What is the line of sight altitude to see hawaii from texas

Question

To see Hawaii from Texas how high in altitude would you have to be? I know the curvature of the earth has a big role in this

Answer 1

TL;DR: It depends on what one means by Hawaii, what one means by Texas, and whether one is concerned with atmospheric effects. The answer is somewhere between less than 2400 km and over 12600 km.

A note regarding the edit

This answer is a significant change from the previous answer where I focused solely on Honolulu and El Paso. I also made a mistake in that previous version of the answer; I used the longitude of El Paso where I should have used its latitude.

Assumptions

I'll be

• Assuming a spherical Earth with a radius of 6371 km
• Completely ignoring atmospheric effects
• Assuming "seeing" means the altitude at which the point in question is barely visible on the limb of the Earth as seen from that altitude.

Locations

The Hawaiian Archipelago is a very long chain of islands, stretching from Hawaii Island to Kure Atoll.I'll be using the following locations in the Hawaii Archipelago :

Location	Coordinates
Mauna Kea	19°49'14.4"N 155°28'5.0"W
Honolulu	21°18'25"N 157°51'30"W
Ni'ihau	21°54'0"N 160°10'0"W
Midway Atoll	28°12'27"N 177°21'0"W
Kure Atoll	28°25'0"N 178°20'0"W

The peak of Mauna Kea on Hawaii Island is about 4.2 km above sea level. I'll be using that elevation for Mauna Kea; for all other locations I'll be using sea level.

I'll be using the following locations in Texas:

Location	Coordinates
El Paso	31°45'33"N 106°29'19"W
Sweetwater	32°28'5"N 100°24'26"W
Texarkana	33°26'14"N 94°4'3"W

EL Paso is about as far west as one can get in Texas; it's closer to Los Angeles than it is to Texarkana. I'm guessing Texarkana is the furthest location in Texas from the Hawaiian islands. Sweetwater is about halfway between El Paso and Texarkana.

The elevation of the Texas locales is irrelevant as I'll be reporting height above sea level (height above 6371 km).

Mathematics

"Barely seeing" a point at sea level from another point at elevation requires calculating the angular separation between the two points via

$$\cos\theta = \sin\phi_1\sin\phi_2 + \cos\phi_1\cos\phi_2\cos(\lambda_1-\lambda_2) \tag{1}$$ where $\theta$ is the angular separation, $\phi_1$ and $\phi_2$ are the latitudes of the two points, and $\lambda_1$ and $\lambda_2$ are the longitudes of the two points.

Since we're just barely seeing the point in Hawaii, the angle between the line segment from the center of the Earth to the point in Hawaii and the line segment from the point in Hawaii to the observation point (the point above the location in Texas) is 90°. This makes the triangle formed by the center of the Earth, the point in Hawaii, and the observation point a right triangle

Denoting $R=6371~\text{km}$ as the radius of the spherical Earth, $h$ as the height above sea level the Earth at the point in question, and $\theta$ as calculated via equation (1), this means $$\cos\theta = \frac R{R+h} \tag{2}$$ or $$h = R\left(\frac1{\cos\theta}-1\right) \tag{3}$$

For locations in Hawaii at sea level (all but Mauna Kea), equations (1) and (3) suffice to find the requisite altitude. A couple of extra steps are needed for Mauna Kea: Use equation (2) to the angular separation at which a point 4.2 km above sea level just becomes visible from sea level. (The answer is 2.0799° with the spherical Earth assumption, which corresponds to about 231.2 km on the surface of the Earth.) This reduces the altitude at the Texas site as all that is needed to be able to see that point at sea level.

Results

The following table portrays the angular separation between the points in Hawaii and the points in Texas (first line in each table cell) and the altitude above sea level for the point in Texas needed to barely see the point in Hawaii (the second line in each cell). For Mauna Kea, the table also reports in parentheses the reduced angular distance (the angular distance via equation (1) less 2.0799°).

	El Paso	Sweetwater	Texarkana
Mauna Kea	45.294° (43.214°) 2371 km	50.462° (48.382°) 3222 km	55.762° (53.682°) 4386 km
Honolulu	46.702° 2919 km	51.833° 3939 km	57.072° 5349 km
Niʻihau	48.424° 3230 km	53.525° 4346 km	58.720° 5899 km
Midway Atoll	60.367° 6514 km	65.057° 8736 km	69.728° 12017 km
Kure Atoll	61.086° 6806 km	65.752° 9142 km	70.394° 12616 km