Equations Explained

1. Extension Factor (m)

m = F_system / F_primary

How much the system focal length is extended by the secondary mirror.

2. Secondary to Primary Focus (A)

A = (F + b) / (m + 1)

Distance from the secondary mirror to the primary focus point.

3. System Focal Plane to Secondary (a)

a = m x A

Distance from the secondary mirror to the image plane.

4. Mirror Separation (B)

B = a - b

Physical separation between primary and secondary mirrors.

5. Radii of Curvature

R₁ = 2 x F

R₂ = (2 x A x m) / (m - 1)

Radii of curvature for the primary and secondary mirrors.

6. Conic Constants

K₁ (Classical) = -1

K₂ (Classical) = -[(4 x m) / (m - 1)²] - 1

K₁ (Ritchey) = (-2*(a/B))/(m**3)-1

K₂ (Ritchey) = ((-4 * m*(m-1)-2*(3+(33/25))) / ((m - 1) ** 3)) - 1

Describes the conic shape of the primary and secondary mirrors.

7. Coma Coefficient

coma (Classical) = 0.5

coma (Ritchey) = (m²/2)*((1/m²)-(1+K1)*(B/(2*B-R1)))

For the true Cassegrain Coma = 0.5, i.e. exactly the same as for a paraboloid of the same diameter .

8. Astigmatism Coefficient

astigmatism = m x [(2B - mR₁) / ((2B - R₁) x m²)]

astigmatism = m x [((2B-mR₁)/(m²(2B-R₁)))-(B/(2B-R₁))²x(1+K₁)]

This coefficient quantifies how much astigmatism is introduced by the geometry of the mirror system.

9. Petzval Curvature

Petzval = m x [1 + (R₁(m - 1)) / (m(2B - R₁))]

Petzval curvature represents the mean field curvature from both mirrors and affects how a flat image plane curves in space.

10. Field Curvature (COF)

Field Curvature (COF) = Petzval - 2 x astigmatism

The total field curvature, taking into account both Petzval and astigmatic effects. A positive curvature indicates a focal surface that is concave towards the incident light.

11. Coma and Astigmatism (Angular)

COMA = (3 / 8) x (D₁ / F_system)² x coma x θ x 103132

ASTIG = (D₁ / F_system) x θ² x astigmatism x (103132 / 4)

Conversion to arcseconds based on system geometry and field angle θ (in radians).

12. Radius of Curvature of Field

R_COF = F_system / COF

How much the image plane curves due to field curvature aberration.

13. Secondary Mirror Axial Diameter

D₂ = (D₁ x A) / F

The physical diamter of the secondary mirror required to capture all the light reflected from the full aperature of the primary.

13. Secondary Mirror Optical Diameter

D_Optical = θ x F_system

The diameter of the light cone from an off-axis object. Uses field angle θ (in radians).

14. Epsilon (Wavefront Error)

E₁ = (b₁ / 32) x (D₁ / 2)⁴ / R₁³

E₂ = (b₂ / 32) x (D_Optical / 2)⁴ / R₂³

The maximum surface deviation from the sphere is given by the above equation.

15. Edge Slope of Mirrors

θ₁ = (b₁ / 4) x (D₁ / 2)³ / R₁³

θ₂ = (b₂ / 4) x (D_Optical / 2)³ / R₂³

The slope at the edge relative to the sphere is given by the derivative of Eq. 14.

References

Texereau, Jean. How to Make a Telescope. Richmond, VA: Willmann-Bell, 1984. A foundational manual for amateur telescope makers, including Newtonian and Cassegrainian design methods.
http://bobmay.astronomy.net/CassNotes/cassformulas.htm - A technical summary of Cassegrain telescope formulas and constants, compiled by Bob May.

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