Aspects: Geometry, Orbs, and Exact Timing #

An aspect is a specific angular relationship between two points on the ecliptic. When astrologers say “Mars squares Saturn,” they mean the ecliptic longitude difference between Mars and Saturn is approximately 90°. This article covers the geometry, the orb system, the applying/separating distinction, and the numerical methods for finding the exact moment an aspect becomes partile (exact).

Angular Separation #

Given two bodies at ecliptic longitudes $\lambda_1$ and $\lambda_2$, the angular separation along the ecliptic is the shortest arc:

$$\Delta\lambda = \min!\left(|\lambda_1 - \lambda_2|,; 360° - |\lambda_1 - \lambda_2|\right)$$

Or equivalently, the formula that handles the 0°/360° wraparound in one step:

$$\Delta\lambda = 180° - \left|,|\lambda_1 - \lambda_2| - 180°,\right|$$

This always yields a value in $[0°, 180°]$. The conjunction (0°) and opposition (180°) are the extreme cases.

Note: this is the ecliptic longitude difference, not the true angular distance on the sphere. Standard aspect theory ignores ecliptic latitude — two planets at the same longitude are considered conjunct regardless of their vertical separation. For the true angular distance including latitude, use the formula from Coordinate Transformations.

Aspect Definitions #

The classical aspects divide the circle harmonically:

The minor aspects extend to higher harmonics:

The harmonic number $n$ relates to the aspect angle $\theta$ by:

$$\theta = \frac{360°}{n} \times k$$

where $k$ is an integer less than $n$. The sextile is $360°/6 = 60°$, the square is $360°/4 = 90°$, etc. The semi-square (45°) and sesquiquadrature (135°) are both 8th-harmonic aspects ($360°/8 = 45°$, $3 \times 45° = 135°$).

Orbs #

An aspect is “in orb” when the angular separation is close to the exact aspect angle but not necessarily exact. The orb is the tolerance:

$$\text{orb} = |\Delta\lambda - \theta_{\text{exact}}|$$

If $\text{orb} \leq \text{orb}_{\text{max}}$, the aspect is active.

Orb assignment approaches #

Fixed orbs per aspect type: a simple global maximum (e.g., 8° for major aspects, 3° for minor). This is the most common approach in software.

Moieties (planet-based orbs): each planet has a “moiety” (half-orb), and the effective orb for a pair is the sum of the two moieties:

$$\text{orb}_{\text{max}} = \text{moiety}(P_1) + \text{moiety}(P_2)$$

Traditional moieties (Lilly):

Under this system, a Sun–Moon aspect has an effective orb of 13.5°, while a Mercury–Saturn aspect has 7.5°.

Software should make the orb model configurable, since different astrological traditions use substantially different orbs.

Applying and Separating #

An aspect is applying if the two bodies are moving toward exact, and separating if they are moving apart. The distinction depends on the relative speed.

Define the relative speed along the ecliptic:

$$v_{\text{rel}} = \dot{\lambda}{\text{faster}} - \dot{\lambda}{\text{slower}}$$

where $\dot{\lambda}$ is the daily motion in longitude (positive = direct, negative = retrograde).

The aspect is:

• Applying if the orb is decreasing over time
• Separating if the orb is increasing over time

To determine this precisely, you need to consider which body is “ahead” in the ecliptic and whether the faster body is approaching or receding from the exact aspect angle. The sign of $d(\text{orb})/dt$ tells you:

$$\frac{d(\text{orb})}{dt} = \text{sgn}(\Delta\lambda - \theta) \times (\dot{\lambda}_2 - \dot{\lambda}_1)$$

If $d(\text{orb})/dt < 0$, the aspect is applying. If $> 0$, separating.

Retrograde motion complicates this: a retrograde planet can turn an applying aspect into a separating one, or cause the same exact aspect to occur three times (direct pass, retrograde pass, direct pass again).

Dexter and Sinister #

In traditional astrology, a dexter aspect is cast backward through the zodiac (in the direction of diurnal motion), and a sinister aspect is cast forward. The distinction is geometric:

• Body A at $\lambda_A$ forms a dexter aspect to body B if B is at $\lambda_A - \theta$ (earlier in zodiacal order)
• Body A forms a sinister aspect to body B if B is at $\lambda_A + \theta$ (later in zodiacal order)

Whether an aspect is dexter or sinister depends on which body you consider as “casting” the aspect. The same physical configuration is a dexter square from Mars’s perspective and a sinister square from Saturn’s perspective.

Declination Parallels #

Aspects can also be measured in declination rather than longitude. A parallel occurs when two bodies have the same declination:

$$|\delta_1 - \delta_2| \leq \text{orb}$$

A contraparallel occurs when they have equal but opposite declinations:

$$|\delta_1 + \delta_2| \leq \text{orb}$$

Parallels are computed from equatorial coordinates, not ecliptic. Typical orb for parallels is 1°.

Finding the Exact Moment of an Aspect #

In transit work and chart progression, you often need the exact date and time when an aspect becomes partile (orb = 0). This is a root-finding problem.

Define the function:

$$f(t) = \Delta\lambda(t) - \theta_{\text{exact}}$$

where $\Delta\lambda(t)$ is the shortest-arc separation between the two bodies at time $t$. The exact aspect occurs when $f(t) = 0$.

Method: linear interpolation #

If you have daily ephemeris positions and you find that $f(t_1)$ and $f(t_2)$ have opposite signs (the aspect crosses exact between day 1 and day 2), the approximate time is:

$$t_{\text{exact}} \approx t_1 - f(t_1) \times \frac{t_2 - t_1}{f(t_2) - f(t_1)}$$

This is the regula falsi (false position) method. For aspects between slow-moving planets, one interpolation step gives accuracy within minutes. For fast-moving bodies (especially the Moon at ~13°/day), you may need to subdivide the interval and iterate.

Method: Newton-Raphson #

If the derivative $f’(t) = \dot{\lambda}_2(t) - \dot{\lambda}_1(t)$ is available (from the ephemeris velocity output):

$$t_{n+1} = t_n - \frac{f(t_n)}{f’(t_n)}$$

This converges quadratically and typically finds the exact time in 2–3 iterations.

Wraparound trap #

Near 0°/360°, the function $f(t)$ can jump discontinuously. For a conjunction between a body at 359° and one at 1°, the separation is 2°, not 358°. The shortest-arc formula handles this correctly in a static calculation, but when tracking $f(t)$ over time, a body crossing 360° → 0° causes an apparent jump. Software must detect and handle this wraparound, typically by working with the signed difference modulo 360° or by tracking the unwrapped (non-modular) longitude.

Worked Example #

Given: Sun at $\lambda_{\odot} = 54.20°$ with $\dot{\lambda}{\odot} = 0.985°$/day, Mars at $\lambda{\mars} = 141.70°$ with $\dot{\lambda}_{\mars} = 0.620°$/day.

Separation: $\Delta\lambda = |141.70 - 54.20| = 87.50°$

Nearest aspect: square (90°). Orb = $|87.50 - 90| = 2.50°$.

Applying or separating? Mars is ahead of the Sun in the zodiac. The Sun is faster ($0.985 > 0.620$). The Sun is closing the gap, so the orb is decreasing. The aspect is applying.

Time to exact: the gap closes at a rate of $0.985 - 0.620 = 0.365°$/day. The remaining gap is $90° - 87.50° = 2.50°$.

$$t_{\text{exact}} = \frac{2.50°}{0.365°/\text{day}} = 6.85 \text{ days}$$

This is a linear approximation. Over nearly 7 days, both speeds will change slightly, so the true exact time will differ by a few hours. For higher precision, evaluate the ephemeris at $t + 6.85$ days and refine.

References #

• Meeus, J. (1998). Astronomical Algorithms, 2nd ed. Willmann-Bell. Chapter 17 (interpolation).
• Ptolemy, C. (c. 150 CE). Tetrabiblos, Book I, Chapter 13 (classical aspect definitions).
• Lilly, W. (1647). Christian Astrology. (Moiety tables.)