Mt. Tam Sunrise

Twenty minutes before sunrise, the sky stops being dark and starts being blue. Not the pale washed-out blue of midday. A saturated, electric blue that deepens toward the zenith and bleeds into amber at the horizon. If you’re standing on a ridge above the fog line, the world below is still asleep, and the color is entirely yours. It lasts fifteen minutes. Then the sun breaches the hills, the gold floods in, and the blue hour is gone.

I’d been wanting to catch that transition from East Peak all winter. The original plan was a pre-dawn trail run: headlamp from the Mountain Home Inn up Hogback Road, summit in time for blue hour, run back down before the parking lot fills. But a torn EHL tendon in October killed that plan. No running for months. Hiking only, and even that carefully.

A slower pace changed the calculus. If I’m hiking anyway, why go alone? I recruited four friends, the kind of people who will, under the right conditions, set an alarm for 4:30am on a Saturday. The conditions were: I promised the light would be worth it, and I promised we wouldn’t have to wake up any earlier than absolutely necessary.

The Climb

Mountain Home Inn parking lot, 6:07am. Five headlamps clicking on in the dark. The Inn has been here since 1912, when a Swiss-German couple named Meyers camped all over the mountain looking for the best combination of weather and views^mhi. They found this ridgeline. The area became known as “Little Switzerland” as more Swiss and German immigrants arrived and built hiking clubs. Jack London slept here. So did the Grateful Dead. It’s the only commercial property still permitted on the mountain.

We started up Hogback Road, a steady fire road climb through Douglas fir and bay laurel. At seven-tenths of a mile, Hogback meets Old Railroad Grade, and you’re walking on history. The “Crookedest Railroad in the World.” From 1896 to 1930, the Mt. Tamalpais and Muir Woods Railway carried tourists from Mill Valley to the summit, 281 curves in 8.2 miles^railroad. Round-trip fare: one dollar. Passengers rode gravity cars back down, open-air coasters freewheeling on the steep grade. A train spark started a fire on July 2, 1929 that burned for three days and shut the line down permanently. The old railbed is now a fire road that hikers and cyclists share with the ghosts of Victorian day-trippers. The grade is gentle here, a relic of the 4-7% gradient the steam locomotives required. At 1.1 miles we took the right onto Fern Creek Trail, trading the wide fire road for single track through redwoods. At two miles, the Plank Trail branches right and climbs steeply to Gardner Lookout.

The redwoods were just sound at first. The creak of branches we couldn’t see, something moving in the duff to our left. My headlamp caught only the first ten feet of trail, the rest dissolving into a black that felt solid. The group spread out on the steep sections and compressed on the flats, the natural accordion of hiking with mixed fitness levels. The sky shifted from black to deep blue somewhere around the 1,500-foot mark, and the eastern horizon was starting to blush. Gardner Fire Lookout was built by the Civilian Conservation Corps in 1936, stone and timber hauled from the mountain itself^gardner. It’s named after Ed Gardner, chief warden of the Tamalpais Forest Fire District, who led the fight against the 1929 fire and died in 1935. The lookout is still active. Inside, a ranger uses an Osborne Firefinder, a sighting device designed in the early 1900s that needs no power. There’s no plaque acknowledging the CCC. There should be.

The question that keeps you moving in the dark is simple: how do you know you’ll make it in time? The answer is two calculations stitched together.

Part I: Where Will the Sun Be?

The question sounds simple: at what time does the sun cross the horizon at a specific point on Earth? But “cross the horizon” glosses over the hard part. Not $0°$, for reasons we’ll get to. And the answer changes every day, because the sun doesn’t rise in the same place twice. You need three things: where you are, the date, and a careful definition of “sunrise” that turns out to be more slippery than it sounds.

Solar Declination

Sunrise time changes through the year. In San Francisco, the earliest sunset is December 7 and the latest is July 3, a spread of nearly four hours. Not December 21. The solstice is the shortest day, but the equation of time shifts the clock so that the earliest sunset falls about two weeks before and the latest sunrise about two weeks after. The asymmetry surprises everyone. The reason is tilt.

Earth’s axis leans $23.44°$ off vertical. As we orbit the Sun, the latitude where the Sun is directly overhead at noon drifts north and south through the year, from $23.44°$N at the summer solstice to $23.44°$S at the winter solstice. This drift is called solar declination, $\delta$, and it’s the single variable that controls how long your day is.

You can approximate it with a cosine:

$$\delta \approx -23.44°\times \cos \left(\frac{360°}{365}(d+10)\right)$$

where $d$ is the day of the year (Jan 1 = 1). The $+10$ shifts the cosine so the minimum lands on the winter solstice (~Dec 21, day 355). On January 26, $d=26$, giving $\delta \approx -19°$. The sun is still well south of the equator, keeping days short and sunrise late.

Good enough for planning, not good enough for a calculator. The real formula accounts for orbital eccentricity (Earth is closest to the Sun in early January, warping the sine wave) and a correction called the equation of time, which shifts solar noon by up to $\pm 16$ minutes through the year. The equation of time is why a sundial and a clock disagree, and why the earliest sunset and latest sunrise don’t land on the solstice. If you plot the sun’s position at the same clock time each day for a year, you get the figure-eight called the analemma. Look for it on old globes. The analemma is often printed in the Pacific Ocean, a figure-eight showing how far the sun drifts from “expected” position through the year. It encodes both axial tilt (north-south drift) and orbital eccentricity (east-west drift) in a single curve.

The intuition is visual: in winter, the sun traces a short, low arc across the southern sky. In summer, it swings a long, high arc that rises far north of east and sets far north of west. If you stood on East Peak every morning for a year, you’d watch the sunrise point migrate along the eastern skyline like a slow metronome.

The Hour Angle

The hour angle ${\omega }_0$ is the sun’s angular distance from the meridian at sunrise, and it answers the question directly. It falls out of spherical trigonometry on the celestial sphere:

$$\cos ({\omega }_0)=\frac{\sin (h_0)-\sin (\varphi )\sin (\delta )}{\cos (\varphi )\cos (\delta )}$$

where $h_0$ is the target altitude ($-0.833°$ for standard sunrise), $\varphi$ is your latitude, and $\delta$ is the solar declination. This follows from the spherical law of cosines applied to the astronomical triangle (pole, zenith, sun). Your latitude $\varphi$ determines how steeply the sun’s arc intersects the horizon: vertical at the equator (fast sunrise), shallow at high latitudes (long twilight), and sometimes never dipping below it at all (midnight sun).

Getting from hour angle to your alarm clock takes a few more corrections:

$$t_{sunrise}=12-\frac{{\omega }_0}{15°}-\frac{\lambda }{15°}+\text{TZ}+\text{EoT}$$

where $\lambda$ is longitude (negative west), TZ is the UTC offset, and EoT is the equation of time in hours. The $÷15°$ converts angular measure to hours (Earth rotates $15°$ per hour). For East Peak on January 26: $\varphi =37.92°$, $\lambda =-122.60°$, $\delta \approx -19°$, TZ = $-8$ (Pacific Standard Time), and EoT $\approx -13$ minutes. The result: sunrise at 7:19am, civil twilight at 6:51am^noaa.

Atmospheric Refraction

Why $-0.833°$ and not $0°$? Because the atmosphere bends light.

At the horizon, sunlight passes through the maximum thickness of atmosphere. The density gradient bends it upward by about 34 arcminutes, over half a degree. Add the sun’s angular radius (16 arcminutes, because “sunrise” means the upper limb crests the horizon, not the center), and you get a total correction of $34^{\prime }+16^{\prime }=50^{\prime }\approx 0.833°$.

The consequence is slightly magical: when you see the first sliver of sun crest a ridge, the physical disk is still entirely below the geometric horizon. You’re seeing a mirage of the sun, bent into view by the atmosphere. Refraction gives you a gift of roughly two minutes of extra sunlight at mid-latitudes.

Refraction also depends on temperature and pressure (cold dense air refracts more), which is why sunrise calculators disagree by a minute or two. At extreme latitudes, where the sun is nearly tangent to the horizon, this uncertainty can stretch to several minutes.

Twilight and the Photographer’s Windows

Civil twilight, nautical twilight, golden hour: photographers treat these as separate phenomena, but they’re all just different values of $h_0$ plugged into the same equation:

One equation, six phenomena. Swap $h_0=-0.833°$ for $h_0=-6°$ and you get civil twilight. Swap for $-4°$ and you get the start of blue hour. The photographers who charge $200 for “golden hour timing guides” are selling a lookup table.

On January 26 at East Peak: blue hour from 6:51 to 7:01 (10 minutes), golden hour from 7:19 to about 7:57 (38 minutes). The blue hour window is tight in winter because the sun’s steep approach to the horizon compresses it. In summer, when the approach angle is shallower, blue hour stretches past 20 minutes. January gives you ten minutes. You don’t want to waste any of them hiking.

Part II: How Long Will It Take?

Flat-ground speed is easy: measure your pace, multiply by distance, set an alarm. Slope ruins everything. A 30% grade cuts your speed in half. A 50% grade cuts it to a third. Mountaineers have argued about the exact numbers since 1892.

Naismith’s Rule (1892)

Scottish mountaineer William Naismith proposed a formula so simple it fits on a napkin: 5 km/hour on flat ground, plus one hour for every 600 meters of ascent^naismith:

$$T=\frac{D}{5}+\frac{\Delta h^{+}}{600}$$

where $D$ is distance in km and $\Delta h^{+}$ is elevation gain in meters. It’s been the standard for 130+ years, which either means it’s brilliant or everyone’s just copying each other’s homework.

It’s implicitly linear: a 10% grade and a 40% grade cost the same per meter of gain. It says nothing about descent. And it assumes you’re as fit as a Victorian Scottish mountaineer carrying a rucksack in the Highlands, which is a real assumption.

Langmuir’s Extension (1984)

Eric Langmuir, writing in Mountaincraft and Leadership, noticed that Naismith’s rule pretends descent doesn’t exist^langmuir. His fix: on gentle downhill slopes (5-12°), subtract 10 minutes per 300 meters of descent. On steep downhill slopes (greater than 12°), add 10 minutes per 300 meters.

That reversal is the insight. Gentle descents are faster than flat ground. But steep descents are slower: every step is a braking action, your quads absorb the full deceleration, and the injury risk climbs. Anyone who has staggered down a rocky trail with shattered quads knows this in their bones. Langmuir just put a number on it.

Tobler’s Hiking Function (1993)

Waldo Tobler, a geographer who apparently thought mountaineering needed more calculus, fit an exponential curve to walking speed data from a 1950 Swiss cartography study^tobler:

$$v=6\cdot e^{-3.5\cdot |s+0.05|}$$

where $v$ is speed in km/h and $s$ is slope (rise/run). The $+0.05$ offset means peak speed occurs not on flat ground but on a slight downhill of about $-5\%$, roughly a $-3°$ grade. That’s right: a gentle descent lets gravity assist without requiring braking.

The payoff: a continuous speed function across all gradients, which means you can integrate it along a GPS track. Total time becomes a sum of segment times:

$$T=\sum _i\frac{d_i}{v(s_i)}$$

The weakness: it’s sensitive to GPS elevation noise, and the training data has a provenance problem. Imhof was a cartographer, not a physiologist. His 1950 book Gelände und Karte was a Swiss military cartography manual with walking speed data on pages 217-220. Tobler fit a curve to it in 1993 and it became the standard. The entire field of computational hiking time estimation rests on a few tables from a Swiss Army field manual seventy years ago. A track with noisy barometric data will produce wildly wrong estimates, because each erroneous spike gets fed through an exponential function.

Tranter’s Corrections (1970s)

Philip Tranter asked the obvious question nobody had bothered with: what if you’re not as fit as a Victorian Scottish mountaineer? His solution was practical: time yourself climbing 305m over 800m horizontal distance. Your time tells you everything^langmuir:

Tranter didn’t change the underlying model. He just added a multiplier based on how fast you actually climb. No VO2 max, no lactate threshold, no lab coat. Just: go climb a hill, time yourself. That number encodes everything about your physiology that matters.

The Munter Method (Swiss Alpine Club, ~1980)

This is the default for peaktime. Here’s why.

Werner Munter, Swiss mountain guide, avalanche researcher, and general-purpose alpine authority, proposed a method that the Swiss Alpine Club (SAC) adopted for standardizing trail times across their hut system^scarf. The idea: calculate horizontal time and vertical time separately, then combine with a min/max trick:

$$T=\max (T_h,T_v)+\frac{\min (T_h,T_v)}{2}$$

where $T_h=D/4$ (horizontal time at 4 km/h) and $T_v=\Delta h^{+}/400+\Delta h^{-}/800$ (vertical time, with descent at twice the ascent rate).

The min/max trick is the key. Whichever component dominates sets the base time; the other adds only half. Steep alpine climb (1000m gain over 3 km)? $T_v=150$ min dominates, total is $150+22.5=172.5$ min. The distance barely matters. Long flat approach (200m gain over 15 km)? $T_h=225$ min dominates, total is $225+15=240$ min. The climbing barely matters. The formula figures out which regime you’re in without you having to think about it.

The base rates (4 km/h horizontal, 400 m/h up, 800 m/h down) come from decades of Swiss Alpine Club hut timing data. Thousands of hikers, hundreds of routes, all conditions. The 800 m/h descent rate is deliberately conservative: you can descend faster on gentle grades, but on the rocky, steep terrain typical of alpine descents, 800 m/h keeps you honest.

Why Munter over everything else? It’s conservative, which is the right bias when being late means missing a sunrise or arriving at a hut after dark. It needs only three numbers (distance, ascent, descent) from any trail description, no GPS required. And it’s backed by real infrastructure. The Swiss didn’t build their trail system on theoretical elegance. They built it on not dying. Every yellow trail sign in Switzerland carries a time estimate derived from Munter’s method. The SAC maintains over 65,000 km of marked paths. When the sign says “2h 45min” to a hut, that number has been tested by decades of hikers arriving before dark. Or not.

Vertical Ascent Rate: The Unifying Metric

Forget the formulas for a minute. On steep trails, elevation gain is all that matters. I’ve switched to thinking in terms of VAR (Vertical Ascent Rate), meters of gain per hour. It’s what VK racers use, what mountain guides measure, and what actually predicts your summit time^vk.

VAR cuts through model differences. Naismith, Tobler, and Munter disagree on how to weight distance versus elevation, but they converge when you calibrate against actual climbing data: VK races, Strava segments, and your own legs on your own trails.

For this morning’s hike (2.9 km, 485m gain), our group clocked 58 minutes from Mountain Home Inn to Gardner Lookout. That’s ~500 m/hr, active pace. The Munter method predicted 61 minutes. Three minutes conservative. When the penalty for being late is missing the sunrise, three minutes conservative is exactly right.

Part III: Working Backward from Light

The Backward Planning Problem

The departure time problem is just arithmetic. Pick your target light condition, compute when it occurs at your summit, subtract the hiking time with adjustments:

$$t_{depart}=t_{target}-T_{ascent}\times k_{night}\times k_{terrain}-t_{buffer}$$

where $k_{night}$ is a darkness penalty (slower pace with a headlamp), $k_{terrain}$ is a surface correction, and $t_{buffer}$ is margin for setup (tripod, layers, finding a windbreak).

Here’s how it played out on January 26. Target: sunrise at Gardner Lookout, 7:19am. Munter ascent time for 2.9 km / 485m gain from Mountain Home Inn:

• $T_h=2.9/4=43.5$ min
• $T_v=485/400=72.8$ min (vertical dominates)
• $T=72.8+43.5/2=94.6$ min. But that’s for the standard hiker.

Adjusted for active fitness (VAR ~500 m/hr, scale factor 0.65): $T_{ascent}=61$ min.

Night penalty $k_{night}=1.10$ (10% slower for pre-dawn ascent; nautical twilight lightens the sky 12 minutes in, headlamps off by civil twilight at 6:51): $61\times 1.10=67$ min. The 10% is conservative. Most of the ascent was before civil twilight, but nautical twilight starts brightening the sky early enough that the truly blind segment is the first 12 minutes: the Hogback junction and initial fire road, where a wrong turn in the dark costs you the whole plan. Your proprioception degrades without peripheral vision cues, and each foot placement takes slightly longer.

Buffer $t_{buffer}=10$ min (find a windbreak, get settled, pull out the thermos).

$$t_{depart}=7\text{:}19-67\text{ min}-10\text{ min}=6\text{:}02\text{am}$$

Peaktime said 6:02. We left the Mountain Home Inn lot at 6:07, five minutes behind schedule. Arrival at Gardner Lookout: 7:05. Fourteen minutes before sunrise, the eastern sky already warming.

Why Munter for Pre-Dawn Planning

Naismith underestimates on steep trails because it treats all grades the same, and East Peak averages 17% with sections above 30%. Tobler needs a GPS track and chokes on elevation noise in the dark (altimeter readings drift overnight with temperature and pressure). Munter needs three numbers, works in the dark, and was designed for exactly this scenario: getting to an alpine objective on time. I’ll take the Swiss method that was calibrated on not dying.

What the Models Don’t Capture

Wind chill at ridgelines. The psychology of darkness. The moment your headlamp flickers and your pace involuntarily doubles. Group dynamics: the slowest member sets the pace, and cold dark mornings have a way of surfacing everyone’s actual commitment level. Trail junctions you can’t read in the dark, where the wrong turn doesn’t just cost time but kills the whole plan.

The night-adjusted model predicted 67 minutes. We did it in 58. Adrenaline, five headlamp batteries that held, and the fact that a January morning on Mt. Tam concentrates the mind wonderfully.

The Light

7:05am. Fourteen minutes to spare. All five of us at Gardner Lookout. Nobody complained about being early.

The model had predicted 67 minutes; we’d done it in 58. That margin is intentional. I’d rather watch the light build than chase it. The blue hour had played out on the trail, the sky shifting overhead as we climbed, and now the pre-sunrise gold was already warming the eastern ridgeline. Below us, the fog moved like something breathing, filling the valleys between ridges, swallowing the Bay and releasing it in slow pulses. The first thing you notice is the cold. Then the silence. Then, slowly, the color.

The Richmond Bridge kept appearing and disappearing. San Francisco was mostly socked in, just the tops of the Transamerica Pyramid and Salesforce Tower poking through like periscopes. When the sun finally crested the East Bay hills, the fog below turned from grey to gold to orange in the space of three minutes. Sunlight at the horizon passes through 38 times more atmosphere than at noon. The short wavelengths scatter out completely, leaving only reds and oranges. Add a fog layer acting as a diffusion screen and every droplet becomes a tiny lamp. The color doesn’t land on the fog. It comes from inside it. No algorithm captures that.

The hour angle formula said 7:19 for sunrise, 6:51 for civil twilight. The Munter method said we’d be there by 7:03. All three were right, and all three were irrelevant the moment the fog caught fire.

Plan Your Own

If you want to do this hike, everything from this post is baked into the planner below: NOAA sun positions, Munter ascent model, VAR calibration, atmospheric refraction corrections. Pick a date, set your fitness level, and it computes your departure time. For other peaks, try the full peaktime tool where you can drop in any GPX file. The source is on GitHub.

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The math, in the end, is in service of a feeling. If you’ve read this far and haven’t yet watched a sunrise from a summit, pick a peak, plug the numbers in, set the alarm. Drag a friend along. It’s better when someone else is also questioning their life choices at 4:45am. Peaktime got five of us there on time. The rest was between us and the sky.