{
  "address": "ec2aa4b27117b3fe32813660477a91f5",
  "category": "other",
  "question": "Using the simple sunrise-equation hour angle, what is the approximate day length at latitude 40 degrees when the solar declination is 23.45 degrees?",
  "answer": "14.85 hours",
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  "madeAt": "2026-09-30T22:39:30.722Z",
  "validUntil": null,
  "validityClass": "forever",
  "sources": [
    {
      "label": "Solar Engineering of Thermal Processes, 4th ed., the sunrise equation",
      "url": null
    }
  ],
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  "source": {
    "kind": "edition_page",
    "value": "Duffie, J.A. and Beckman, W.A. Solar Engineering of Thermal Processes, 4th ed. Wiley, 2013, eq. 1.6.10.",
    "retrieved_utc": "2026-09-30T19:43:00Z"
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  "cardContractVersion": "card_contract/1",
  "deck_id": "method-deck",
  "deck_title": "Method Deck",
  "method": {
    "name": "Day length from latitude and solar declination",
    "version": "1",
    "source_url_or_file": "methods/day-length-from-latitude-and-declination.cjs",
    "sha256_of_method_file": "91d37a59633272b64f87b69a2376436373f6c95074886b536dd4220b53e41742"
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  "method_deck": {
    "inputs": [
      {
        "name": "latitude_deg",
        "type": "number",
        "unit": "degrees",
        "required": true,
        "description": "Latitude, positive north, negative south.",
        "min": -90,
        "max": 90
      },
      {
        "name": "declination_deg",
        "type": "number",
        "unit": "degrees",
        "required": true,
        "description": "Solar declination for the date (see the sun-declination-by-day-of-year card).",
        "min": -23.45,
        "max": 23.45
      }
    ],
    "outputs": [
      {
        "name": "day_length_hours",
        "unit": "hours",
        "description": "Estimated hours of daylight from sunrise to sunset."
      }
    ],
    "steps": [
      "Sunrise equation: cos(H) = -tan(latitude) * tan(declination), where H is the hour angle of sunrise/sunset in degrees.",
      "If the product makes cos(H) <= -1, the location has continuous daylight (midnight sun); if cos(H) >= 1, continuous night (polar night).",
      "Otherwise H = arccos(cos(H)) in degrees.",
      "Day length in hours = 2 * H / 15, because Earth's rotation moves the hour angle 15 degrees per hour.",
      "Round to 2 decimals.",
      "This is the simple, geometric-horizon classroom version (it ignores atmospheric refraction and the sun's angular radius, both of which the estate's own sunrise card handles with the full NOAA method); treat this card's output as an approximation good to a few minutes."
    ],
    "constants": [
      {
        "name": "degrees_per_hour",
        "value": 15,
        "unit": "degrees/hour",
        "source": 0
      }
    ],
    "examples": [
      {
        "name": "worked example",
        "inputs": {
          "latitude_deg": 40,
          "declination_deg": 23.45
        },
        "expected": {
          "answer": "14.85 hours"
        },
        "check": "By hand: tan(40deg)=0.8391, tan(23.45deg)=0.4335, product=0.3638, so cos(H)=-0.3638, H=acos(-0.3638)=111.34 degrees, day length=2*111.34/15=14.85 hours, consistent with the well known fact that New York City (about 40N) has roughly 15 hours of daylight at the June solstice."
      },
      {
        "name": "test vector: winter solstice at the same latitude",
        "inputs": {
          "latitude_deg": 40,
          "declination_deg": -23.45
        },
        "expected": {
          "answer": "9.15 hours"
        },
        "check": "By hand: tan(40deg)*tan(-23.45deg)=-0.3638, so cos(H)=0.3638, H=acos(0.3638)=68.66 degrees, day length=2*68.66/15=9.15 hours, consistent with the well known fact that New York City has roughly 9 hours of daylight at the December solstice."
      }
    ],
    "disclaimer": null,
    "deck_category": "astronomy"
  },
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          "value": "Duffie, J.A. and Beckman, W.A. Solar Engineering of Thermal Processes, 4th ed. Wiley, 2013, eq. 1.6.10.",
          "retrieved_utc": "2026-09-30T19:43:00Z"
        },
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        "supersedes": null,
        "added_utc": "2026-09-30T22:39:30.722Z",
        "added_by": "method:Day length from latitude and solar declination",
        "why": "the method that computed this answer",
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        "status_label": "pending",
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}