each term

| Want to double $R_\text{max}$ by changing… | Required change | Practical? |
| --- | --- | --- |
| Transmit power $P_t$ | ×16 (+12 dB) | Expensive; thermal and prime-power limits. |
| Antenna gain $G_t$ or $G_r$ | ×16 each (+12 dB) | Huge apertures; physically bounded. |
| Minimum signal $S_\text{min}$ | ÷16 | Limited by noise floor; integration helps a little. |
| Target RCS $\sigma$ | ÷16 | This is the **target's** lever — low-observable design. |

The table is the whole survivability argument in one place: the radar pays $\times 16$ for every range doubling, and there is no cheap term to push.

## A worked example

Take an S-band fire-control radar:

- $P_t = 1$ MW $= 90$ dBW
- $G_t = G_r = 30$ dBi
- $\lambda = 0.1$ m (3 GHz, S-band)
- $S_\text{min}$ set so a $\sigma = 1\ \text{m}^2$ target is detected at the design range

Running the equation for a **1 m² target** gives a maximum detection range of roughly

$$
R_\text{max} \approx 472\ \text{km}.
$$

Now make the target stealthy. Drop the RCS to $\sigma = -30$ dBsm (i.e. $10^{-3}\ \text{m}^2$, a thousandfold reduction). Scaling by the fourth root,

$$
R_\text{max,new} = 472\ \text{km}\times(10^{-3})^{1/4} = 472\times 0.178 \approx 84\ \text{km}.
$$

A factor-of-1000 RCS reduction cuts the detection range from 472 km to about 84 km — roughly a **5.6× reduction in range** for a **1000× reduction in RCS**. That is the fourth root at work.

::::{admonition} Quick Exercise
:class: quick-exercise

A radar detects a target at $R_\text{max} = 200$ km.

1. The target reduces its RCS by **12 dB**. What is the new detection range?
2. Instead, the radar quadruples its transmit power. What is the new detection range?
3. Which is the easier lever to pull, and for whom?

:::{admonition} Solution
:class: dropdown

1. 12 dB is a factor of ~16 in power; $16^{1/4} = 2$, so range **halves** to **100 km**.
2. ×4 power → $4^{1/4} = \sqrt{2} \approx 1.41$, so range grows to about **283 km**.
3. The fourth root means both sides pay dearly. The target's RCS lever (shaping, materials) is one-time engineering; the radar's power/gain lever runs into hard physical and thermal limits. Stealth wins because the target's lever is cheaper to keep pulling.
:::

::::

## Wrap-Up

The radar range equation, $P_r = P_t G_t G_r \lambda^2 \sigma / [(4\pi)^3 R^4]$, is power bookkeeping over a two-way path, and its $1/R^4$ forces $R_\text{max} = K\,\sigma^{1/4}$. Every term sits under a fourth root, so doubling range costs a 16-fold change in power, gain, or sensitivity — while the target's RCS lever, though equally fourth-rooted, is the one with room left to push.

So far the radar has been a continuous power budget. Real radars send **pulses**, and that changes everything about how range is measured — and introduces ambiguities the next lesson is built around.