{
  "id": "telescope-magnification",
  "version": "df990bcd9671",
  "status": "published",
  "name": "Telescope Magnification Calculator",
  "question": "What is the telescope magnification?",
  "summary": "Telescope magnification calculator: M = telescope focal length ÷ eyepiece focal length, with a Barlow, plus the exit pupil, true field of view, focal ratio and resolving limit.",
  "category": "physics",
  "subcategory": "optics",
  "url": "https://www.acalculator.org/physics/telescope-magnification-calculator",
  "markdown": "https://www.acalculator.org/physics/telescope-magnification-calculator.md",
  "kind": "function",
  "method": "M = telescope focal length × Barlow factor ÷ eyepiece focal length; exit pupil = D ÷ M; true field = apparent field ÷ M; focal ratio = focal length ÷ D; resolving limit = 1.22 × 550 nm ÷ D radians.",
  "assumptions": [
    "The final image is at infinity (a relaxed eye), and the image is upside down; the page gives the size of M.",
    "The true field is the apparent field ÷ M, the small-angle rule; the resolving limit is the diffraction (Rayleigh) limit, before the air and the optics blur it."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "f": {
        "title": "Telescope focal length",
        "description": "The focal length of the main lens or mirror (the objective).",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.0001,
        "maximum": 1000
      },
      "fe": {
        "title": "Eyepiece focal length",
        "description": "The focal length printed on the eyepiece.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.0001,
        "maximum": 10
      },
      "b": {
        "title": "Barlow or reducer factor",
        "description": "Multiplies the focal length: 1 for none, 2 for a 2× Barlow, 0.63 for an f/6.3 reducer.",
        "type": "number",
        "minimum": 0.1,
        "maximum": 10
      },
      "D": {
        "title": "Aperture (D)",
        "description": "Optional: the diameter of the main lens or mirror, for the exit pupil, focal ratio and resolving limit.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.0001,
        "maximum": 100
      },
      "afov": {
        "title": "Eyepiece apparent field (°)",
        "description": "Optional: the apparent field of view of the eyepiece, in degrees (often 50° to 82°).",
        "type": "number",
        "minimum": 1,
        "maximum": 120
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Magnification",
      "description": "M to 6 significant figures, written as a times sign: 40×.",
      "format": "text"
    },
    "magnification": {
      "label": "Magnification (M)",
      "description": "Telescope focal length × Barlow factor ÷ eyepiece focal length.",
      "format": "number"
    },
    "effective": {
      "label": "Focal length with the Barlow (mm)",
      "description": "The telescope focal length times the Barlow factor.",
      "format": "number"
    },
    "exitPupil": {
      "label": "Exit pupil (mm)",
      "description": "D ÷ M: the width of the beam of light that leaves the eyepiece.",
      "format": "number"
    },
    "trueField": {
      "label": "True field of view (°)",
      "description": "The apparent field ÷ M: how much sky you see.",
      "format": "number"
    },
    "trueFieldArcmin": {
      "label": "True field of view (′)",
      "description": "The true field in minutes of arc (60 to a degree).",
      "format": "number"
    },
    "focalRatio": {
      "label": "Focal ratio (f/)",
      "description": "The telescope focal length ÷ D, without the Barlow.",
      "format": "number"
    },
    "resolution": {
      "label": "Resolving limit at 550 nm (″)",
      "description": "1.22 × λ ÷ D in seconds of arc: the closest pair of stars it can split.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "1000 mm",
      "fe": "25 mm",
      "b": 1,
      "D": "100 mm",
      "afov": 52
    },
    "outputs": {
      "result": "40×",
      "magnification": 40,
      "effective": 1000,
      "exitPupil": 2.5,
      "trueField": 1.3,
      "trueFieldArcmin": 78,
      "focalRatio": 10,
      "resolution": 1.3840368499180167
    },
    "text": "The magnification is 40×."
  },
  "examples": [
    {
      "given": {
        "f": 1,
        "fe": 0.025,
        "b": 1
      },
      "expect": {
        "magnification": 40,
        "result": "40×"
      },
      "source": "OpenStax, University Physics Volume 3, §2.8 Microscopes and Telescopes (Equations 2.39 and 2.40, M = θ_image ÷ θ_object = −f_obj ÷ f_eye; a 1 m objective gives 40× with a 2.5 cm eyepiece and 80× with 1.25 cm), https://openstax.org/books/university-physics-volume-3/pages/2-8-microscopes-and-telescopes (retrieved 2026-10-19); hand calculation in content.mdx: M = 100 cm ÷ 2.5 cm = 40"
    },
    {
      "given": {
        "f": 1,
        "fe": 0.0125,
        "b": 1
      },
      "expect": {
        "magnification": 80
      },
      "source": "OpenStax, University Physics Volume 3, §2.8 Microscopes and Telescopes (Equations 2.39 and 2.40, M = θ_image ÷ θ_object = −f_obj ÷ f_eye; a 1 m objective gives 40× with a 2.5 cm eyepiece and 80× with 1.25 cm), https://openstax.org/books/university-physics-volume-3/pages/2-8-microscopes-and-telescopes (retrieved 2026-10-19); hand calculation in content.mdx: M = 100 ÷ 1.25 = 80"
    },
    {
      "given": {
        "f": 1,
        "fe": 0.025,
        "b": 2,
        "D": 0.1,
        "afov": 52
      },
      "expect": {
        "magnification": 80,
        "exitPupil": 1.25,
        "trueField": 0.65,
        "trueFieldArcmin": 39,
        "focalRatio": 10,
        "effective": 2000
      },
      "source": "OpenStax, University Physics Volume 3, §2.8 Microscopes and Telescopes (Equations 2.39 and 2.40, M = θ_image ÷ θ_object = −f_obj ÷ f_eye; a 1 m objective gives 40× with a 2.5 cm eyepiece and 80× with 1.25 cm), https://openstax.org/books/university-physics-volume-3/pages/2-8-microscopes-and-telescopes (retrieved 2026-10-19); hand calculation in content.mdx: M = 1,000 × 2 ÷ 25 = 80; exit pupil 100 ÷ 80 = 1.25 mm; true field 52 ÷ 80 = 0.65°"
    },
    {
      "given": {
        "f": 57.6,
        "fe": 0.025,
        "b": 1,
        "D": 2.4
      },
      "expect": {
        "resolution": 0.05766820207991736,
        "magnification": 2304
      },
      "source": "OpenStax, University Physics Volume 3, §4.5 Circular Apertures and Resolution (Equation 4.5, θ = 1.22λ ÷ D; Example 4.6: the Hubble Space Telescope, D = 2.40 m at 550 nm, 2.80 × 10⁻⁷ rad), https://openstax.org/books/university-physics-volume-3/pages/4-5-circular-apertures-and-resolution (retrieved 2026-10-19): 2.80 × 10⁻⁷ rad = 0.0577″; hand calculation in content.mdx: 1.22 × 550 × 10⁻⁹ ÷ 2.4 × 648,000 ÷ π"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 3, §2.8 Microscopes and Telescopes: Equation 2.39 (angular magnification M = θ_image ÷ θ_object) and Equation 2.40 (M = −f_obj ÷ f_eye); a 1 m objective with a 2.5 cm eyepiece gives 40×, and with 1.25 cm gives 80×, retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-3/pages/2-8-microscopes-and-telescopes",
    "OpenStax, University Physics Volume 3, §2.4 Thin Lenses: Equation 2.19 (1 ÷ d_o + 1 ÷ d_i = 1 ÷ f) and Equation 2.22 (m = −d_i ÷ d_o), used for the exit pupil, retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-3/pages/2-4-thin-lenses",
    "OpenStax, University Physics Volume 3, §4.5 Circular Apertures and Resolution: Equation 4.5 (Rayleigh criterion θ = 1.22λ ÷ D); Example 4.6 (the Hubble Space Telescope, D = 2.40 m, λ = 550 nm: θ = 2.80 × 10⁻⁷ rad), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-3/pages/4-5-circular-apertures-and-resolution",
    "NIST Special Publication 811 (2008), Appendix B.8: 1 in = 0.0254 m, retrieved 2026-10-19. https://www.nist.gov/pml/special-publication-811"
  ],
  "related": [
    "field-of-view",
    "wavelength",
    "frequency"
  ],
  "changelog": []
}